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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf,len=27312
+page_content='Objectivity in continuum mechanics, an introduction  Motions, Eulerian and Lagrangian variables and functions, deformation gradient,  Lie derivatives, velocity-addition formula, Coriolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Gilles Leborgne, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='isima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='fr/leborgne  January 4, 2023  In classical mechanics, there are two objectivities: 1- The covariant objectivity concerns the universal  laws of physics required to be observer independent (true in any reference frame);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This is a main topic in  this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 2- The isometric objectivity concerns the constitutive laws of materials once expressed  in a reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Covariant objectivity in continuum mechanics follows Maxwell’s requirements, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [13] page 1: “2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  The formula at which we arrive must be such that a person of any nation, by substituting for the different  symbols the numerical value of the quantities as measured by his own national units, would arrive at  a true result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The introduction of coordinate axes into geometry by Des Cartes was one  of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations  performed on numerical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The position of a point is made to depend on the length of three lines  which are always drawn in determinate directions (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') But for many purposes in physical reasoning, as  distinguished from calculation, it is desirable to avoid explicitly introducing the Cartesian coordinates,  and to fix the mind at once on a point of space instead of its three coordinates, and on the magnitude  and direction of a force instead of its three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This mode of contemplating geometrical and  physical quantities is more primitive and more natural than the other,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..”  And see the (short) historical note given in the introduction of Abraham and Marsden book “Foun-  dations of Mechanics” [1], about qualitative versus quantitative theory: “Mechanics begins with a long  tradition of qualitative investigation culminating with Kepler and Galileo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Following this is the period  of quantitative theory (1687-1889) characterized by concomitant developments in mechanics, mathemat-  ics, and the philosophy of science that are epitomized by the works of Newton, Euler, Lagrange,  Laplace, Hamilton, and Jacobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') For celestial mechanics (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') resolution we owe to the genius of  Poincaré, who resurrected the qualitative point of view (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') One advantage (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') is that by suppressing  unnecessary coordinates the full generality of the theory becomes evident.”  After having defined motions, Eulerian and Lagrangian variables and functions, we give the definition  of the deformation gradient as a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We then obtain a simple understanding of the Lie derivatives  of vector fields which meet the needs of engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then we get the velocity addition formula and verify  that the Lie derivatives are objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Note that Cauchy would certainly have used the Lie derivatives if  they had existed during his lifetime: To get a stress, Cauchy had to compare two vectors, whereas one  vector is enough when using the derivatives of Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We systematically start with qualitive definitions (observer independent), before quantifying with  bases and/or Euclidean dot products (observer dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A fairly long appendix tries to give in one  manuscript the definitions, properties and interpretations, usually scattered across several books (and  not always that easy to find).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='01056v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='class-ph]  3 Jan 2023  2  CONTENTS  Contents  I  Motions, Eulerian and Lagrangian descriptions, flows  11  1  Motions  11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Referential .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='2  Einstein’s convention (duality notation)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='2  Lagrangian variables and functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  24  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='4  Computation of d⃗v called L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1 wih Lagrangian variables .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='6  Time Taylor expansion of Φt0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  26  2  3  CONTENTS  4  Deformation gradient F := dΦ  26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definitions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of the deformation gradient F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='2  Push-forward (values of F)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='5  The ambiguous notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='1  Linear forms = “Covariant vectors” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='4  Matrix representation of a linear form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' 119  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='8 Small deformations (linearization): The infinitesimal strain tensor ε  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='1  Landau notations big-O and little-o  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' 167  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' 168  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' 175  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Gradients (subjective) associated with a differential through inner dot products .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='1  Coordinate system, and coordinate system basis  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' 177  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='5  Application 3: Differential of a vector field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 191  10  11  A quantity f being given then: g defined by « g equals f » is noted g := f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Part I  Motions, Eulerian and Lagrangian  descriptions, flows  1  Motions  The framework is classical mechanics, time being decoupled from space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R3 is the classical geometric  affine space (the space we live in), and ( ⃗R3, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = { ⃗pq : p, q ∈ R3} =noted ⃗R3 is the associated vector  space of bipoint vectors equipped with its usual rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We also consider R and R2 as subspaces of R3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  we consider Rn and ⃗Rn, n = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Referential  Origin:  An observer chooses an origin O ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus a point p ∈ Rn can be located by the observer  thanks to the bipoint vector −→  Op = ⃗x ∈ ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence p = O + ⃗x, and ⃗x = −→  Op =noted p − O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Another observer chooses an origin �  O ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the point p can also be located by this observer  with the bipoint vector  −→  �  Op = �⃗x ∈ ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So p = O + ⃗x = �  O + �⃗x, and �⃗x =  −−→  O �  O + ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cartesian coordinate system:  A Cartesian coordinate system in the affine space Rn is a set RCart =  (O, (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n), where O is an origin and (⃗ei) := (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n is a basis in ⃗Rn chosen by the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus the location of a point p ∈ Rn can quantified by the observer ∃⃗x ∈ ⃗Rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  p = O + ⃗x  with  ⃗x =  n  �  i=1  xi⃗ei,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [−→  Op]|⃗e = [⃗x]|⃗e =  �  �  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xn  �  � ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  [⃗x]|⃗e = [−→  Op]|⃗e being the column matrix containing the components xi ∈ R of −→  Op = ⃗x in the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Another observer with his origin Ob and his Cartesian basis (⃗bi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n make the Cartesian coordinate  system RCart,b = (Ob, (⃗bi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n), and gets for the same position p in Rn,  p = Ob + ⃗y  with  ⃗y =  n  �  i=1  �yi�⃗bi,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [−−→  Obp]|⃗b = [⃗y]|⃗b =  �  �  �  y1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  yn  �  �  � ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  [⃗y]|⃗b = [−−→  Obp]|⃗b being the column matrix containing the components yi ∈ R of −−→  Obp = ⃗y in the basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And −−→  Obp = −−→  ObO + −→  Op, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗y =  −−→  �  OO + ⃗x, gives the relation between ⃗x and ⃗y (drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Chronology:  A chronology (or temporal coordinate system) is a set Rtime = (t0, (∆t)) chosen by an  observer, where t0 ∈ R is the time origin, and (∆t) is the time unit (a basis in ⃗R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Referentiel:  A referential R is the set  R = (Rtime, RCart) = (t0, (∆t), O, (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n) = (“chronologie”,“Cartesian coordinate system”),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  made of a chronology and a Cartesian coordinate system, chosen by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In the following, to simplify the writings, the same implicit chronology is used by all observers, and  a referential R = (Rtime, RCart) will simply be noted as the reference frame R = (O, (⃗ei)) (so := RCart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Einstein’s convention (duality notation)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Einstein’s convention (duality notation)  Starting point: The classical notation xi for the components of a vector ⃗x relative to a basis, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then the duality notion is introduced: xi =noted xi (enables to see the difference between a vector and a  function when using components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  ⃗x =  n  �  i=1  xi⃗ei  � �� �  classic not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  xi⃗ei  � �� �  duality not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ,  and  [⃗x]|⃗e  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  �  �  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xn  �  � dual  =  �  �  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xn  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  The duality notation is part of the Einstein’s convention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Moreover Einstein’s convention uses the notation  �n  i=1xi⃗ei =noted xi⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the sum sign �n  i=1 can be omitted when an index (i here) is used twice, once  up and once down, details at § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' However this omission of the sum sign � will not be made in this  manuscript (to avoid ambiguities): The TEX-LATEX program makes it easy to print �n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The height of a child is represented on a wall by a vertical bipoint vector ⃗x starting from  the ground up to a pencil line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Question: What is the size of the child ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: It depends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  on the observer (quantitative value = subjective result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', an English  observer chooses a vertical basis vector ⃗a1 which length is one English foot (ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So he writes ⃗x = x1⃗a1,  and for him the size of the child (size of ⃗x) is x1 in foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' x1 = 4 means the child is 4 ft tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A  French observer chooses a vertical basis vector ⃗b1 which length is one metre (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So he writes ⃗x = y1⃗b1,  and for him the size of the child (size of ⃗x) is y1 metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if x1 = 4 then y1 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22, since 1 ft :=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3048 m: The child is both 4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in foot or metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This quantification is written ⃗x = 4 ft  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 m, where ft means ⃗a1 and m means ⃗b1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' NB: The qualitative vector ⃗x is the same vector for  all observers, not the quantitative values 4 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 (depends on a choice of a unit of measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With duality notation: ⃗x = x1⃗a1 = y1⃗b1, so if x1 = 4 then y1 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  This manuscript insists on covariant objectivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus an English engineer (and his foot) and a French  engineer (and his metre) will be able to work together .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and be able to avoid crashes like that of the Mars  Climate Orbiter probe, see remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And they will be able to use the results of Galileo, Descartes,  Newton, Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  who used their own unit of length, and knew nothing about the metre defined in  1793 and adopted in 1799 in France (after 6 years of measurements), and considered by the scientific  community at the end of the ninetieth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and couldn’t explicitly use the “Euclidean dot products”  either (which seems to have been defined mathematically by Grassmann around 1844).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Motion of an object  Let Obj be a “real object”, or “material object”, made of particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', the Moon: Exists independently  of an observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t1, t2 ∈ R, t1 < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The motion of Obj in Rn is the map  �Φ :  �  �  �  �  �  [t1, t2] × Obj → Rn  (t, PObj)  � �� �  particle  →  p = �Φ(t, PObj)  �  ��  �  its position at t in the Universe  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  And t is the time variable, p is the space variable, and (t, p) ∈ R × Rn is the time-space variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  �Φ is supposed to be C2 in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With an origin O (observer dependent), the motion can be described with the bi-point vector  ⃗x =  −−−−−−−→  O�Φ(t, PObj) = −→  Op noted  =  �⃗ϕ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  But then, two observers with different origins O and Ob have different description of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' There-  fore, in the following we won’t use �⃗ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then (quantification) with a Cartesian basis (⃗ei) to make a  referential R, we get (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Virtual and real motion  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 A virtual (or possible) motion of Obj is a function �Φ “regular enough for the calculations  to be meaningful”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Among all the virtual motions, the observed motion is called the real motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  12  13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Hypotheses (Newton and Einstein)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Hypotheses (Newton and Einstein)  Hypotheses of Newtonian mechanics (Galileo relativity) and general relativity (Einstein):  1- You can describe a phenomenon only at the actual time t and from the location p you are at (you  have no gift of ubiquity in time or space);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- You don’t know the future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- You can use your memory, so use some past time t0 and some past position pt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- You can use someone else memory (results of measurements) if you can communicate objectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Configurations  Fix t ∈ [t1, t2], and define �Φt :  �  Obj → Rn  PObj �→ p = �Φt(PObj) := �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 The “configuration at t” of Obj is the range (or image) of �Φt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is the subset of Rn  (affine space) defined by  Ωt := {p ∈ Rn : ∃PObj ∈ Obj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' p = �Φt(PObj)} noted  =  �Φt(Obj) noted  =  Im(�Φt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  If t is the actual time then Ωt is the actual (or current or Eulerian) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If t0 is a time in the past then Ωt0 is the past (or initial or Lagrangian) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Hypothesis: At any time t, Ωt is supposed to be a “smooth domain” in Rn, and the map �Φt is assumed  to be one-to-one (= injective): Obj does not crash onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Definition of the Eulerian and Lagrangian variables  If t is the actual time, then pt = �Φt(PObj) ∈ Ωt is called the Eulerian variable relative to PObj and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If t0 is a time in the past, then pt0 = �Φt0(PObj) ∈ Ωt0 is called the Lagrangian variable relative to PObj  and t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A Lagrangian variable is a “past Eulerian variable”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Two observers with two different origin of  time t0 and t0′ get two different Lagrangian variable while they have the same Eulerian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Trajectories  Let �Φ be a motion of Obj, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5), and PObj ∈ Obj (a particle in Obj = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the Moon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The (parametric) trajectory of PObj is the function  �ΦPObj :  �  [t1, t2] → Rn,  t �→ p(t) = �ΦPObj (t) := �Φ(t, PObj)  (position of PObj at t in the Universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Its geometric trajectory is the range (image) of �ΦPObj , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  geometric trajectory of PObj := {q ∈ Rn : ∃t ∈ [t1, t2] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' q = �ΦPObj (t)} = Im(�ΦPObj ) = �ΦPObj ([t1, t2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  Pointed vector, tangent space, fiber, vector field, bundle  (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Abraham–Marsden [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') To deal with surfaces S in R3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with S = a sphere (and more  generally with manifolds in Rn), a vector cannot simply be a “bi-point vector connecting two points of S”  (would get “through the surface”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A vector is defined to be tangent to S: Consider a “regular” curve  c : s ∈] − ε, ε[→ c(s) ∈ S where S is a surface in an affine space, and the vector tangent to S at c(0) is  ⃗w(c(0)) = limh→0  c(h)−c(0)  h  (it is defined with a parametrization of c in a general manifold);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Considering  all the possible curves, we get “all possible vectors on S”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation:  TpS := {tangent vectors ⃗wp at S at p} = The tangent space at p ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if S is a sphere in R3 and p ∈ S, then TpS is its usual tangent plane at p at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', particular case: If S = Ω is an open set in Rn, then TpS = TpΩ = ⃗Rn is independent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  13  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The set of configurations  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  The fiber at p := {p} × TpS = {  (p, ⃗wp)  � �� �  pointed vector  ∈ {p} × TpS},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', the fiber at p is the set of “pointed vectors at p”, a pointed vector being the couple (p, ⃗wp) made of  the “base point” p and the vector ⃗wp defined at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Drawing: A vector in ⃗Rn can be drawn anywhere in Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While a “pointed vectors at p” has to be  drawn at the point p in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If the context is clear, a pointed vector is simply noted �⃗w(p) =noted ⃗w(p) (lighten the writing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Particular case: If S = Ω is an open set in Rn, then the fiber at p is TpΩ = {p} × ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  The tangent bundle TS :=  �  p∈S  ({p} × TpS),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  that is, is the union of the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 A vector field �⃗w in S is a C∞ function (or at least C2 in the following)  �⃗w :  �  S → TS  p → �⃗w(p) = (p, ⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  If the context is clear, a vector field is simply noted �⃗w =noted ⃗w (lighten the writing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2  Eulerian description (spatial description at actual time t)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  The set of configurations  Let �Φ be a motion of Obj, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5), and Ωt = �Φt(Obj) ⊂ Rn be the configuration at t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The set  of configurations is the subset C ⊂ R × Rn (the “time-space”) defined by  C :=  �  t∈[t1,t2]  ({t} × Ωt)  (= set in which you find particles in “time-space”)  = {(t, p) ∈ R × Rn : ∃(t, PObj) ∈ [t1, t2] × Obj, p = �Φ(t, PObj)},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  Question: Why don’t we simply use �  t∈[t1,t2] Ωt instead of C = �  t∈[t1,t2]({t} × Ωt)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: C gives the film of the life of Obj = the succession of the photos Ωt taken at each t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And Ωt  is obtained from C thanks to the pause feature at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Whereas �  t∈[t1,t2] Ωt ⊂ Rn is the superposition of  all the photos on the image �  t∈[t1,t2] Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and we don’t distinguish the past from the present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Eulerian variables and functions  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 In short: A Eulerian function relative to Obj is a function, with m ∈ N∗,  Eul :  �  C → ⃗  Rm (or more generally a suitable set of tensors)  (t, p) → Eul(t, p),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  the spatial variable p being the Eulerian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In details: A function Eul being given as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2), the associated Eulerian function �  Eul is the function  �  Eul :  �  C → C × ⃗  Rm (or C× some suitable set of tensors)  (t, p) → �  Eul(t, p) = ((t, p), Eul(t, p)) = (time-space position , value),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  and is called “a field of functions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So �  Eul(t, p) is the “pointed Eul(t, p)” at (t, p) (in time-space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, the range Im( �  Eul) = �  Eul(C) of an Eulerian function �  Eul is the graph of Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Recall: The graph of  a function f : x ∈ A → f(x) ∈ B is the subset {(x, f(x)) ∈ A × B} ⊂ A × B: gives the “drawing of f”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If there is no ambiguity, �  Eul =noted Eul for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  14  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Eulerian velocity (spatial velocity) and speed  At t, the Eulerian vector field at t is �  Eult :  �  Ωt → Ωt × ⃗Rn  p → �  Eult(p) := (p, Eult(p)) = (position , value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Eul(t, p) = θ(t, p) ∈ R = temperature of the particle PObj which is at t at p = �Φ(t, PObj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Eul(t, p) = ⃗u(t, p) ∈ ⃗Rn = force applied on the particle PObj which is at t at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Eul(t, p) = d⃗u(t, p) ∈ L(⃗Rn : ⃗Rn) = the differential at t at p of a Eulerian function ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Question: Why introduce �  Eul?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Isn’t Eul sufficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: The “pointed value” �  Eul(t, p) = ((t, p), Eul((t, p))) is drawn on the graph of Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', at t at p the velocity vector ⃗v(t, p) ∈ ⃗R3 can be drawn anywhere, while the “pointed vector”  �⃗v(t, p) = ((t, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, p)) is ⃗v(t, p) drawn at t at p (and �⃗v is called the velocity field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Moreover (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) emphasizes the difference between a Eulerian vector field and a Lagrangian vector  function, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', the initial framework of Cauchy for his description of forces is Eulerian: The Cauchy  stress vector ⃗t = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n is considered at the actual time t at a point p ∈ Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (It is not Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Eulerian velocity (spatial velocity) and speed  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 In short: Consider a particle PObj and its (regular) trajectory �ΦPObj : t → p(t) = �ΦPObj (t),  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its Eulerian velocity at t at p(t) = �ΦPObj (t) is  ⃗v(t, p(t)) := �ΦPObj  ′(t) noted  =  ∂�Φ  ∂t (t, PObj),  when  p(t) = �ΦPObj (t),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗v(t, p(t)) is the tangent vector at t at p(t) = �ΦPObj (t) to the trajectory �ΦPObj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines the vector  field (in short) ⃗v :  �  C → ⃗Rn  (t, pt) → ⃗v(t, pt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In details: cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3), the Eulerian velocity is the function �⃗v :  �  C → C × ⃗  Rm  (t, p) → �⃗v(t, p) = ((t, p),⃗v(t, p))  �  (pointed vector) where ⃗v(t, p) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  d�ΦPObj  dt  (t) = ⃗v(t, �ΦPObj (t)), with p(t) = �ΦPObj (t), is often written  dp  dt (t) = ⃗v(t, p(t)),  or  d⃗x  dt (t) = ⃗v(t, ⃗x(t)),  or  d⃗x  dt = ⃗v(t, ⃗x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  the two last notations when an origin O is chosen and ⃗x(t) = −−−→  Op(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Such an equation is the pro-  totype of an ODE (ordinary differential equation) solved with the Cauchy–Lipschitz theorem, see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A Lagrangian velocity does not produce an ODE, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 If an observer chooses a Euclidean dot product (·, ·)g (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' foot or metre built), the  associated norm being ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g, then the length ||⃗v(t, p)||g is the speed (or scalar velocity) of PObj (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in  ft/s or in m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the context must remove the ambiguities: the “velocity” is either the vector velocity  ⃗v(t, p) = �ΦPObj  ′(t) or the speed (the scalar velocity) ||⃗v(t, p)||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Euclidean dot product (·, ·)g, ⃗x(t) = −−−→  Op(t), ⃗T(t) =  ⃗x ′(t)  ||⃗x ′(t)||g , and f(t) = ||⃗x ′(t)||g (speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Prove :  df  dt(t) = (⃗x ′′(t), ⃗T(t))g =noted ⃗x ′′(t) • ⃗T(t) (= tangential acceleration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2-D and Euclidean basis:  ⃗x(t)  =  � x(t)  y(t)  �  gives f(t)  =  (x′(t)2 + y′(t)2)  1  2 , thus f ′(t)  =  x′(t)x′′(t)+y′(t)y′′(t)  f(t)  = ⃗r ′(t) • ⃗r ′′(t)  ||⃗r ′(t)||  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem in n-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Spatial derivative of the Eulerian velocity  t ∈ [t1, t2] is fixed, Eul is a given Eulerian function, and Eult :  �  Ωt → ⃗  Rm  p → Eult(p) := Eul(t, p)  �  is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  15  16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Spatial derivative of the Eulerian velocity  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Recall: If Ω is an open set in Rn and if f : Ω → R is differentiable at p, then its differential at p is the  linear form df(p) ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) (linear map with real values) defined by, for all ⃗u ∈ ⃗Rn (vector at p),  df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = lim  h→0  f(p+h⃗u) − f(p)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  This expression is the same for all observers (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': There is no inner dot product here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 The space derivative of Eul at (t, p) is the differential dEult at p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all t ∈ [t1, t2],  all p ∈ Ωt and all ⃗wp ∈ ⃗Rn  t (vector at p),  (dEult(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp =)  dEul(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp = lim  h→0  Eul(t, p+h⃗wp) − Eul(t, p)  h  noted  =  ∂Eul  ∂p (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  In Ωt (the photo at t), dEul(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp gives the rate of variations of Eult at p in the direction ⃗wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', at t, the space derivative d⃗v of the Eulerian velocity field is defined by  d⃗v(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp = lim  h→0  ⃗v(t, p+h⃗wp) − ⃗v(t, p)  h  (= d⃗vt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 In differential geometry, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) is also written ⃗u(f)(p) =  d  dhf(p+h⃗u)|h=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Don’t use this  notation if you are not at ease with differential geometry (where a vector is defined to be a derivation,  so ⃗u[f] is the derivation of f by ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The convective derivative dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 If ⃗v is the Eulerian velocity field, then dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v is called the convective derivative of Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Quantification in a basis: df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u is written (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)f  Quantification: Let f : p ∈ Rn → f(p) ∈ R be C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ei) be a basis in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (usual definition)  ∂f  ∂xi  (p) := df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei  and  [df(p)]|⃗e = ( ∂f  ∂x1 (p)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂f  ∂xn (p) ) (line matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  (Recall: The matrix which represents a linear form is a line matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') And [df(p)]|⃗e is the Jacobian matrix  of f at p relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, with ⃗u = �n  i=1ui⃗ei a vector at p, and with the usual matrix multiplication  rule, we have  df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = [df(p)]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗e =  n  �  i=1  ∂f  ∂xi  (p)ui =  n  �  i=1  ui  ∂f  ∂xi  (p) noted  =  (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)|ef(p),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  where (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)|e : C1(��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is the differential operator defined relative to a basis (⃗ei) by  (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)|e(f) =  n  �  i=1  ui  ∂f  ∂xi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  If the basis (⃗ei) is unambiguously imposed, then (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)|e =noted ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad  For vector valued functions ⃗f : Ω → ⃗  Rm, the above steps apply to the components of ⃗f in a basis (⃗bi)  in ⃗  Rm: If ⃗f = �m  i=1fi⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗f(p) = �m  i=1fi(p)⃗bi, then  (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)|e(⃗f) =  m  �  i=1  (dfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)⃗bi =  m  �  i=1  ((⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)|efi)⃗bi =  m  �  i=1  n  �  j=1  (uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂fi  ∂xj  )⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  16  17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Streamline (current line)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Representation relative to a Euclidean dot product:  ⃗  gradf  An observer chooses a distance unit (foot, metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') and uses the associated Euclidean dot product (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let Ω be an open set in Rn, f ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) (scalar valued function), and p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the (·, ·)g-Riesz  representation vector of the differential form df(p) is called the gradient of f at p relative to (·, ·)g, and  named  ⃗  gradgf(p) ∈ ⃗Rn: It is defined by  ∀⃗u ∈ ⃗Rn,  ( ⃗  gradgf(p), ⃗u)g = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  written  ⃗  gradf • ⃗u = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  the last notation iff a Euclidean dot product (·, ·)g is imposed to all observer (quite subjective: foot,  metre ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (The first order Taylor expansion f(p+h⃗u) = f(p) + h df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u + o(h) can therefore, after a choice of  an Euclidean dot product, be written f(p+h⃗u) = f(p) + h  ⃗  gradgf(p) •g ⃗u + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Quantification: Let (⃗ei) be a Cartesian basis in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) gives [df].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u] = [ ⃗  gradf]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u], for all  ⃗u ∈ ⃗Rn  t (more precisely [df]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗e = [ ⃗  gradgf]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗e), thus (since [g]|⃗e is symmetric)  [ ⃗  gradf] = [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [df]T  (column matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if  ⃗  gradf = �n  i=1ai⃗ei then ai = �n  j=1gij  ∂f  ∂xj for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, if (⃗ei) is a (·, ·)g-orthonormal  basis then [ ⃗  gradf] = [df]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With duality notations,  ⃗  gradf = �n  i=1ai⃗ei and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14) gives ai = �n  j=1gij  ∂f  ∂xj : The Einstein convention  is not satisfied (the index j is twice bottom), which is expected since the definition of  ⃗  gradgf depends on  a subjective choice (unit of length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In comparison, df = �n  i=1  ∂f  ∂xi dxi satisfies the Einstein convention (a  differential is objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Mind the notations: The gradient  ⃗  gradgf =noted  ⃗  gradf depends on (·, ·)g, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14), while  (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)f does not (only depends on a basis), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11) (historical notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Vector valued functions  For vector valued functions ⃗f : Ω → ⃗  Rm, the above steps apply to the components fi of ⃗f relative to a  basis (⃗bi) in ⃗  Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But, depending on the book you read:  1- Ambiguous: d⃗f, the differential of ⃗f, is unfortunately also sometimes called the “gradient matrix”  (although no Euclidean dot product is required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Ambiguous: It could mean the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' or the Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' or its transposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' because  an orthonormal basis relative to an imposed Euclidean dot product is chosen (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') and then  [ ⃗  gradfi] = [dfi]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And calculations confuses [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='] and [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- Non ambiguous: In the objective framework of this manuscript, we will use the differential d⃗f  (objective) to begin with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And only after an explicit choice of bases (⃗ei) for quantitative purposes, the  Jacobian matrix, which is [df]|⃗e, will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 A Euclidean framework being chosen, prove: (⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)⃗v = 1  2 ⃗  grad(||⃗v||2) + ⃗  rot⃗v ∧ ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Euclidean basis ( ⃗Ei), Euclidean dot product (·, ·)g =noted (·, ·), associated norm ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g =noted ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  ⃗v = �n  i=1vi ⃗Ei gives ||⃗v||2 =  �  i  v2  i , thus ∂||⃗v||2  ∂xk  =  �  i  2vi ∂vi  ∂xk , for any k = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And, the first component  of ⃗  rot⃗v is ( ⃗  rot⃗v)1 = ∂v3  ∂x2 − ∂v2  ∂x3 , idem for ( ⃗  rot⃗v)2 and ( ⃗  rot⃗v)3 (circular permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (first component)  ( ⃗  rot⃗v∧⃗v)1 = ( ∂v1  ∂x3 − ∂v3  ∂x1 )v3−( ∂v2  ∂x1 − ∂v1  ∂x2 )v2, idem for ( ⃗  rot⃗v∧⃗v)2 and ( ⃗  rot⃗v∧⃗v)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ( 1  2 ⃗  grad(||⃗v||2)+ ⃗  rot⃗v∧⃗v)1 =  v1  ∂v1  ∂x1 + v2  ∂v2  ∂x1 + v3  ∂v3  ∂x1 + ∂v1  ∂x3 v3 − ∂v3  ∂x1 v3 − ∂v2  ∂x1 v2 + ∂v1  ∂x2 v2 = v1  ∂v1  ∂x1 + v2  ∂v1  ∂x2 + v3  ∂v1  ∂x3 = (⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  grad)v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for the  other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Streamline (current line)  Fix t ∈ R, and consider the photo Ωt = �Φt(Obj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let pt ∈ Ωt, ε > 0, and consider the spatial curve in Ωt  at pt defined by:  cpt :  �  ] − ε, ε[ → Ωt  s → q = cpt(s)  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  cpt(0) = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  17  18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Material time derivative (dérivées particulaires)  So s is a curvilinear spatial coordinate (dimension of a length), and the graph of cpt is drawn in the photo  Ωt at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 ⃗v : (t, p) → ⃗v(t, p) being the Eulerian velocity field of Obj, a streamline through a point  pt ∈ Ωt is a (parametric) spatial curve cpt solution of the differential equation  dcpt  ds (s) = ⃗vt(cpt(s))  with  cpt(0) = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  And Im(cpt) is the geometric associated streamline (⊂ Ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) cannot be confused with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5): In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) the variable is the time variable t, while in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  the variable is the space variable s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Usual notation: If an origin O is chosen at t by an observer and ⃗x(s) := −−−−−→  Ocpt(s) , then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) is written  d⃗x  ds (s) = ⃗vt(⃗x(s))  with  ⃗x(0) = −−→  Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  Moreover, with a Cartesian basis (⃗ei)) chosen at t by the observer, with ⃗x(s) = �n  i=1xi(s)⃗ei we get  d⃗x  ds (s) = �n  i=1  dxi  ds (s)⃗ei, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17) reads as the differential system of n equations in ⃗Rn  ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  dxi  ds (s) = vi(t, x1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn(s))  with  xi(0) = (−−→  Opt)i  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  (the n functions xi : s → xi(s) are the unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Also written  dx1  v1  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' = dxn  vn  = ds,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  which means: It is the differential system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) of n equations and n unknowns which must be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (With duality notations, dxi  ds (s) = vi(t, x1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn(s)) and xi(0) = (−−→  Opt)i for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Material time derivative (dérivées particulaires)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Usual definition  Goal: To compute the variations of a Eulerian function Eul along the trajectory �ΦPObj of a particle PObj  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the temperature of a particle along its trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So consider the function gPObj giving the values  of Eul relative to a PObj along its trajectory:  gPObj (t) := Eul(t, p(t))  when  p(t) := �ΦPObj (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 The Material time derivative of Eul at (t, p(t)) is gPObj  ′(t) =noted DEul  Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So:  DEul  Dt (t, p(t)) := gPObj  ′(t)  (= lim  h→0  Eul(t+h, p(t+h)) − Eul(t, p(t))  h  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Since gPObj (t) := Eul(t, �ΦPObj (t)) we get gPObj  ′(t) = ∂Eul  ∂t (t, �ΦPObj (t))+dEul(t, �ΦPObj (t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�Φ′  PObj (t), thus, having  �Φ′  PObj (t) = ⃗v(t, p(t)) (Eulerian velocity), DEul  Dt (t, p(t)) = ∂Eul  ∂t (t, p(t)) + dEul(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, p(t)):  DEul  Dt  := ∂Eul  ∂t  + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 Prove, if Eul is C2:  D2Eul  Dt2  = ∂2Eul  ∂t2  + 2d∂Eul  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t + d2Eul(⃗v,⃗v) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  D2Eul  Dt2  = D DEul  Dt  Dt  = g′′  PObj (t) = ∂( ∂Eul  ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)  ∂t  + d(∂Eul  ∂t  + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v  = ∂2Eul  ∂t2  + ∂(dEul)  ∂t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t + d∂Eul  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + d2Eul(⃗v,⃗v) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v,  and Eul C2 gives  ∂  ∂t ◦ d = d ◦ ∂  ∂t (Schwarz theorem), hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  18  19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Material time derivative (dérivées particulaires)  Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 Prove, if Eul is C2, for any vector field ⃗w,  D(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  Dt  = d∂Eul  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂ ⃗w  ∂t + d2Eul(⃗v, ⃗w) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  Dt  = ∂(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  ∂t  + d(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = ∂dEul  ∂t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂ ⃗w  ∂t + (d(dEul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the  Schwarz theorem gives ∂(dEul)  ∂t  = d( ∂Eul  ∂t ) since Eul ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Remark: About notations  The notation  d  dt (lowercase letters) concerns a function of one variable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dgPObj  dt (t) := gPObj  ′(t) :=  limh→0  gPObj (t+h))−gPObj (t)  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The  notation  ∂  ∂t  concerns  a  function  with  more  than  one  variable,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂Eul  ∂t (t, p)  =  limh→0  Eul(t+h,p)−Eul(t,p)  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The notation  D  Dt (capital letters) concerns a Eulerian function differentiated along a motion,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Other notations, often practical but might be ambiguous if composed functions are considered:  dEul(t, p(t))  dt  := DEul  Dt (t, p(t)),  and  dEul(t, p(t))  dt  |t=t0  := DEul  Dt (t0, p(t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Definition bis: Time-space definition  Consider a C1 time-space function f : (t, p) ∈ R × Rn → f(t, p) where t = time and p = space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 The differential of f : (t, p) ∈ Rn+1 → f(t, p) considered as a function on the Cartesian  (time×space) product R × Rn is called the “total differential”, or “total derivative”, and is written Df  (here time and space are of a different nature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So if p+ = (t, p) ∈ R×Rn and ⃗w+ = (w0, ⃗w) ∈ ⃗R×⃗Rn (time×space) then, by definition of a differential,  Df(p+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w+ := lim  h→0  f(p+ + h⃗w+) − f(p+)  h  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Df(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (w0, ⃗w) := lim  h→0  f(t+hw0, p+h⃗w) − f(t, p)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Thus  Df(t, p) = ∂f  ∂t (t, p) dt + df(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  Along a trajectory �ΦPObj : t → p(t) = �ΦPObj (t) with f = Eul a Eulerian function: Consider the  time-space trajectory  �ΨPObj :  �  [t1, t2] → R × Rn  t → �ΨPObj (t) := (t, �ΦPObj (t))  (= (t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  (So Im(�ΨPObj ) = graph(�ΦPObj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The tangent vector to this curve at t is  �ΨPObj  ′(t) = (1, �ΦPObj  ′(t)) = (1,⃗v(t, p(t)) ∈ ⃗R × ⃗Rn  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  where ⃗v(t, p(t)) the Eulerian velocity at p+ = (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) reads  gPObj (t) = (Eul ◦ �ΨPObj )(t) = Eul(�ΨPObj (t)),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  thus  g′  PObj (t) = DEul(�Ψ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ΨPObj  ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  And we recover (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22): g′  PObj (t) =(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28) ∂Eul  ∂t (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1+dEul(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, p(t)) =noted DEul  Dt (t, p(t)) : The ma-  terial time derivative is the “total derivative” DEul along the time-space trajectory �ΨPObj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  19  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Eulerian acceleration  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  The material time derivative is a derivation  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19 All the functions are Eulerian and supposed C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Linearity:  D(Eul1 + λEul2)  Dt  = DEul1  Dt  + λDEul2  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Product rules: If Eul1, Eul2 are scalar valued functions then  D(Eul1Eul2)  Dt  = DEul1  Dt  Eul2 + Eul1  DEul2  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  And if ⃗w is a vector field and T a compatible tensor (so T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w is meaningful) then  D(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  Dt  = DT  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let i = 1, 2, and gi defined by gi(t) := Euli(t, p(t)) where p(t) = �ΦPObj (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (g1 + λg2)′ = g′  1 + λg′  2 gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  On the one hand D(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  Dt  = ∂(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  ∂t  + d(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = ∂T  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂ ⃗w  ∂t + (dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v), and on the  other hand DT  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w  Dt = ( ∂T  ∂t + dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='( ∂ ⃗w  ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Commutativity issue  The Schwarz theorem that, when Eul is C2, the derivatives ∂Eul  ∂t  and dEul commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 The material time derivative  D  Dt does not commute with the temporal derivation  ∂  ∂t  or with the spatial derivation d: We have ∂( DEul  Dt )  ∂t  ̸= D( ∂Eul  ∂t )  Dt  and d( DEul  Dt ) ̸= D(dEul)  Dt  in general (because  the variables t and p are not independent along a trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In facts:  ∂( DEul  Dt )  ∂t  = D( ∂Eul  ∂t )  Dt  + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t  = ∂2Eul  ∂t2  + d∂Eul  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t  �  �  �  �  �  �  �  ,  and  �  �  �  �  �  d(DEul  Dt ) = D(dEul)  Dt  + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  = ∂(dEul)  ∂t  + d2Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂ DEul  Dt  ∂t  = ∂( ∂Eul  ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)  ∂t  = ∂( ∂Eul  ∂t )  ∂t  + ∂dEul  ∂t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t  Schwarz  =  ∂( ∂Eul  ∂t )  ∂t  + d∂Eul  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t ,  thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dDEul  Dt  = d(∂Eul  ∂t  + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) = ∂(dEul)  ∂t  + d(dEul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v, thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So ∂( DEul  Dt )  ∂t  ̸= D( ∂Eul  ∂t )  Dt  and d(DEul  Dt ) ̸= D(dEul)  Dt  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 Prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗ei) is a Cartesian basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂ DEul  Dt  ∂t  =  ∂( ∂Eul  ∂t +�  i  ∂Eul  ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vi)  ∂t  = ∂2Eul  ∂t2  + �  i  ∂2Eul  ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vi + �  i  ∂Eul  ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂vi  ∂t = ∂2Eul  ∂t2  +  �  i  ∂2Eul  ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vi + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂⃗v  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  D( ∂Eul  ∂t )  Dt  = ∂2Eul  ∂t2  + �  i  ∂ ∂Eul  ∂t  ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vi = ∂2Eul  ∂t2  + �  i  ∂2Eul  ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And d( DEul  Dt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = �  j  ∂ DEul  Dt  ∂xj wj = �  j  ∂( ∂Eul  ∂t +�  i  ∂Eul  ∂xi vi)  ∂xj  wj = �  j  ∂2Eul  ∂t∂xj wj +�  ij  ∂2Eul  ∂xi∂xj viwj +�  ij  ∂Eul  ∂xi  ∂vi  ∂xj wj =  �  j  ∂2Eul  ∂t∂xj wj + d2Eul(⃗v, ⃗w) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And  D(dEul)  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ( ∂(dEul)  ∂t  + d(dEul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w =  ∂(dEul)  ∂t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d2Eul(⃗v, ⃗w) =  �  i  ∂2Eul  ∂xi∂t wi + d2Eul(⃗v, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus d( DEul  Dt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = D(dEul)  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w for all ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Eulerian acceleration  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 In short: If �ΦPObj is C2, then the Eulerian acceleration of the particle PObj which is at t  at pt = �Φ(t, PObj) is  ⃗γ(t, pt) := �ΦPObj  ′′(t) noted  =  ∂2�Φ  ∂t2 (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  In details: as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3), the Eulerian acceleration (vector) field �⃗γ is defined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37) by  �⃗γ(t, pt) = ((t, pt),⃗γ(t, pt)) ∈ C × ⃗Rn  t  (pointed vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  20  21  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Time Taylor expansion of �Φ  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23  ⃗γ = D⃗v  Dt = ∂⃗v  ∂t + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  And if ⃗v is C2 then  d⃗γ = ∂(d⃗v)  ∂t  + d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = D(d⃗v)  Dt  + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With g(t) = ⃗v(t, p(t)) = �ΦPObj  ′(t) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22) we get ⃗γ(t, p(t)) = g′(t) =  D⃗v  Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And ⃗v  being C2, the Schwarz theorem gives d ∂⃗v  ∂t = ∂(d⃗v)  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24 If an observer chooses a Euclidean dot product (·, ·)g (based on a foot, a metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='), the  associated norm being ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g, then the length ||⃗γ(t, pt)||g is the (scalar) acceleration of PObj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Time Taylor expansion of �Φ  Let PObj ∈ Obj and t ∈]t1, t2[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Suppose �ΦPObj ∈ C2(]t1, t2[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its second-order (time) Taylor expansion  of �ΦPObj is, in the vicinity of a t ∈]t1, t2[,  �ΦPObj (τ) = �ΦPObj (t) + (τ−t)�Φ′  PObj (t) + (τ−t)2  2  �Φ′′  PObj (t) + o((τ−t)2),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  p(τ) = p(t) + (τ−t)⃗v(t, p(t)) + (τ−t)2  2  ⃗γ(t, p(t)) + o((τ−t)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  3  Motion from an initial configuration: Lagrangian description  Instead of working on Obj, an observer may prefer to work with an initial configuration Ωt0 = �Φ(t0, Obj)  of Obj (essential for elasticity): This is the “Lagrangian approach”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This Lagrangian approach is not  objective: Two observers may choose two different initial (times and) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Initial configuration and Lagrangian “motion”  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Obj is a material object, �Φ : [t1, t2[×Obj → Rn is its motion, Ωτ = �Φτ(Obj) is its configuration at τ,  t0 ∈]t1, t2[ is an “initial time”, and Ωt0 is the initial configuration for the observer who chose t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The motion of Obj relative to the initial configuration Ωt0 = �Φ(t0, Obj) is the function  Φt0 :  �  [t1, t2] × Ωt0 → Rn  (t, pt0) �→ pt = Φt0(t, pt0) := �Φ(t, PObj)  when  pt0 = �Φ(t0, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  So, pt = Φt0(t, pt0) := �Φ(t, PObj) is the position at t of the particle PObj which was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular  pt0 = Φt0(t0, pt0) := �Φ(t0, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Marsden and Hughes notations:  Once an initial time t0 has been chosen by an observer, then  Φt0 =noted Φ, then pt0 =noted P (capital letter for positions at t0) and pt =noted p (lowercase letter for  positions at t), so  p = Φ(t, P) ∈ Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  (When objectivity is under concern, we need to switch back to the notations Φt0, pt0 and pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  NB: • Talking about the motion of a position pt0 is absurd: A position in Rn does not move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  Φt0 has no existence without the definition, at first, of the motion �Φ of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The domain of definition of Φt0 depends on t0 through Ωt0: The superscript t0 recalls it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And a late  observer with initial time t0′ > t0 defines Φt0  ′ which domain of definition is [t1, t2]×Ωt0′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And Φt0  ′ ̸= Φt0  in general because Ωt0′ ̸= Ωt0 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  21  22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Initial configuration and Lagrangian “motion”  The following notation is also used:  Φt0(t, pt0) = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  (The couple (t0, pt0) is “the initial condition”, or t0 and pt0 are the initial conditions, see the § on flows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If a origin O ∈ Rn is chosen by the observer, we may also use, with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6),  ⃗xt0 = −−→  Opt0 = ⃗ϕ t0(t0, ⃗xt0) = ⃗X = −−→  OP  and  ⃗xt = −−→  Opt = ⃗ϕ t0(t, ⃗xt0) = ⃗x = −→  Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Diffeomorphism between configurations  With (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), define  Φt0  t :  �  Ωt0 → Ωt  pt0 → pt = Φt0  t (pt0) := Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Hypothesis: For all t0, t ∈]t1, t2[, the map Φt0  t  : Ωt0 → Ωt is a Ck diffeomorphism (a Ck invertible  function whose inverse is Ck), where k ∈ N∗ depends on the required regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) gives �Φt(PObj) = Φt0  t (�Φt0(PObj)), true for all PObj ∈ Obj, thus Φt0  t ◦ �Φt0 = �Φt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Φt0  t := �Φt ◦ (�Φt0)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Thus, Φt0  t0 = I and Φt  t0 ◦ Φt0  t = (�Φt ◦ (��Φt0)−1) ◦ (�Φt0 ◦ (�Φt)−1) = I give  Φt  t0 = (Φt0  t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Trajectories  Let (t0, pt0) ∈ [t1, t2] × Ωt0 (initial conditions) and with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) define  Φt0  pt0 :  � [t1, t2] → Rn  t �→ p(t) = Φt0  pt0 (t) := �ΦPObj (t) = Φt0(t, pt0)  when  pt0 = �ΦPObj (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Φt0  pt0 is called the (parametric) “trajectory of pt0”, which means: Φt0  pt0 is the trajectory  of the particle PObj that is located at pt0 = �Φ(t, PObj) at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the geometric “trajectory of pt0” is  Im(Φt0  pt0 ) = Φt0  pt0 ([t1, t2]) =  �  t∈[t1,t2]  {Φt0  pt0 (t)}  (= Im(�ΦPObj )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  NB: The terminology “trajectory of pt0” is awkward, since a position pt0 does not move: It is indeed  the trajectory �ΦPObj of a particle PObj which is at pt0 at t0 that must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Streaklines (lignes d’émission)  Take a film between t0 and T (start and end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Let Q be a fixed point in Rn (you see the point Q on each photo that make up the film).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The streakline through Q is the set  Et0,T (Q) = {p ∈ Ω : ∃τ ∈ [t0, T] : p = Φτ  T (Q) = (ΦT  τ )−1(Q)}  = {p ∈ Ω : ∃u ∈ [0, T−t0] : p = ΦT −u  T  (Q) = (ΦT  T −u)−1(Q)}  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  = the set at T of the positions (a line in Rn) of all the particles which were at Q at a τ ∈ [t0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Smoke comes out of a chimney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Fix a camera nearby, choose a point Q at the top of  the chimney where the particles are colored, and make a film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' At T stop filming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (at time T)  superimpose the photos in the film: The colored curve we see is the streakline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In other words = �  τ∈[t0,T ]{Φτ  Q(T)} = �  u∈[0,T −t0]{ΦT −u  Q  (T)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  22  23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lagrangian variables and functions  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Lagrangian variables and functions  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Consider a motion �Φ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' An observer chose (subjective) a t0 ∈ [t1, t2] (“in the past”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So Ωt0 =  �Φ(t0, Obj) is his initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let m ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 In short: A Lagrangian function, relative to Obj, �Φ and t0, is a function  Lagt0 :  �  [t1, t2] × Ωt0 → ⃗  Rm (or, more generally, some adequat set)  (t, pt0) → Lagt0(t, pt0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  and pt0 is called the Lagrangian variable relative to the (subjective) choice t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (To compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2): A Eulerian function does not depend on any t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 Scalar values: Lagt0(t, pt0) = Θt0(t, pt0) = temperature at t at pt = Φt0  t (pt0) = �Φ(t, PObj)  of the particle PObj that was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (So, continuing example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2, Θt0(t, pt0) = θ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 Vectorial values: Lagt0(t, pt0) = ⃗U t0(t, pt0) = force at t at pt = Φt0  t (pt0) = �Φ(t, PObj)  acting on the particle PObj that was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (So, continuing example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3, ⃗U t0(t, pt0) = ⃗u(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  If t is fixed or if pt0 ∈ Ωt0 is fixed, then we define  Lagt0  t :  �  Ωt0 → ⃗  Rm (or, more generally, some adequat set)  pt0 → Lagt0  t (pt0) := Lagt0(t, pt0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Lagt0  pt0 :  �  [t1, t2] → ⃗  Rm (or, more generally, some adequat set)  t → Lagt0  pt0 (t) := Lagt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 The position pt0 is also sometimes called a “material point”, which is counter intuitive:  PObj (objective) is the material point, and pt0 is just its spatial position at t0 (subjective);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And a Eulerian  variable pt is not called a “material point” at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  By the way, the variable pt is also called the “updated Lagrangian variable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  A Lagrangian function is a two point tensor  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 In details: Lagt0 being defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11), a Lagrangian function is a function  �  Lag  t0 :  � [t1, t2] × Ωt0 → C × ⃗  Rm  (t, pt0) → �  Lag  t0(t, pt0) = ((t, pt), Lagt0(t, pt0))  when  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  Lag  t0(t, pt0) = ((t, Φt0  t (pt0)), Lagt0(t, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (And ⃗  Rm can be replaced by some set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 (Marsden and Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') A Lagrangian function is a “two point vector field” (or  more generally a “two point tensor”) in reference to the points pt0 ∈ Ωt0 (departure set) and pt ∈ Ωt  (arrival set) where the value Lagt0(t, pt0) is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14) tells that Lagt0(t, pt0) is not represented at (t, pt0), but at (t, pt): That is, having  graph(Lagt0) = {((t, pt0), Lagt0(t, pt0))  and  Im(�  Lag  t0) = {((t, pt), Lagt0(t, pt0))},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  we have  Im(�  Lag  t0) ̸= graph(Lagt0) :  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  So a Lagrangian function does not define a tensor in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To compare with the Eulerian  function Eul which defines a tensor (in particular Im( �  Eul) = graph(Eul)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  23  24  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lagrangian function associated with a Eulerian function  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Lagrangian function associated with a Eulerian function  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Let �Φ be a motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Eul be a Eulerian function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t0 ∈ [t1, t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 The Lagrangian function Lagt0 associated with the Eulerian function Eul is defined by,  for all (t, PObj) ∈ [t1, t2] × Obj,  Lagt0(t, �Φ(t0, PObj)) := Eul(t, �Φ(t, PObj)),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all (t, pt0) ∈ [t1, t2] × Ωt0,  Lagt0(t, pt0) := Eul(t, pt),  when  pt = �Φ(t, PObj) = Φt0  t (pt)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Lagt0(t, pt0) := Eul(t, pt) when pt0 = (Φt0  t )−1(pt) for all (t, pt) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other words:  Lagt0  t := Eult ◦ Φt0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Remarks  If you have a Lagrangian function, then you can associate the function  Eult0  t := Lagt0  t ◦ (Φt0  t )−1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  which thus a priori depends on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But, a Eulerian function is independent of any initial time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For one measurement, there is only one Eulerian function Eul, while there are as many associated  Lagrangian function Lagt0 as they are t0 (as many as observers): The Lagrangian function Lagt0  ′ of a  late observer who chooses t0′ > t0 is different from Lagt0 since the domains of definition Ωt0 and Ωt0′ are  different (in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Lagrangian velocity  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 In short: The Lagrangian velocity at t at pt = �Φ(t, PObj) of the particle PObj is the  function  ⃗V t0 :  �  R × Ωt0 → ⃗Rn  (t, pt0) → ⃗V t0(t, pt0) := �ΦPObj  ′(t)  when  pt0 = �Φ(t0, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  In details: With (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21), the Lagrangian velocity is the two point vector field given by  �  ⃗V t0(t, pt0) :  �  �  �  R × Ωt0 → C × ⃗Rn  (t, pt0) → �  ⃗V t0(t, pt0) := ((t, pt), ⃗V t0(t, pt0)),  when  pt = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Thus ⃗V t0(t, pt0) = �ΦPObj  ′(t) = ⃗v(t, pt) is the velocity at t at pt = �Φ(t, PObj) of the particle PObj which  was at pt0 = �Φ(pt0, PObj) at t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗V t0(t, pt0) is not tangent to graph(⃗V t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16): It is tangent to  graph(⃗v) at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If t is fixed, or if pt0 ∈ Ωt0 is fixed, then we define  ⃗V t0  t (pt0) := ⃗V t0(t, pt0),  or  ⃗V t0  pt0 (t) := ⃗V t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  Remark: A usual definition is given without explicit reference to a particle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is, instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21),  ⃗V t0(t, pt0) := ∂Φt0  ∂t (t, pt0),  ∀(t, pt0) ∈ R × Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Lagrangian velocity versus Eulerian velocity  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4) give (alternative definition), with pτ = �Φ(τ, PObj),  ⃗V t0(t, pt0) = ⃗v(t, pt)  (= ∂Φt0  ∂t (t, pt0) = �ΦPObj  ′(t) = velocity at t at pt of PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  In other words,  ⃗V t0  t  = ⃗vt ◦ Φt0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  24  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lagrangian acceleration  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Relation between differentials  For C2 motions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) gives  d⃗V t0  t (pt0) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0  t (pt0)  when  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', with  F t0  t  = dΦt0  t  noted  =  the deformation gradient relative to t0 and t,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  d⃗V t0  t (pt0) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0)  when  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  Abusively written (dangerous notation: At what points, relative to what times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  d⃗V = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Computation of d⃗v called L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1 wih Lagrangian variables  The Lagrangian approach can be introduced before the Eulerian approach: ⃗V t0 being given, define  ⃗vt0(t, pt) := ⃗V t0(t, pt0),  when  pt = Φt0  t (pt0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗vt0(t, pt) := ⃗V t0(t, Φt0  t  −1(pt))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So ⃗vt0(t, Φt0  t (pt0)) = ∂Φt0  ∂t (t, pt0), thus  d⃗vt0(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0(t, pt0) = d(∂Φt0  ∂t )(t, pt0) = ∂(dΦt0)  ∂t  (t, pt0) = ∂F t0  ∂t (t, pt0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  when Φt0 is C2 and F t0 := dΦt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  d⃗vt0(t, pt) = ∂F t0  ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0(t, pt0)−1,  written in short  d⃗v = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1  (points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  And d⃗vt0  t  can be written Lt0  t in classical mechanics books, so you can find  Lt0  t (pt) := F t0  pt0 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0)−1,  written in short  L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1  (at what points, what times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  Here it is not obvious that Lt0  t (pt) does not depend on t0, which is indeed the case, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29):  Lt0  t (pt) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Reminder: if possible, use Eulerian quantities as long as possible1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Lagrangian acceleration  Let PObj ∈ Obj, t0, t ∈ R, pt0 = �ΦPObj (t0) and pt = �ΦPObj (t) (positions of PObj at t0 and t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 In short, the Lagrangian acceleration at t at pt of the particle PObj is  ⃗Γ t0(t, pt0) := �ΦPObj  ′′(t)  when  pt0 = �ΦPObj (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  In other words  ⃗Γ t0(t, pt0) := ⃗γ(t, pt)  when  pt = Φt0(t, pt0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  where ⃗γ(t, pt) = �ΦPObj  ′′(t) is the Eulerian acceleration at t at pt = �Φ(t, PObj), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In details, the Lagrangian acceleration is the “two point vector field” defined on R × Ωt0 by  �  ⃗Γ t0(t, pt0) = ((t, pt), �ΦPObj  ′′(t)),  when  pt = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  (To compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') In particular ⃗Γ t0(t, pt0) is not drawn on the graph of ⃗Γ t0 at (t, pt0), but on  the graph of ⃗γ at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1To get Eulerian results from Lagrangian computations can make the understanding of a Lie derivative quite difficult: To  introduce the “so-called” Lie derivatives in classical mechanics you can find the following steps: 1- At t consider the Cauchy  stress vector ⃗t (Eulerian), 2- then with a unit normal vector ⃗n, define the associated Cauchy stress tensor σ (satisfying  ⃗t = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 3- then use the virtual power and the change of variables in integrals to be back into Ωt0 to be able to work  with Lagrangian variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 4- then introduce the first Piola–Kirchhoff (two point) tensor PK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 5- then introduce the second  Piola–Kirchhoff tensor SK (endomorphism in Ωt0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 6- then differentiate SK in Ωt0 (in the Lagrangian variables although the  initials variables are the Eulerian variables in Ωt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 7- then back in Ωt to get back to Eulerian functions (change of variables  in integrals),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 8- then you get some Jaumann or Truesdell or other so called Lie derivatives type terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the appropriate choice  among all these derivatives being quite obscure because the covariant objectivity has been forgotten en route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While, with  simple Eulerian considerations, it requires a few lines to understand the (real) Lie derivative (Eulerian concept) and its  simplicity, see § 9, and deduce second order covariant objective results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  25  26  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Time Taylor expansion of Φt0  If t is fixed, or if pt0 ∈ Ωt0 is fixed, then define  ⃗Γ t0  t (pt0) := ⃗Γ t0(t, pt0),  and  ⃗Γt0  pt0 (t) := ⃗Γ t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  Thus  ⃗Γ t0  t  = ⃗γt ◦ Φt0  t ,  and  d⃗Γ t0  t (pt0) = d⃗γt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  when pt = Φt0  t (pt0) and F t0  t  := dΦt0  t (the deformation gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Risky notation: d⃗Γ = d⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F (points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Time Taylor expansion of Φt0  Let pt0 ∈ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, at second order,  ��t0  pt0 (τ) = Φt0  pt0 (t) + (τ−t)Φt0  pt0  ′(t) + (τ−t)2  2  Φt0  pt0  ′′(t) + o((τ−t)2),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  that is, with p(τ) = �ΦPObj (τ) = Φt0  τ (pt0),  p(τ) = p(t) + (τ−t)⃗V t0(t, pt0) + (τ−t)2  2  ⃗Γ t0(t, pt0) + o((τ−t)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  NB: There are three times involved: t0 (observer dependent), t and τ (for the Taylor expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To  compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42): p(τ) = p(t)+(τ−t)⃗v(t, p(t))+ (τ−t)2  2  ⃗γ(t, p(t))+o((τ−t)2), independent of t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  A vector field that let itself be deformed by a motion  Consider a C0 Eulerian vector field ⃗w :  �  C → ⃗Rn  (t, pt) → ⃗w(t, pt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let t0 ∈ [t1, t2[ and let ⃗wt0  :  �  Ωt0 → ⃗Rn  pt0 → ⃗wt0(pt0) := ⃗w(t0, pt0)  �  (vector field in Ωt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then define the (virtual) vector field  ⃗wt0∗ :  �  C → ⃗Rn  (t, pt) → ⃗wt0∗(t, pt) := dΦt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0),  when  p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  (The push-forward = result of the deformation of ⃗wt0 by the motion, see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 For C2 motions, we have (time variation rate along a virtual trajectory)  D ⃗wt0∗  Dt  = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L⃗v ⃗wt0∗ = ⃗0, where L⃗v⃗u := D⃗u  Dt −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u (= ∂⃗u  ∂t +d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) is the Lie derivative of a (unsteady) vector  field ⃗u : C → ⃗Rn along ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation:  We will see that L⃗v ⃗w(t0, pt0) = limt→t0  ⃗w(t,p(t))− ⃗wt0∗(t,p(t))  h  measures the “re-  sistance of ⃗w to a motion”, see § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus the result L⃗v ⃗wt0∗(t0, pt0)  = ⃗0 is “obvious” (=  limt→t0  ⃗wt0∗(t,p(t))− ⃗wt0∗(t,p(t))  h  ): If ⃗w = ⃗wt0∗ then the vector (“force”) field ⃗w does not oppose any re-  sistance to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' pt0 being fixed, with dΦt0(t, pt0) =noted F(t) we have ⃗wt0∗(t, p(t)) =(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43) F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0), thus  D ⃗wt0∗  Dt  (t, p(t)) = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗(t, p(t)) =(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33) d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗(t, p(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4  Deformation gradient F := dΦ  Consider a motion �Φ :  �  R × Obj → Rn  (t, PObj) → pt = �Φ(t, PObj)  �  , Ωt := �Φ(t, Obj) the configuration of Obj at any t,  fix t0, t in R, and let Φt0  t  :  �  Ωt0 → Ωt  pt0 = �Φ(t0, PObj) → pt = Φt0  t (pt0) := �Φ(t, PObj)  �  , supposed to be a C1  diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Notations for calculations (quantification), to comply with practices:  26  27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definitions  1- Classical (unambiguous) notations as in Arnold, Germain: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (⃗ai) and (⃗bi) are bases resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in ⃗Rn  t0  and ⃗Rn  t , ⃗wt0(pt0) = �  i wt0,i(pt0)⃗ai ∈ ⃗Rn  t0, ⃗wt,i(pt) = �  i wt,i(pt)⃗bi ∈ ⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  2- Marsden–Hughes duality notations: Capital letters at t0, lower case letters at t, duality notation,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ( ⃗EI) and (⃗ei) are bases resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in ⃗Rn  t0 and ⃗Rn  t , ⃗W(P) = �  I W I(P) ⃗EI ∈ ⃗Rn  t0, ⃗w(p) = �  i wi(p)⃗ei ∈ ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definitions  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of the deformation gradient F  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The differential dΦt0  t =noted F t0  t  :  �  Ωt0 → L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  pt0 → F t0  t (pt0) := dΦt0  t (pt0)  �  is called “the covari-  ant deformation gradient between t0 and t”, or simply “the deformation gradient”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And “the covariant  deformation gradient at pt0 between t0 and t”, or in short “the deformation gradient at pt0” is the linear  map F t0  t (pt0) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ), so defined by, for all ⃗wt0(pt0) ∈ ⃗Rn  t0 (vector at pt0),  F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) := lim  h→0  Φt0  t (pt0+h⃗wt0(pt0)) − Φt0  t (pt0)  h  noted  =  (Φt0  t )∗(⃗wt0)(pt) noted  =  ⃗wt0∗(t, pt),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  with pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Marsden–Hughes notations: Φ := Φt0  t , F := dΦ, P := pt0, ⃗W(P) := ⃗wt0(pt0), p = Φ(P), thus  F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P) := lim  h→0  Φ(P+h ⃗W(P)) − Φ(P)  h  noted  =  Φ∗ ⃗W(p) noted  =  ⃗w∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1: ⃗w is a Eulerian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' At t0 define vector field ⃗wt0 in Ωt0 by ⃗wt0(pt0) := ⃗w(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The  (spatial) curve ct0 : s → pt0 = ct0(s) in Ωt0 is an integral curve of ⃗wt0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' satisfies ct0  ′(s) = ⃗wt0(ct0(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  It is transformed by Φt0  t into the (spatial) curve ct = Φt0  t ◦ ct0 : s → pt = ct(s)=Φt0  t (ct0(s)) in Ωt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence  ct′(s) = dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ct0  ′(s) = dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) =noted ⃗wt0∗(t, pt) is the tangent vector at ct at pt (the  push-forward of ⃗wt0 by Φt0  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗w(t, p(t)) (actual value) is also drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: The “deformation gradient” F t0  t  = dΦt0  t  is not a “gradient” (its definition does not need a  Euclidean dot product);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This lead to confusions when covariance-contravariance and objectivity are at  stake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It would be simpler to stick to the name “F t0  t  = the differential of Φt0  t ”, but it is not the standard  usage, except in thermodynamics: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', the differential dU of the internal energy U is not called “the  gradient of U” (there is no meaningful inner dot product): It is just called “the differential of U”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Push-forward (values of F)  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Let ⃗wt0 :  �  Ωt0 → ⃗Rn  t0  pt0 → ⃗wt0(pt0)  �  be a vector field in Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its push-forward by Φt0  t  is the  vector field (Φt0  t )∗(⃗wt0) in Ωt defined by  (Φt0  t )∗ ⃗wt0(pt) = F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) noted  =  ⃗wt0∗(t, pt)  when  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Marsden notation: Φ∗ ⃗W(p) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P) =noted ⃗w∗(p) when p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  27  042  w(p  Cto  Ct28  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definitions  In other words  (Φt0  t )∗ ⃗wt0 := (F t0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0) ◦ (Φt0  t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  Marsden notation: Φ∗ ⃗W = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W) ◦ Φ−1 = ⃗w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  F is a two point tensors  With (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), “the tangent map” is  �  F t0  t  :  � Ωt0 → Ωt × L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  pt0 → �  F t0  t (pt0) = (pt, F t0  t (pt0))  when  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 (Marsden–Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The function �  F t0  t  is called a two point tensor, referring to the  points pt0 ∈ Ωt0 (departure set) and pt = Φt0  t (pt0) ∈ Ωt (arrival set where ⃗wt0∗(t, pt) = F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0)  is drawn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And in short �  F t0  t  =noted F t0  t  is said to be a two point tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 The name “two point tensor” is a shortcut than can create confusions and errors when  dealing with the transposed: F t0  t  is not immediately a “tensor”: A tensor is a multilinear form, so gives  scalar results (∈ R), while F(P) := F t0  t (P) =noted FP ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) gives vector results (in ⃗Rn  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' However  FP can be naturally and canonically associated with the bilinear form �FP ∈ L(⃗Rn∗  t , ⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) defined by,  for all ⃗uP ∈ ⃗Rn  t0 and ℓp ∈ ⃗Rn∗  t , with p = Φt0  t (P),  �FP (ℓp, ⃗uP ) := ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uP (∈ R),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13, and it is �FP which defines the so-called “two point tensor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  But don’t forget that the transposed of a linear form (FP here) is not deduced from the transposed  of the associated bilinear form ( �FP here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So be careful with the word “transposed” and its two distinct  definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Indeed, the transposed of a bilinear form b(·, ·) is intrinsic to b(·, ·) (is objective), given by  bT (⃗u, ⃗w) = b(⃗w, ⃗u), while the transposed of a linear function L is not intrinsic to L (is subjective), given  by (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u, ⃗w)g = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, ⃗u)h where (·, ·)g and (·, ·)h are inner dot products chosen by human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (details  in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 and § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 More generally for manifolds, the differential of Φ := Φt0  t  at P ∈ Ωt0 is F(P) := dΦ(P) :  �  TP Ωt0 → TpΩt  ⃗W(P) → ⃗w∗(p) := dΦ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P)  �  with p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the tangent map is  TΦ :  �  TΩt0 → TΩt  (P, ⃗W(P)) → TΦ(P, ⃗W(P)) := (p, dΦ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P)) = (p, ⃗w∗(p)),  where  p = Φt0  t (P),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  called the associated two point tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Evolution: Toward the Lie derivative (in continuum mechanics)  Consider a Eulerian vector field ⃗w :  �  �  �  C =  �  t  ({t} × Ωt) → ⃗Rn  (t, p) → ⃗w(t, p)  �  �  �, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a “force field”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, at t0  consider ⃗wt0 :  �  Ωt0 → ⃗Rn  t0  pt0 → ⃗wt0(pt0) := ⃗w(t0, pt0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The push-forward of ⃗wt0 by Φt0  t is, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2),  ⃗wt0∗(t, p(t)) = F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0),  where  p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, without any ubiquity gift, at t at p(t) we can compare ⃗w(t, p(t)) (real value of ⃗w  at t at p(t)) with ⃗wt0∗(t, p(t)) (transported memory along the trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the rate  ⃗w(t, p(t)) − ⃗wt0∗(t, p(t))  t − t0  =  actual(t, p(t)) − memory(t, p(t))  t − t0  is meaningful at (t, p(t))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  (no ubiquity gift required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This rate gives, as h → 0, the Lie derivative L⃗v ⃗w (the rate of stress), and we  will see at § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 that L⃗v ⃗w = D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w (the d⃗v term tells that a “non-uniform flow” acts on the stress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  28  29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification with bases  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Pull-back  Formally the pull-back is the push-forward with (Φt0  t )−1:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 The pull-back (Φt0  t )∗ ⃗wt of a vector field ⃗wt defined on Ωt is the vector field defined on Ωt0  by, with pt0 = (Φt0  t )−1(pt),  ⃗w∗  t (t0, pt0) = (Φt0  t )∗ ⃗wt(pt0) := (F t0  t )−1(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt(pt),  written  ⃗W ∗(P) = F −1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification with bases  (Simple Cartesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') (⃗ai) is a Cartesian basis in ⃗  Rn  t0, (⃗bi) is a Cartesian basis in ⃗  Rn  t , ot is an  origin in Rn at t, Φt0  t =noted Φ supposed C1, ϕi : Ωt0 → R is its components in the referential (ot, (⃗bi)):  Φ(pt0) = ot +  n  �  i=1  ϕi(pt0)⃗bi,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  −−−−−→  otΦ(pt0) =  n  �  i=1  ϕi(pt0)⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Thus, with the classic notation dϕi(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =noted ∂ϕi  ∂Xj (pt0) since (⃗ai) is a Cartesian basis, and (⃗bi) being  a Cartesian basis,  dΦ(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  (dϕi(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj)⃗bi =  n  �  i=1  ∂ϕi  ∂Xj  (pt0)⃗bi,  thus  [dΦ(pt0)][⃗a,⃗b] = [ ∂ϕi  ∂Xj  (pt0)] = [F(pt0)][⃗a,⃗b],  [dΦ(pt0)][⃗a,⃗b] = [F(pt0)][⃗a,⃗b] being the Jacobian matrix of Φ at pt0 relative to the chosen bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In short:  dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  ∂ϕi  ∂Xj  ⃗bi,  thus  [dΦ][⃗a,⃗b] = [ ∂ϕi  ∂Xj  ] = [F][⃗a,⃗b] = [Fij],  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Thus, if ⃗W ∈ ⃗Rn  t0 is a vector at pt0 and ⃗W = �n  j=1Wj⃗aj then, by linearity of differentials,  dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W =  n  �  i=1  FijWj⃗bi,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W]|⃗b = [F]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W]|⃗a  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  (more precisely: F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(pt0) = �n  i=1Fij(pt0)Wj(pt0)⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Similarly, for the second order derivative d2Φ = dF (when Φ is C2): With ⃗U = �n  j=1Uj⃗aj and  ⃗W = �n  k=1Wk⃗ak, and with (⃗ai) and (⃗bi) Cartesian bases, we get  dF(⃗U, ⃗W) = d2Φ(⃗U, ⃗W) =  n  �  i=1  d2ϕi(⃗U, ⃗W)⃗bi =  n  �  i,j,k=1  ∂2ϕi  ∂Xj∂Xk  UjWk⃗bi =  n  �  i=1  �  [⃗U]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[d2ϕi]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W]|⃗a  �  ⃗bi,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  [d2ϕi(pt0)]|⃗a = [  ∂2ϕi  ∂Xj∂Xk (pt0)] j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n being the Hessian matrix of ϕi at pt0 relative to the basis (⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With Marsden duality notations:  p = Φ(P) = ot +  n  �  i=1  ϕi(P)⃗ei,  F i  J(P) = ∂ϕi  ∂XJ (P)  (= dϕi(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ),  F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W =  n  �  i,J=1  F i  J(P) W J⃗ei,  [F] = [F i  J] = [dΦ],  dF(⃗U, ⃗W) = d2Φ(⃗U, ⃗W) =  n  �  i,J,K=1  ∂2ϕi  ∂XJ∂XK U JW K⃗ei =  n  �  i=1  �  [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d2ϕi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W]  �  ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 J, j are dummy variables when used in a summation: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = �n  j=1  ∂f  ∂Xj W j =  �n  J=1  ∂f  ∂XJ W J = �n  α=1  ∂f  ∂Xα W α =  ∂f  ∂X1 W 1 +  ∂f  ∂X2 W 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (there is no uppercase for 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And  Marsden–Hughes notations (capital letters for the past) are not at all compulsory, classical notations  being just as good and even preferable if you hesitate (because they are not misleading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  29  30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The unfortunate notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  The unfortunate notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Issue  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗w∗(p) := F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P), is sometimes written  d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X : “a very unfortunate and misleading notation”  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  which amounts to “confuse a length and a speed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And you also the phrase “(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) is still true if  ||d ⃗X|| = 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' while d ⃗X is supposed to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Where does this unfortunate notation come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) comes from the first order Taylor expansion Φ(Q) = Φ(P) + dΦt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Q−P) +  o(||Q−P||), where P, Q ∈ Ωt0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', with p = Φt0  t (P) and q = Φt0  t (Q) and h = ||Q−P||,  q − p = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Q−P) + o(h),  written  δ⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δ ⃗X + o(δ ⃗X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  or −→  pq = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−→  PQ + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So as Q → P we get 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Quite useless, isn’t it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  While  q − p  h  = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q − P  h  + o(1)  is useful:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  As Q → P we get ⃗w∗ = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W which relates tangent vectors, see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 Details:  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Interpretation: Vector approach  Consider a spatial curve ct0 :  �  [s1, s2] → Ωt0  s → P := ct0(s)  �  in Ωt0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is deformed by Φt0  t  to  become the spatial curve defined by ct := Φt0  t ◦ ct0 :  �  [s1, s2] → Ωt  s → p := ct(s) = Φt0  t (ct0(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  in Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence,  relation between tangent vectors:  dct  ds (s) = dΦt0  t (ct0(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dct0  ds (s), ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗w∗(p) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P)  written  d⃗x  ds = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X  ds ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  But you can’t simplify by ds to get d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X: It is absurd to confuse “a slope d ⃗  X  ds (s)” and “a length δ ⃗X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: || dct  ds (s)|| = || d ⃗  X  ds (s)|| = 1 is meaningful in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19): It means that the parametrization of the  curve ct0 in Ωt0 uses a spatial parameter s such that ||ct0  ′(s)|| = 1 for all s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' || ⃗WP || = 1 in  figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' You cannot simplify by ds: ||d ⃗X|| = 1 is absurd together with d ⃗X “small”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Interpretation: Differential approach  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) is a relation between differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if you adopt the correct notations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let us do it: With (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11),  ⃗x = −→  otp = −−−−−−→  otΦt0  t (P) =  n  �  i=1  ϕi(P)⃗bi  noted  =  n  �  i=1  xi(P)⃗bi,  where  ϕi  noted  =  xi  (function of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Thus, with (πai) = (dXi) the (covariant) dual basis of (⃗ai) we get the system of n equations (functions):  dΦ = F,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  dϕ1(P) = �n  j=1  ∂ϕ1  ∂Xj (P) dXj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dϕn(P) = �n  j=1  ∂ϕn  ∂Xj (P) dXj  �  �  �  �  �  �  �  �  �  ,  which is noted  d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  this last notation being often misunderstood2: It is nothing more than dΦ = F (coordinate free notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2Spivak [17] chapter 4: Classical differential geometers (and classical analysts) did not hesitate to talk about “infinitely  small” changes dxi of the coordinates xi, just as Leibnitz had.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' No one wanted to admit that this was nonsense, because  true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the  right way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Eventually it was realized that the closest one can come to describing an infinitely small change is to describe  a direction in which this change is supposed to occur, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since df is supposed to be the infinitesimal  change of f under an infinitesimal change of the point, df must be a function of this change, which means that df should  be a function on tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The dXi themselves then metamorphosed into functions, and it became clear that they  must be distinguished from the tangent vectors ∂/∂Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Once this realization came, it was only a matter of making new  definitions, which preserved the old notation, and waiting for everybody to catch up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  30  31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Tensorial notations, warnings, remarks  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  The ambiguous notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X  The bad notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X gives the unfortunate and misunderstood notations d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X, and then d⃗x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗x  where  L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Question: What is the meaning (and legitimate notation) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: d⃗x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗x means  D ⃗wt0∗  Dt  = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗  = evolution rate of tangent vectors along a trajectory  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Indeed, ⃗wt0∗(t, p(t)) =(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) F t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) gives  D ⃗wt0∗  Dt  (t, p(t)) = ∂F t0  ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) = ∂F t0  ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t0  t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗(t, p(t))),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗wt0∗  Dt  (t, p(t)) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗(t, p(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular D ⃗wt0∗  Dt  (t0, pt0) = d⃗v(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0)  is the evolution rate of tangent vectors at t0 at pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Tensorial notations, warnings, remarks  As already noted, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6), the linear map F := dΦt0  t (pt0) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) is naturally canonically associated  with the bipoint tensor �F ∈ L(⃗Rn∗  t , ⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) defined by, for all (ℓ, ⃗W) ∈ ⃗Rn∗  t  × ⃗Rn  t0,  �F(ℓ, ⃗W) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  Quantification of �F: basis (⃗ai) with dual basis (πai) in ⃗Rn  t0, basis (⃗bi) in ⃗Rn  t :  if  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  ∂ϕi  ∂Xj  ⃗bj  then  �F =  n  �  i,j=1  ∂ϕi  ∂Xj  ⃗bi ⊗ πaj =  n  �  i=1  ⃗bi ⊗ dϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  And similarly  d �F =  n  �  i,j,k=1  ∂2ϕi  ∂Xj∂Xk  ⃗bi ⊗ (πaj ⊗ πak) =  n  �  i=1  ⃗bi ⊗ d2ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Warning: The tensorial notation can be misleading, in particular if you use the transposed, see re-  mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, you should always use the standard notation for the linear form F ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) to begin  with, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' use F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �n  j=1Fij⃗bi or F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ = �n  i,j=1F i  J⃗ei (Marsden notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And only use the tensorial  notations for calculations purposes at the end (after application of the proper definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 In some manuscripts you find the notation F = dΦ =noted Φ ⊗ ∇X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It does not help to  understand what F is (it is the differential dΦ), and should not be used as far as objectivity is concerned:  A differentiation is not a tensorial operation, see example R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1, so why use the tensor product  notation Φ ⊗ ∇X, when the standard notation dΦ ≃ �F = �n  i=1⃗ei ⊗ dϕi is legitimate, explicit, objective  and easy to manipulate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And it could be misinterpreted, since, in mechanics, ∇f is often understood to be the vector �  i  ∂f  ∂xi⃗ei  (contravariant) which needs a Euclidean dot product to be defined (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ), while the differential df  is covariant (a differential is unmissable in thermodynamics because you can’t use gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  It gives the confusing notation Φ ⊗ ∇X ⊗ ∇X, instead of the legitimate d2Φ = �n  i=1⃗bi ⊗ d2ϕi which  is explicit, objective and easy to manipulate: d2Φ(⃗U, ⃗W) = �n  i=1d2ϕi(⃗U, ⃗W)⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Use Marsden duality notations for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Cartesian bases, with (dXi) the (covariant) dual basis of ( ⃗Ei): with F i  J =  ∂ϕi  ∂XJ , we get dΦ =noted �F =  �n  i=1⃗ei ⊗ dϕi = �n  i,J=1F i  J ⃗ei ⊗ dXJ, and d2Φ = �n  i=1⃗ei ⊗ d2ϕi = �n  i,J,K=1  ∂2ϕi  ∂XJ ∂XK ⃗ei ⊗ (dXJ ⊗ dXK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  31  32  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of coordinate system at t for F  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Change of coordinate system at t for F  Let pt0 ∈ Ωt0, pt = Φt0  t (pt0) ∈ Ωt, ⃗W(pt0) ∈ ⃗Rn  t0, ⃗w(pt) = F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(pt0) ∈ ⃗Rn  t (its push-forward),  written ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The observer at t0 used a basis (⃗ai) in ⃗Rn  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' At t, in ⃗Rn  t , a first observer  chooses a Cartesian basis (⃗bold,i), and a second observer chooses a Cartesian basis (⃗bnew,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let P = [Pij]  be the transition matrix from (⃗bold,i) to (⃗bnew,i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗bnew,j = �n  i=1Pij⃗bold,i for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The change of basis  formula for vectors from (⃗bold,i) to (⃗bnew,i) (in ⃗Rn  t ) gives  [⃗w]|⃗bnew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗bold,  thus  [F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W]|⃗bnew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  Thus  [F]|⃗a,⃗bnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W]|⃗a = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[F]|⃗a,⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W]|⃗a,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  true for all ⃗W, thus  [F]|⃗a,⃗bnew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]|⃗a,⃗bold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  NB: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30) is not the change of basis formula [L]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P for endomorphisms, which would  be nonsense since F := F t0  t (pt0) : ⃗Rn  t0 → ⃗Rn  t is not an endomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30) is just the usual change of  basis formula for vectors ⃗w in ⃗Rn  t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Spatial Taylor expansion of Φ and F  Φt0  t =noted Φ is supposed to be C2 for all t0, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let P ∈ Ωt0, dΦ = F, and ⃗W ∈ ⃗Rn  t0 vector at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then,  in Ωt,  Φ(P+h ⃗W) = Φ(P) + h F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W + h2  2 dF(P)( ⃗W, ⃗W) + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Time Taylor expansion of F  The motion �Φ is supposed to be C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  t0 ∈ R, Φt0 be the associated motion, p(t) = �Φ(t, PObj) and  pt0 = �Φ(t0, PObj), with ⃗v(t, pt) = ∂�Φ  ∂t (t, PObj) the Eulerian velocity and ⃗V t0(t, pt0) := ∂Φt0  ∂t (t, pt0) = ⃗v(t, pt)  the Lagrangian velocity, and F t0  pt0 (t) = F t0(t, pt0) = dΦt0(t, pt0) =noted F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have  ∂F t0  ∂t (t, pt0) = ∂(dΦt0)  ∂t  (t, pt0) = d(∂Φt0  ∂t )(t, pt0)  = d⃗V t0(t, pt0) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t),  in short F = d⃗V = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  Then ∂2Φt0  ∂t2 (t, pt0) = ⃗At0(t, pt0) = ⃗γ(t, p(t)) (Lagrangian and Eulerian accelerations), hence  ∂2F t0  ���t2 (t, pt0) = d ⃗At0(t, pt0) = d⃗γ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t),  in short  ••  F = d ⃗A = d⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Thus, the second order time Taylor expansion of F t0  pt0 =noted F is, in the vicinity of t,  F(t+h) = F(t) + h d⃗V (t) + h2  2 d ⃗A(t) + o(h2)  =  �  I + h d⃗v + h2  2 d⃗γ  �  (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) + o(h2)  when  p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  NB: They are three times are involved: t and t+h as usual, and t0 through F := F t0  pt0 , ⃗V := ⃗V t0  pt0 and  ⃗A := ⃗At0  pt0 (observer dependent), as for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular F(pt0) := F t0  t0 (pt0) = I gives, in the vicinity of t0,  F(t0+h) =  �  I + h d⃗v + h2  2 d⃗γ  �  (t0, pt0) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 γ = ∂⃗v  ∂t + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v is not linear in ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem,  d⃗γ = d(D⃗v  Dt ) = d(∂⃗v  ∂t + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) = d∂⃗v  ∂t + d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  (= D(d⃗v)  Dt  + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  is non linear in ⃗v, and gives F t0  pt0  ′′(t) = (d ∂⃗v  ∂t + d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  pt0 (t), non linear in ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  32  33  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Homogeneous and isotropic material  Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Directly check that F ′ = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F gives F ′′ = d⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) gives F ′′(t) =  D(d⃗v)  Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) + d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ′(t) with  D(d⃗v)  Dt  = d⃗γ − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36), thus F ′′(t) = (d⃗γ − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) + d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) = d⃗γ(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Homogeneous and isotropic material  Let P ∈ Ωt0, let F t0  t (P) := dΦt0  t (P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Suppose that the “Cauchy stress vector” ⃗ft(pt) à t at pt = Φt0  t (P)  only depends on P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' there exists a function ⃗  fun such that  ⃗ft(pt) = ⃗  fun(P, F t0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 A material is homogeneous iff ⃗  fun doesn’t depend on the first variable P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', iff, for  all P ∈ Ωt0,  ⃗  fun(P, F t0  t (P)) = ⃗  fun(F t0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  (Same mechanical property at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 (Isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Consider a Euclidean dot product, the same at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A material is  isotropic at P ∈ Ωt0 iff ⃗  fun is independent of the direction you consider, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', iff, for any rotation Rt0(P)  in ⃗Rn  t0,  ⃗  fun(P, F t0  t (P) = ⃗  fun(P, F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Rt0(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  (Mechanical property unchanged when rotating the material first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 A material is isotropic homogeneous iff it is isotropic and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  The inverse of the deformation gradient  ((Φt0  t )−1 ◦ Φt0  t )(P) = P gives, with p = Φt0  t (P),  d(Φt0  t )−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0  t (P) = It0,  thus  d(Φt0  t )−1(p) = dΦt0  t (P)−1 = F t0  t (P)−1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  where F t0  t  = dΦt0  t is the deformation gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have thus define the two point tensor  Ht0  t := (F t0  t )−1 :  �  �  �  Ωt → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0)  p → Ht0  t (p) = (F t0  t )−1(p) := (F t0  t (P))−1  when  p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  So  Ht0  t (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(p) = (F t0  t )−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(p) := F t0  t (P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(p) ∈ ⃗Rn  t0,  in short  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  for all ⃗w(p) ∈ ⃗Rn  t vector at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines, with pt = Φt0(t, P),  Ht0 :  �  �  �  C =  �  t  ({t} × Ωt) → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0)  (t, pt) → Ht0(t, pt) := Ht0  t (pt) = (F t0(t, P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  NB: Ht0 looks like a Eulerian map, but isn’t: Ht0 depends on a initial time t0 and is a two point tensor  (starts in ⃗Rn  t , arrives in ⃗Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We will however use the material time derivative  D  Dt notation in this case,  that is, we define, along a trajectory t → p(t) = Φt0(t, P),  DHt0  Dt (t, p(t)) := ∂Ht0  ∂t (t, p(t)) + dHt0(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, p(t)),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  DHt0  Dt  = ∂Ht0  ∂t  + dHt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  which is the time derivative g′(t) of the function g : t → g(t) = Ht0(t, Φt0(t, P)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' g(t) = Ht0(t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Hence, with p(t) = Φt0(t, P) and Ht0(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0(t, P) = It0, written H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = I, we get  DH  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂F  ∂t = 0,  thus  DH  Dt = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  since ∂F  ∂t (t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1(t, p(t)) = d⃗v(t, p(t)) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  33  34  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Introduction: Motion versus flow  Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 With ⃗wt0∗(t, p(t)) = F t0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ht0(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗(t, p(t)) = ⃗W(P), when p(t) =  Φt0(t, P), prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  D ⃗wt0∗  Dt  (t, p(t)) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗(t, p(t)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (Ht0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗)(t, p(t)) = ⃗W(P) gives DHt0  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗ +  Ht0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  D ⃗wt0∗  Dt  = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus DHt0  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗ + Ht0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0∗ = 0, thus DH  Dt = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 Prove: Ht0  t  = Ht0  t1 ◦ Ht1  t  and DHt0  Dt (t, p(t)) = Ht0  t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' DHt1  Dt (t, p(t)) for all t0, t1 with  pt1 = Φt0  t1(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have Φt0  t (pt0) = Φt1  t (Φt0  t1(pt0)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18), hence F t0  t (pt0) = F t1  t (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t1 (pt0), thus F t0  t (pt0)−1 =  F t0  t1 (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t1  t (pt1)−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ht0  t (pt) = Ht0  t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ht1  t (p(t)), thus, Ht0(t, p(t)) = Ht0  t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ht1(t, p(t)), thus  DHt0  Dt (t, p(t)) = Ht0  t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' DHt1  Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5  Flow  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Introduction: Motion versus flow  Motion: A motion �Φ : (t, PObj) → pt = �Φ(t, PObj) locates at t a particle PObj in the affine space Rn,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' From which the Eulerian velocity field ⃗v is deduced: ⃗v(t, pt) :=  d�ΦPObj  dt  (t, PObj), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Flow: A flow starts with a Eulerian velocity field ⃗v, from which we deduce a motion by solving the  ODE (ordinary differential equation) dΦ  dt (t) = ⃗v(t, Φ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Definition  Let ⃗v :  �  R × Rn → ⃗Rn  (t, p) → ⃗v(t, p)  �  be a unstationary vector field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a Eulerian velocity ��eld which definition  domain is C = �  t∈[t1,t2]({t} × Ωt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We look for maps Φ :  �  R → Rn  t → p = Φ(t)  �  which are locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in  the vicinity of some t0) solutions of the ODE (ordinary differential equation)  dΦ  dt (t) = ⃗v(t, Φ(t)),  also written  dp  dt (t) = ⃗v(t, p(t)),  or  d⃗x  dt (t) = ⃗v(t, ⃗x(t))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  where ⃗x(t) = −−−→  Op(t) after a choice of an origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Also written dp  dt = ⃗v(t, p) or d⃗x  dt = ⃗v(t, ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 A solution Φ of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) is a flow of ⃗v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Also called an integral curve of ⃗v since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) also  reads Φ(t) =  � t  τ=t1 ⃗v(τ, Φ(τ)) dτ + Φ(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Improper notation for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1):  dp  dt (t) noted  =  dp(t)  dt  (= ⃗v(t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Question: If the notation dp(t)  dt  is used, then what is the meaning of dp(f(t))  dt  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: It means, either dp  dt (f(t)), or d(p◦f)  dt  (t) = dp  dt (f(t)) df  dt(t): Ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So it is better to use  dp  dt (t), and to avoid dp(t)  dt , unless the context is clear (no composite functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Cauchy–Lipschitz theorem  Let (t0, pt0) be in the definition domain of ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We look for Φ solution of “the ODE with initial condition  (t0, pt0)”, in some vicinity of t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' such that  dΦ  dt (t) = ⃗v(t, Φ(t))  and  Φ(t0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  (The couple (t0, pt0) is the initial condition, and the values t0 and pt0 are the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Let t1, t2 ∈ R, t1 < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Ω be an open set in Rn and Ω its closure supposed to be a  regular domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='|| be a norm in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A continuous map ⃗v : [t1, t2] × Ω → ⃗Rn is Lipschitzian iff it is  “space Lipschitzian, uniformly in time”, that is, iff  ∃k > 0, ∀t ∈ [t1, t2], ∀p, q ∈ Ω, ||⃗v(t, q) − ⃗v(t, p)|| ≤ k||q − p||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  So, ||⃗vt(q)−⃗vt(p)||  ||q−p||  ≤ k, for all t and all p ̸= q (the variations of ⃗v are bounded in space, uniformly in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  34  35  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cauchy–Lipschitz theorem  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 (and definifion) (Cauchy–Lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If ⃗v : [t1, t2] × Ω → ⃗Rn is Lipschitzian and  (t0, pt0) ∈]t1, t2[×Ω, then there exists ε = εt0,pt0 > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) has a unique solution Φ :]t0−ε, t0+ε[→ Rn,  noted Φt0  pt0 :  dΦt0  pt0  dt  (t) = ⃗v(t, Φt0  pt0 (t))  and  Φt0  pt0 (t0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Moreover, if ⃗v is Ck then Φ is Ck+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Arnold [2], or any ODE course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular ||⃗v||∞ :=  sup  t∈]t0−ε,t0+ε[, p∈Ω  ||⃗v(t, p)||Rn  (maximum speed) exists since ⃗v ∈ C0 on the compact [t1, t2]×Ω), see definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3, hence we can choose  ε = min(t0−t1, t2−t0, d(pt0,∂Ω)  ||⃗v||∞  ) (the time needed to reach the border ∂Ω from pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We have thus defined the function, also called “a flow”,  Φ :  � ]t1, t2[×]t1, t2[× Ωt0 → Ω  (t, t0, pt0) → p = Φ(t, t0, pt0) := Φt0  pt0 (t) noted  =  Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  And (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) reads  ∂Φ  ∂t (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0) = ⃗v(t, Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0)),  with  Φ(t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  We have thus defined the function, also called “a flow”,  Φt0 :  �  [t0−ε, t0+ε] × Ωt0 → Rn  (t, pt0) → p = Φt0(t, pt0) := Φt0  pt0 (t) :  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  And (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) reads  ∂Φt0  ∂t (t, pt0) = ⃗v(t, Φt0(t, pt0)),  and  Φt0(t0, pt0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Other definition and notation (can be ambiguous): Φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0 = Φt0  t : Ωt0 → Rn, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) is written  dΦt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0)  dt  = ⃗v(t, Φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0)),  and  Φt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Let ⃗v be Lipschitzian, let t0 ∈]t1, t2[, and let Ωt0 be an open set s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ωt0 ⊂⊂ Ω (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' there  exists a compact set K ∈ Rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ωt0 ⊂ K ⊂ Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then there exists ε > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a flow Φt0 exists on  ]t0−ε, t0+ε[×Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let d = d(K, Rn−Ω) (la distance of K to the border of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let ||⃗v||∞ :=  sup  t∈[t1,t2],p∈Ω  ||⃗v(t, p)||Rn (exists since ⃗v ∈ C0 on the compact [t1, t2] × Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let ε = min(t0−t1, t2−t0,  d  ||⃗v||∞ ) (less that the minimum time to reach the border from K at maximum  speed ||v||∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let pt0 ∈ K and t ∈]t0−ε, t0+ε[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then Φt0  pt0 exists, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4, and ||Φt0  pt0 (t) − Φt0  pt0 (t0)||Rn ≤  [t−t0| supτ∈]t0−ε,t0+ε[(||(Φt0  pt0 )′(τ)||Rn) (mean value theorem since, ⃗v being C0, Φ is C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ||Φt0  pt0 (t)−  Φt0  pt0 (t0)||Rn ≤ [t − t0| ||v||∞, thus Φt0  pt0 (t) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus Φt0  pt0 exists on ]t0−ε, t0+ε[, for all pt0 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 The definition of a flow starts with a Eulerian velocity (independent of any initial time),  and then, due to the introduction of initial conditions, leads to the Lagrangian functions Φt0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Once again, Lagrangian functions are the result of Eulerian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  35  36  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Examples  Example 1  R2 with an origin O, a Euclidean basis (⃗e1,⃗e2) and Ω = [0, 2]×[0, 1] (observation window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let p ∈ R2, −→  Op =noted ⃗x = x⃗e1 + y⃗e2 =noted (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t1 = −1, t2 = 1, t0 ∈]t1, t2[, a, b ∈ R, a ̸= 0, and  ⃗v(t, p) =  �  v1(t, x, y) = ay,  v2(t, x, y) = b sin(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  (b = 0 corresponds to the stationary case = shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') ⃗x(t0) =  �  x0  y0  �  , ⃗x(t) =  �  x(t)  y(t)  �  = −−−−−−→  OΦt0  pt0 (t) and  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) give  �  �  �  �  �  dx  dt (t) = v1(t, x(t), y(t)) = ay(t),  dy  dt (t) = v2(t, x(t), y(t)) = b sin(t−t0),  with  �  x(t0) = x0,  y(t0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Thus  ⃗x(t) = −−−→  Op(t) = −−−−−−→  OΦt0  pt0 (t) =  �  x(t) = x0 + a(y0 + b)(t−t0) − ab sin(t−t0)  y(t) = y0 + b − b cos(t−t0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Example 2  Similar framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ω > 0 and consider (spin vector field)  ⃗v(t, x, y) =  �  −ωy  ωx  �  = ω  �  0  −1  1  0  � �  x  y  �  noted  =  ⃗v(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  With −−→  Opt0 = ⃗xt0 =  �  xt0  yt0  �  , rt0 =  �  x2  t0 + y2  t0, and θ0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗xt0 =  �  xt0 = rt0 cos(ωt0)  yt0 = rt0 sin(ωt0)  �  , the solution Φt0  pt0  of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) is  ⃗x(t) = −−−→  Op(t) = −−−−−−→  OΦt0  pt0 (t) =  �  x(t) = rt0 cos(ωt)  y(t) = rt0 sin(ωt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  Indeed,  � ∂x  ∂t (t, ⃗x0)  ∂y  ∂t (t, ⃗x0)  �  =  �  v1(t, x(t, ⃗x0), y(t, ⃗x0))  v2(t, x(t, ⃗x0), y(t, ⃗x0))  �  =  �  −ωy(t, ⃗x0)  ωx(t, ⃗x0)  �  , thus  ∂x  ∂t (t, ⃗x0) = −ωy(t, ⃗x0)  and  ∂y  ∂t (t, ⃗x0) = ωx(t, ⃗x0), thus  ∂2y  ∂t2 (t, ⃗x0) = −ω2y(t, ⃗x0), hence y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here d⃗v(t, x, y) =  ω  �  0  −1  1  0  �  = ω  �  cos π  2  − sin π  2  sin π  2  cos π  2  �  is the π/2-rotation composed with the homothety with ratio ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Composition of flows  Let ⃗v be a vector field on R × Ω and Φt0  pt0 solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We use the notations  pt = Φt0  t (pt0) = Φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0) := Φt0  pt0 (t) = Φt0(t, pt0) = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0) = Φt0,pt0 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Law of composition of flows (determinism)  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 For all t0, t1, t2 ∈ R, we have (determinism)  Φt1  t2 ◦ Φt0  t1 = Φt0  t2,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t1 ◦ Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0 = Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  (“The composition of the photos gives the film”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So,  pt2 = Φt1  t2(pt1) = Φt0  t2(pt0)  when  pt1 = Φt0  t1(pt0),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  pt2 = Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t1(pt1) = Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0)  when  pt1 = Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  Thus  dΦt1  t2(pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0  t1(pt0) = dΦt0  t2(pt0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t1(pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0) = dΦt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Summary with commutative diagrams:  pt1  Φt1  t2  �  pt0  Φt0  t1  �  Φt0  t2  � pt2  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  pt1  Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t1  �  pt0  Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0  �  Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0  � pt2  36  37  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Composition of flows  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let pt1 = Φt0  pt0 (t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) gives  �  �  �  �  �  �  �  dΦt0  pt0  dt  (t) = ⃗v(t, Φt0  pt0 (t)),  dΦt1  pt1  dt  (t) = ⃗v(t, Φt1  pt1 (t)),  �  �  �  �  �  �  �  with  pt1 = Φt0  pt0 (t1) = Φt1  pt1 (t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus Φt0  pt0 and Φt1  pt1 satisfy the same ODE with the same value at t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus they are equal (uniqueness  thanks to Cauchy–Lipschitz theorem), thus Φt1  pt1 (t) = Φt0  pt0 (t) when pt1 = Φt0  t1(pt0), that is, Φt1  t (pt1) =  Φt0  t (pt0) when pt1 = Φt0  t1(pt0), which is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17) for any t = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 A flow is compatible with the motion �Φ of an object Obj: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) gives Φt1  t2 ◦ Φt0  t1 = (�Φt2 ◦  (�Φt1)−1) ◦ (�Φt1 ◦ (�Φt0)−1) = �Φt2 ◦ (�Φt0)−1 = Φt0  t2, that is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Stationnary case  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 ⃗v is a stationary vector field iff ∂⃗v  ∂t = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And then ⃗v(t, p) =noted ⃗v(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the associated  flow Φt0 which satisfies  ∂Φt0  ∂t (t, pt0) = ⃗v(pt)  when  pt = Φt0(t, pt0),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  is said to be stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 If ⃗v is a stationary vector field then, for all t0, t1, h, when meaningful (h small enough  and t1 close enough to t0),  Φt1  t1+h = Φt0  t0+h,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Φt1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t1 = Φt0+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Φt1  t1+h(q) = Φt0  t0+h(q), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Φ(t1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t1, q) = Φ(t0+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, q) for all q ∈ Ωt0 (see theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other  words,  Φt0+h  t1+h = Φt0  t1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Φt1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0+h = Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Φt0+h  t1+h(q) = Φt0  t1(q), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Φ(t1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0+h, q) = Φ(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, q) for all q ∈ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let q ∈ Ωt0, α(h) = Φt0  t0+h(q) = Φt0  q (t0+h) and β(h) = Φt1  t1+h(q) = Φt1  q (t1+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus α′(h) =  dΦt0  q  dt (t0+h) = ⃗v(t0+h, Φt0  q (t0+h)) = ⃗v(Φt0  q (t0+h)) = ⃗v(α(h)) (stationary flow), and  β′(h) = dΦt1q  dt (t1+h) = ⃗v(t1+h, Φt1  q (t1+h)) = ⃗v(Φt1  q (t1+h)) = ⃗v(β(h)) (stationary flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus α and β satisfy the same ODE with the same initial condition α(0) = β(0) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, with h = t1−t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with t1 = t0+h and t0+h = t1, we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 If ⃗v is a stationary vector field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21), then  dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(pt0) = ⃗v(pt)  when  pt = Φt0  t (pt0),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  that is, if ⃗v is stationary, then ⃗v is transported (push-forwarded by Φt0  t ) along itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18), t2 = t1+s and t1 = t0+s give Φt0+s  t1+s(Φt0  t0+s(pt0)) = Φt0  t1+s(pt0), and ⃗v is stationary, thus  Φt0  t1(Φt0  t0+s(pt0)) = Φt0  t1+s(pt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Φ(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, Φt0,pt0 (t0+s)) = Φt0,pt0 (t1+s), thus (s derivative)  dΦ(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, Φ(t0+s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Φt0,pt0  ′(t0+s) = Φt0,pt0  ′(t1+s),  thus dΦt0  t1(Φ(t0+s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t0+s, Φt0,pt0 (t0+s)) = ⃗v(t1+s, Φt0,pt0 (t1+s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus with s = 0, and ⃗v being  stationary, dΦt0  t1(Φ(t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(Φt0,pt0 (t0)) = ⃗v(Φt0,pt0 (t1)), thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  37  38  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Velocity on the trajectory traveled in the opposite direction  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Velocity on the trajectory traveled in the opposite direction  Let t0, t1 ∈ R, t1 > t0, and pt0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the trajectory Φt0  pt0 :  �  [t0, t1] → Rn  t → p(t) = Φt0  pt0 (t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So pt0  is the beginning of the trajectory, pt1 = Φt0  t1(pt0) at the end, ⃗v(t, p(t)) =  dΦt0  pt0  dt  (t) being the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Define the trajectory traveled in the opposite direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' define  Ψt1  pt1 :  �  [t0, t1] → Rn  u → q(u) = Ψt1  pt1 (u) := Φt0  pt0 (t0+t1−u) = Φt0  pt0 (t) = p(t)  when  t = t0+t1−u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  In particular q(t0) = Ψt1  pt1 (t0) = Φt0  pt0 (t1) = p(t1) and q(t1) = Ψt1  pt1 (t1) = Φt0  pt0 (t0) = p(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 The velocity on the trajectory traveled in the opposite direction is the opposite of  the velocity on the initial trajectory:  dΨt1  pt1  du  (u) = q′(u) = −p′(t) = −⃗v(t, p(t))  when  t = t0+t1−u,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ψt1  pt1 (u) = Φt0  pt0 (t0+t1−u) gives  dΨt1  pt1  du (u) = −  dΦt0  pt0  dt  (t0+t1−u) = −⃗v(t0+t1−u, Φt0  pt0 (t0+t1−u)) =  −⃗v(t, Φt0  pt0 (t)) when t = t0+t1−u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Variation of the flow as a function of the initial time  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Ambiguous and non ambiguous notations  Let Φ : (t, u, p) ∈ R × R × Rn → Φ(t, u, p) ∈ Rn be a C1 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The partial derivatives are  ∂1Φ(t, u, p) := lim  h→0  Φ(t+h, u, p) − Φ(t, u, p)  h  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  ∂2Φ(t, u, p) := lim  h→0  Φ(t, u+h, p) − Φ(t, u, p)  h  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  and ∂3Φ(t, u, p), defined for all ⃗w ∈ ⃗Rn (a vector at p) by,  ∂3Φ(t, u, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w := lim  h→0  Φ(t, u, p+h⃗w) − Φ(t, u, p)  h  noted  =  dΦ(t, u, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  When the name of the first variable is systematically noted t, then  ∂1Φ(t, u, p) noted  =  ∂Φ  ∂t (t, u, p) noted  =  ∂Φ(t, u, p)  ∂t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  NB: This notation can be ambiguous: What is the meaning of ∂Φ  ∂t (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t, p)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In ambiguous situations, use  the notation ∂1Φ, or (if no composed functions inside) use ∂Φ(t,u,p)  ∂t  |u=t (so t is the derivation variable,  and after the calculation you take u = t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  When the name of the second variable is systematically noted u, then  ∂2Φ(t, u, p) noted  =  ∂Φ  ∂u (t, u, p) noted  =  ∂Φ(t, u, p)  ∂u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  NB: Idem this notation can be ambiguous: What is the meaning of ∂Φ  ∂u (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' u, p)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In ambiguous situations,  use the notation ∂2Φ, or use ∂Φ(t,u,p)  ∂u  |t=u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  When the name of the third variable is systematically a space variable noted p, then  ∂3Φ(t, u, p) noted  =  dΦ(t, u, p) noted  =  ∂Φ  ∂p (t, u, p) noted  =  ∂Φ(t, u, p)  ∂p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  38  39  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Variation of the flow as a function of the initial time  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Variation of the flow as a function of the initial time  The law of composition of the flows gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) gives Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' u, Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0)) = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the derivative  in u gives  ∂2Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' u, Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0)) + dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' u, Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂1Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0) = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂2Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' u, p(u)) = −dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(u, p(u))  when  p(u) = Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  In particular u = t0 gives, for all (t, t0, p0) ∈ R2 × Ωt0,  (∂Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0)  ∂t0  =)  ∂2Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0) = −dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  In particular  (dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0)  dt0  |t=t0  =)  ∂2Φ(t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, p0) = −⃗v(t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  39  40  Part II  Push-forward  6  Push-forward  The general tool to describe “transport” is “push-forward by a motion” (the “take with you” operator),  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 and figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The push-forward also gives the tool needed to understand the velocity addition  formula: In that case, the push-forward is the translator between observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The push-forward can also  be used to write coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' As usual, we start with qualitative results (observer independent  results);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, quantitative results are deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  E and F are affine spaces, E and F are the associated vector spaces equipped with norms ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||E and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||F  with dim E = dim F = n, UE and UF are open sets in the affine space E and F, or possibly the vector  spaces E and F, and  Ψ :  �  UE → UF  pE → pF = Ψ(pE)  is a diffeomorphism  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  (a C1 invertible map which inverse is C1), called the push-forward, and Ψ−1 is the pull-back (push-forward  with Ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1: cE : s → pE = cE(s) is a curve in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Push-forwarded by Ψ it becomes the curve cE∗ := Ψ ◦ cE  in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The tangent vector at pE = cE(s) is ⃗wE(pE) = cE ′(s), and the tangent vector at pF = cF(s) =  Ψ(cE(s)) is ⃗wE∗(pF) = cF ′(s) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Other illustation: See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example: Ψ = Φt0  t : Ωt0 → Ωt, the motion that transforms Ωt0 into Ωt, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example: Ψ : UE → UF a coordinate system, see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example: Ψ = Θt : RB → RA, a change of referential at t (change of observer), see § 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Push-forward and pull-back of points  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If pE ∈ UE (a point in UE) then its push-forward by Ψ is the point  pF = Ψ∗pE := Ψ(pE) = pE∗ ∈ UF,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1, the last notation if Ψ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if pF ∈ UF then its pull-back by Ψ is the point  pE = Ψ∗pF := Ψ−1(pF) = pF  ∗ ∈ UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  We immediately have Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The notations ∗ for push-forward and ∗ for pull-back have been proposed by Spivak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Also see Abraham  and Marsden [1] (second edition) who adopt this notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  40  Us  亚  we(pe  p =C(s  P = 亚(p)  Im(c*  Im(C41  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Push-forward and pull-back of curves  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Push-forward and pull-back of curves  We push-forward (and pull-back) the points on a curve:  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Let cE :  �  ] − ε, ε[ → UE  s → pE = cE(s)  �  be a curve in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its push-forward by Ψ is the curve  Ψ∗cE := Ψ ◦ cE :  � ] − ε, ε[ → UF  s → pF = Ψ∗cE(s) := Ψ(cE(s)) noted  =  cE∗(s)  (= Ψ(pE)),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗cE =noted cE∗ when Ψ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') This defines  Ψ∗ :  � F(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' UE) → F(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' UF)  cE → Ψ∗(cE) := Ψ ◦ cE  noted  =  Ψ∗cE = cE∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Let cF :  �  ] − ε, ε[ → UF  s → pF = cF(s)  �  is a curve in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its pull-back by Ψ is  Ψ∗cF := Ψ−1 ◦ cE  � ] − ε, ε[ → UE  s → pE = Ψ∗cF(s) := Ψ−1(cF(s)) noted  =  cF  ∗(s)  (= Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  We have thus defined  Ψ∗ :  � F(C1(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' UF) → F(C1(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' UE)  cF → Ψ∗(cF) := Ψ−1 ◦ cF  noted  =  Ψ∗cF = cF  ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Push-forward and pull-back of scalar functions  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definitions  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Let fE :  �  UE → R  pE → fE(pE)  �  (scalar valued function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its push-forward by Ψ is the (scalar  valued) function  Ψ∗fE := fE ◦ Ψ−1 :  � UF → R  pF → Ψ∗fE(pF) := fE(pE) noted  =  fE∗(pF)  when  pE = Ψ−1(pF),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  (noted fE∗ when Ψ is implicit), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ψ∗fE(Ψ∗pE) := fE(pE), or fE∗(pE∗) := fE(pE) when pE∗ = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We  have thus defined  Ψ∗ :  � F(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → F(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  fE → fF := Ψ∗(fE) = fE ◦ Ψ−1 noted  =  Ψ∗fE,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  the notation Ψ∗(fE) = Ψ∗fE since Ψ∗ is linear: ((fE + λgE) ◦ Ψ−1)(pF) = (fE + λgE)(pE) = fE(pE) +  λgE(pE) = (fE ◦ Ψ−1)(pF) + λ(gE ◦ Ψ−1)(pF) gives Ψ∗(fE + λgE) = Ψ∗(fE) + λΨ∗(gE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Let fF :  �  UF → R  pF → fF(pF)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its pull-back by Ψ is the push-forward by Ψ−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is  Ψ∗fF := fF ◦ Ψ :  � UE → R  pE → Ψ∗fF(pE) := fF(pF) noted  =  fF  ∗(pE)  when  pF = Ψ(pE),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ψ∗fF(Ψ∗pF) := fF(pF), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' fF ∗(pF ∗) := fF(pF) when pF = Ψ∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have thus defined  Ψ∗ :  � F(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → F(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  fF → Ψ∗(fF) = fF  ∗ := fF ◦ Ψ noted  =  Ψ∗fF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  We immediately have Ψ∗ ◦ Ψ∗ = I  and  Ψ∗ ◦ Ψ∗ = I (the first I is the identity in F(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), the  second I is the identity in F(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: We used the same notations Ψ∗ and Ψ∗ than for the push-forward and pull-backs of points: The  context removes ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  41  42  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Push-forward and pull-back of vector fields  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Interpretation: Why is it useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : Let �Φ : R × Obj → Rn be a motion of an object Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' An observer records the temperature θ at  all t ∈ [t0, T] and all p = �Φ(t, Obj): He gets θ :  �  �  �  C =  �  t  ({t} × Ωt) → R  (t, p) → θ(t, p)  �  �  � a Eulerian scalar valued  function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then he chooses an initial time t0 and considers the associated motion Φt0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1),  and considers θt0 :  �  Ωt0 → R  pt0 → θt0(pt0) := θ(t0, pt0)  �  (snapshot of the temperatures at t0 in Ωt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The  push-forward of θt0 by Φt0  t is (Φt0  t )∗θt0 := θt0 ◦ (Φt0  t )−1 defines the “memory function”  (Φt0  t )∗θt0 :  �  Ωt → R  pt → (Φt0  t )∗θt0(pt) := θt0(pt0)  when  pt = Φt0  t (pt0),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  And he writes (Φt0  t )∗θt0(pt) =noted θt0∗(t, pt), so the memory transported is at t at pt (along a trajectory)  by  θt0∗(t, p(t)) = θt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Question: Why do we introduce θt0∗ since we have θt0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: An observer does not have the gift of temporal and/or spatial ubiquity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' He has to do with  values at the actual time t and position pt where he is (Newton and Einstein’s point of view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, when  he was at t0 at pt0 the observer wrote the value θt0(pt0) on a piece of paper (for memory), puts the piece  of paper is his pocket, then once at t at p(t) = Φt0(t, pt0), he takes the paper out of his pocket, and  renames the value he reads as θt0∗(t, pt) because he is now at t at pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And, now at t at pt, he can compare  the past and present value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular the rate  θ(t, p(t)) − θt0∗(t, p(t))  t − t0  =  actual(t, p(t)) − memory∗(t, p(t))  t − t0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  is physically meaningful for one observer at t at pt (no ubiquity gift required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' For scalar value functions,  we get the usual rate θ(t,p(t))−θ(t0,p(t0))  t−t0  , but it isn’t that simple for vector valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And the limit t → t0 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14) defines the Lie derivative for scalar valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Push-forward and pull-back of vector fields  This is one of the most important concept for mechanical engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  A definition by approximation  Elementary introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let pE and qE be points in UE, and let pF = pE∗ = Ψ(pE) and qF = qE∗ = Ψ(qE)  in UF be the push-forwards by Ψ cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The first order Taylor expansion gives  (Ψ(qE) − Ψ(pE) =)  qF − pF = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (qE − pE) + o(||qE − pE||E),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  thus,  −−→  pFqF  ||−−→  pEqE||E  = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  −−→  pEqE  ||−−→  pEqE||E  + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  And the definition of the push-forward is obtained by “neglecting” the o(1) (limit as qE → pE):  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 If ⃗wE(pE) ∈ E is a vector at pE ∈ U then its push-forward by Ψ is the vector  ⃗wF(pF) =noted ⃗wE∗(pF) =noted Ψ∗ ⃗wE(pF) ∈ F defined at pF = pE∗ = Ψ(pE) ∈ UF by  ⃗wF(pF) = ⃗wE∗(pF) := dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE) noted  =  Ψ∗ ⃗wE(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The definition of the push-forward of a vector field  To fully grasp the definition, and to avoid making interpretation errors as in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 (the unfortunate  notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X), we use the following definition of “a vector”: It is a “tangent vector to a curve”  (needed for surfaces and manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Details:  42  43  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Push-forward and pull-back of vector fields  Let cE :  �  ] − ε, ε[ → UE  s → pE = cE(s)  �  be a C1 curve in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its tangent vector at pE = cE(s) is  ⃗wE(pE) := cE  ′(s)  (= lim  h→0  cE(s + h) − cE(s)  h  ),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines the function ⃗wE :  �  Im(cE) → E  pE → ⃗wE(pE)  �  called a vector field along Im(cE)⊂UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The push-forward of cE by Ψ being the image curve cE∗ = Ψ ◦ cE (the curve transformed by Ψ)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4), its tangent vector at pF = cE∗(s) is  ⃗wE∗(pF) := cE∗  ′(s)  thus  = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='cE  ′(s) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  Thus we have defined the vector field ⃗wE∗ along Im(cE∗) called the push-forward of ⃗wE by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With all the integral curves of a vector field defined in UE, we get:  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 The push-forward by Ψ of a C0 vector field ⃗wE :  �  UE → E  pE → ⃗wE(pE)  �  is the vector field  Ψ∗ ⃗wE = ⃗wE∗ :  �  �  �  UF → F  pF → Ψ∗ ⃗wE(pF) := dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE) noted  =  ⃗wE∗(pF)  when  pF = Ψ(pE),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗ ⃗wE =noted ⃗wE∗ if Ψ is implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other words,  Ψ∗ ⃗wE := (dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE) ◦ Ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  This defines the map Ψ∗ :  �  C∞(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) → C∞(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  ⃗wE → Ψ∗(⃗wE) := Ψ∗ ⃗wE = ⃗wE∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (We use the same notation Ψ∗ as  in definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 for scalar valued functions: The context removes ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Unlike scalar functions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2: At t0 at pt0 you cannot just draw a vector ⃗wt0(pt0)  on a piece of paper, put the paper in your pocket, then let yourself be carried by the flow Ψ = Φt0  t  (push-forward), then, once arrived at t at pt, take the paper out of your pocket and read it to get the  push-forward: The direction and length of the vector ⃗wt0∗(t, pt) are modified by the flow (a vector is not  just a collection of scalar components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Prove:  ⃗cE  ′′(s) = d⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  and  d⃗wE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wE(pE) + d2Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  and  cE∗  ′′(s) = d⃗wE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE∗(pF)  (= dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′′(s) + d2Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s) = ⃗wE(cE(s)) gives ⃗cE  ′′(s) = d⃗wE(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s), hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗wE∗(Ψ(pE)) = dΦ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE) by definition of ⃗wE∗, hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  cF(s) = Ψ(cE(s)) gives ⃗cF  ′(s) = dΨ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s) = dΨ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(cE(s)) = ⃗wE∗(cF(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus ⃗cF  ′′(s) =  (d2Ψ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′(s) + dΨ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cE  ′′(s) = d⃗wE∗(cF(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗cF  ′(s), hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Pull-back of a vector field  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 If ⃗wF :  �  UF → F  pF → ⃗wF(pF)  �  is a vector field on UF, then its pull-back by Ψ is the  push-forward by Ψ−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is the vector field on UE defined by  Ψ∗ ⃗wF :  �  �  �  UE → E  pE → Ψ∗ ⃗wF(pE) := dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) noted  =  ⃗wF  ∗(pE),  when  pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  In other words,  Ψ∗ ⃗wF := (dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF) ◦ Ψ noted  =  ⃗wF  ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  43  44  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification with bases  And we get  Ψ∗ ◦ Ψ∗ = I  and  Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Indeed, Ψ∗(Ψ∗ ⃗wE)(pE) = dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ψ∗ ⃗wE(pF) = dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE) = ⃗wE(pE), for all pE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for  the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Quantification with bases  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Usual result  (⃗ai) is a Cartesian basis in E, OF and (⃗bi) are an origin in F and a Cartesian basis in F, pE ∈ UE,  pF = Ψ(pE) = OF +  n  �  i=1  ψi(pE)⃗bi,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [−−−→  OFpF]|⃗b =  �  �  �  ψ1(pE)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ψn(pE)  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  Then, if ⃗wE is a vector field in UE and ⃗wE  = �  i wj⃗ai, we get Ψ∗ ⃗wE(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE) =  �n  i=1(dψi(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE))⃗bi = �n  i,j=1wj(pE)(dψi(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj)⃗bi = �n  i,j=1  ∂ψi  ∂xj (pE)wj(pE)⃗bi, so  [Ψ∗ ⃗wE(pF)]|⃗b = [dΨ(pE)]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗wE(pE)]|⃗a,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  where [dΨ(pE)]|⃗a,⃗b = [dψi(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj] =noted [ ∂ψi  ∂xj (pE)] is the Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Example: Polar coordinate system  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Change of coordinate system interpreted as a push-forward: Paradigmatic example of  the polar coordinate system (model generalized for the parametrization of any manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Parametric Cartesian vector space R × R =noted ⃗R2  p = {⃗q = (r, θ)}, with its canonical basis (⃗a1,⃗a2),  and ⃗q = r⃗a1 + θ⃗a2 =noted (r, θ), so [⃗q]|⃗a =  �  r  θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Geometric affine space R2 (of positions), p ∈ R2,  associated vector space ⃗R2, O ∈ R2 (origin), ⃗x = −→  Op, and a Euclidean basis (⃗b1,⃗b2) in ⃗R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The “polar  coordinate system” is the associated map Ψ :  � ⃗R∗  + × R ⊂ ⃗R2  p → ⃗R2  ⃗q = (r, θ) → ⃗x = Ψ(⃗q) = Ψ(r, θ),  �  defined by  ⃗x = Ψ(⃗q) := r cos θ⃗b1 + r sin θ⃗b2,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗x]|⃗b =  �  x = r cos θ  y = r sin θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  The i-th coordinate line at ⃗q in ⃗R2  p (parametric space) is the straight line ⃗c⃗q,i :  �  R → ⃗R2  p  s → ⃗c⃗q,i(s) = ⃗q + s⃗ai  �  ,  and its tangent vector at ⃗c⃗q,i(s) is ⃗c⃗q,i′(s) = ⃗ai for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This line is transformed by Ψ into the curve  Ψ∗(cq,i) = Ψ ◦ ⃗c⃗q,i =noted c⃗x,i :  �  R → R2  s → c⃗x,i(s) = Ψ(⃗q + s⃗ai)  �  (in particular c⃗x,i(0) = ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  [−−−−−→  Oc⃗x,1(s)]|⃗b =  �  (r+s) cos θ  (r+s) sin θ  �  (straight line),  and  [−−−−−→  Oc⃗x,2(s)]|⃗b =  �  r cos(θ+s)  r sin(θ+s)  �  (circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  And the tangent vector at c⃗x,i(s) is c⃗x,i′(s) =noted ⃗ai∗(⃗x) (push-forward by Ψ), so  ⃗a1∗(⃗x) := Ψ∗⃗a1(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1 = lim  h→0  Ψ(⃗q+h⃗a1) − Ψ(⃗q)  h  = lim  h→0  Ψ(r+h, θ) − Ψ(r, θ)  h  = ∂Ψ  ∂r (⃗q),  ⃗a2∗(⃗x) := Ψ∗⃗a2(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2 = lim  h→0  Ψ(⃗q+h⃗a2) − Ψ(⃗q)  h  = lim  h→0  Ψ(r, θ+h) − Ψ(r, θ)  h  = ∂Ψ  ∂θ (⃗q),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  Thus  ⃗a1∗(⃗x) = cos θ⃗b1 + sin θ⃗b2  and  ⃗a2∗(⃗x) = −r sin θ⃗b1 + r cos θ⃗b2,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗a1∗(⃗x)]|⃗b =  �  cos θ  sin θ  �  and  [⃗a2∗(⃗x)]|⃗b =  �  −r sin θ  r cos θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  The basis (⃗a1∗(⃗x),⃗a2∗(⃗x)) is called the basis of the polar coordinate system at ⃗x (it is orthogonal  but not orthonormal since ||⃗a2∗(⃗x)|| = r ̸= 1 in general);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And [dΨ(⃗q)]|⃗a,⃗b =  � [ ∂Ψ  ∂r (⃗q)]|⃗b  [ ∂Ψ  ∂θ (⃗q)]|⃗b  �  =  � [⃗a1∗(⃗x)]|⃗b  [⃗a2∗(⃗x)]|⃗b  �  =  �  cos θ  −r sin θ  sin θ  r cos θ  �  = [ ∂Ψi  ∂qj (⃗q)] is the Jacobian matrix of Ψ at ⃗q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  44  45  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification with bases  And the dual basis of the polar system basis (⃗a1∗(⃗x),⃗a2∗(⃗x)) is called (dq1(⃗x), dq2(⃗x)) (defined by  dqi(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(⃗x) = δij), so  dq1(⃗x) = cos θ dx1 + sin θ dx2  and  dq2(⃗x) = −1  r sin θ dx1 + 1  r cos θ dx2,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dq1(⃗x)]|⃗b = ( cos θ  sin θ ) and [dq2(⃗x)]|⃗b = − 1  r ( sin θ  cos θ ) (row matrices) when ⃗x = Ψ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 The components γk  ij(⃗x) of the vector d⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai∗(⃗x) ∈ ⃗R2 in the basis (⃗ai∗(⃗x)) are the  Christoffel symbols of the polar coordinate system (with duality notations as it is usually presented):  d⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai∗(⃗x) =  n  �  k=1  γk  ij(⃗x)⃗ak∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  At ⃗x = Ψ(⃗q), with ⃗aj∗(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗aj∗ ◦ Ψ)(⃗q) = ∂Ψ  ∂qj , we get  d⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai∗(⃗x) =  ∂2Ψ  ∂qi∂qj (⃗q) = d⃗ai∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(⃗x),  so  γk  ij = γk  ji  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  for all i, j (symmetry of the bottom indices as soon as Ψ is C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here for the polar coordinates,  ∂Ψ  ∂r (⃗q) = cos θ⃗b1 + sin θ⃗b2 gives  ∂2Ψ  ∂r2 (⃗q) = ⃗0, thus γ1  11 = γ2  11 = 0,  and  ∂2Ψ  ∂θ∂r(⃗q) = − sin θ⃗b1 + cos θ⃗b2 = 1  r⃗a2∗(⃗x), thus γ1  12 = 0 = γ1  21 and γ2  12 = 1  r = γ2  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ∂Ψ  ∂θ (⃗q) =  −r sin θ⃗b1 + r cos θ⃗b2 gives ∂2Ψ  ∂θ2 (⃗q) = −r cos θ⃗b1 − r sin θ⃗b2 = −r⃗a1∗(⃗x), thus γ1  22 = −r and γ2  22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 The (widely used) normalized polar coordinate basis (⃗n1(⃗x),⃗n2(⃗x)) = (⃗a1∗(⃗x), 1  r⃗a2∗(⃗x))  is not holonomic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is not the basis of a coordinate system (and its use makes higher deriva-  tion formulas complicated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Indeed ⃗n2(⃗x) =  1  r⃗a2∗(⃗x) gives d⃗n2(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n1(⃗x) = (d( 1  r)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n1(⃗x))⃗a2∗(⃗x) +  1  rd⃗a2∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n1(⃗x), and ⃗n1(⃗x)  = ⃗a1∗(⃗x) gives d⃗n1(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n2(⃗x)  =  d⃗a1∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ( 1  r⃗a2∗), thus d⃗n2(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n1(⃗x) −  d⃗n1(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n2(⃗x)  =  (d( 1  r)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n1(⃗x))⃗a2∗(⃗x)  ̸=  ⃗0,  since  1  r  =  (x2 + y2)− 1  2  gives d( 1  r)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n1(⃗x)  =  ( −x(x2 + y2)− 3  2  −y(x2 + y2)− 3  2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  cos θ  sin θ  �  =  1  r3 (−r cos2 θ − r sin2 θ) = −1  r2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 (Pay attention to the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Let f : ⃗q ∈ ⃗R2  p → f(⃗q) ∈ R be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Call g its push-  forward by Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' g : ⃗x ∈ R2 → g(⃗x) = f(⃗q) ∈ R when ⃗x = Ψ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So f(⃗q) = (g ◦ Ψ)(⃗q)and  df(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = dg(Ψ(⃗q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  With df(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =noted  ∂f  ∂qj (⃗q) and dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj =noted  ∂g  ∂xj (⃗x) and ⃗aj∗(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �  i  ∂Ψi  ∂qj (⃗q)⃗aj, we get  ∂f  ∂qj (⃗q) =  �  i  ∂g  ∂xi (⃗x)∂Ψi  ∂qj (⃗q) noted  =  ∂g  ∂qj (⃗x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  Mind this notation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' g is a function of ⃗x, not of ⃗q, so ∂g  ∂qi (⃗x) means  =  ∂f  ∂qi (⃗q), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂g  ∂qi (⃗x) means  =  ∂(g ◦ Ψ)  ∂qi  (⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  which is [df(⃗q)] = [dg(⃗x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then (with f and Ψ C2)  ∂ ∂g  ∂qi  ∂qj (⃗x) means  =  ∂ ∂(g◦Ψ)  ∂qi  ∂qj  (⃗q) = d(dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai∗)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = d(dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai∗)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(⃗x)  = d((dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(⃗x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai∗(⃗x) + dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗ai∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj(⃗x)) noted  =  ∂2g  ∂qi∂qj (⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  So  ∂2g  ∂qi∂qj (⃗x) means  =  d2g(⃗x)(⃗ai∗(⃗x),⃗aj∗(⃗x)) +  n  �  k=1  ∂g  ∂xk (⃗x)γk  ij(⃗x)⃗ak(⃗x),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  and  ∂2g  ∂qi∂qj (⃗x) is not reduced to d2g(⃗x)(⃗ai∗(⃗x),⃗aj∗(⃗x)) (the Christoffel symbols have appeared): First  order derivatives  ∂g  ∂xk are still alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Contrary to  ∂2g  ∂xi∂xj (⃗x) = d2g(⃗x)(⃗bi,⃗bj) with a Cartesian basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  NB: The independent variables r and θ don’t have the same dimension (a length and an angle): There  is no physical meaningful inner dot product in the parameter space ⃗R2  p = R×R = {(r, θ)}, but this space  is very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (As in thermodynamics: No meaningful inner dot product in the (T, P) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  45  46  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition  7  Push-forward and pull-back of differential forms  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Setting of § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider a differential form αE :  �  UE → E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  pE → αE(pE)  �  on UE (a field of linear forms),  and a vector field ⃗wE :  �  UE → E  pE → ⃗wE(pE)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence  fE = αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE :  �  UE → R  pE → fE(pE) = (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE)(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE)  is a scalar valued function (value of ⃗wE given by αE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) gives (push-forward fE = αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE by Ψ)  Ψ∗(αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE)(pF) = (αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE)(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE)  when  pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  With ⃗wE∗(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) (push-forward of ⃗wE), we get  Ψ∗(αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE)(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)−1  �  ��  �  =noted αE∗(pF)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF)  when  pF = Ψ(pE) :  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The push-forward of a differential form αE ∈ Ω1(UE) is the differential form ∈ Ω1(UF)  given by  Ψ∗αE :  �  �  �  UF → F ∗ = L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  pF → Ψ∗αE(pF) := αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)−1  noted  =  αE∗(pF)  when  pF = Ψ(pE),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  the last notation when Ψ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other words, Ψ∗αE(pF) = αE(Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Ψ∗αE := (αE ◦ Ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  (Once again, we used the same notation Ψ∗ than for the push-forward of vector fields and functions: The  context removes any ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 We cannot always see a vector field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' we can’t see an internal force field): To know it we  need to measure it with a well defined tool, the tool being here a differential form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  is a compatbility definition so that we can recover the push-forward of the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 The pull-forward of a a differential form αF ∈ Ω1(UF) is the differential form  Ψ∗αF :  � UE → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  pE → Ψ∗αF(pE) := αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE) noted  =  αF  ∗(pE)  when  pF = Ψ(pE),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  In other words,  Ψ∗αF := (αF ◦ Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  (For an alternative definition, see remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 For all αE ∈ Ω1(UE) and αF ∈ Ω1(UF) (differential forms), and ⃗wE ∈ Γ(UE) and  ⃗wF ∈ Γ(UF) (vector fields), we have (objectivity result)  (Ψ∗αE)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗ ⃗wF)(pE)  when  pF = Ψ(pE),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' αE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF ∗(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular with αE = df (exact differential form) where  f ∈ C1(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R),  d(Ψ∗f) = Ψ∗(df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  (This commutativity result is very particular to the case α = df: In general d(Ψ∗T) ̸= Ψ∗(dT) for a  tensor of order ≥ 2, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  46  47  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Incompatibility: Riesz representation and push-forward  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' αE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) = (αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF)) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w∗  F(pE),  for all pF = Ψ(pE) ∈ UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And Ψ∗f(pF) := f(pE) = f(Ψ−1(pF)), thus d(Ψ∗f)(pF) = df(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF) = Ψ∗(df)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And we have  Ψ∗ ◦ Ψ∗ = I  and  Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Indeed Ψ∗(Ψ∗αE)(pE) = Ψ∗αE(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The pull-back αF ∗ can also be defined thanks to the natural canonical isomorphism  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗)  L → L∗  �  given by L∗(ℓF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE = ℓF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE) for all (⃗uE, ℓF ) ∈ E×F ∗, and L∗(ℓF ) = ℓF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L  is called the pull-back of ℓF by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular with ℓF  = αF(pF) and L = dΨ(pE) we get  dΨ(pE)∗(αF(pF)) = αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Incompatibility: Riesz representation and push-forward  A push-forward is independent of any inner dot product: It is objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  But here we introduce inner dot products (·, ·)g in E and (·, ·)h in F, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Euclidean dot products  in ⃗Rn  t0 and ⃗Rn  t (observer dependent therefore subjective), because some mechanical engineers can’t begin  with their beloved Euclidean dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let αE ∈ Ω1(UE) and call βF := Ψ∗αE its push-forward by Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  βF(pF) := αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)−1  when  pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Then call ⃗ag(pE) ∈ E and ⃗bh(pF) ∈ F the (·, ·)g and (·, ·)h-Riesz representation vectors of αE and βF, so,  for all ⃗uE ∈ Γ(UE) and all ⃗wF ∈ Γ(UF), in short,  αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE = (⃗ag, ⃗uE)g,  and  βF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF = (⃗bh, ⃗wF)h,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  which means αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE(pE) = (⃗ag(pE), ⃗uE(pE))g and βF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) = (⃗bh(pF), ⃗wF(pF))h, for all pE ∈ UE  and pF ∈ UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines the vector fields ⃗ag ∈ Γ(UE) and ⃗bh ∈ Γ(UF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 ⃗bh ̸= Ψ∗⃗ag in general (although βF = Ψ∗αE), because  ⃗bh(pF) = dΨ(pE)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ag(pE)  ̸= dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ag(pE) in general  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  (unless dΨ(pE)−T = dΨ(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)−1 = I, as a rigid body motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So the Riesz representation vector of the push-forwarded linear form is not the push-forwarded rep-  resentation vector of the linear form push-forwarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  This is not a surprise: A push-forward is independent of any inner dot product, while a Riesz repre-  sentation vector depends on a chosen inner dot product (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Euclidean foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, as long as possible (not before you need to quantify), you should avoid using a Riesz representation  vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' you should use the original (the qualitative differential form) as long as possible, and delay  the use of a representative (quantification with which dot product?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') as late as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Recall: The transposed relative to (·, ·)g and (·, ·)h of the linear map dΨ(pE) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the  linear map dΨ(pE)T  gh =noted dΨ(pE)T ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) defined by, for all ⃗uE ∈ E and ⃗wF ∈ F vectors at pE  and pF, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68),  (dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF, ⃗uE)g = (⃗wF, dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11) gives, with pF = Ψ(pE),  (⃗ag(pE), ⃗uE)g = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE =  �  βF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE = βF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE  �  = (⃗bh(pF), dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uE)h = (dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bh(pF), ⃗uE)g,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  true for all ⃗uE, thus ⃗ag(pE) = dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bh(pF), thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  47  48  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Push-forward and pull-back of order 1 tensors  8  Push-forward and pull-back of tensors  To lighten the presentation, we only deal with order 1 and 2 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Similar approach for any tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Push-forward and pull-back of order 1 tensors  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If T is either a vector field or a differential form, then its push-forward satisfies, for  all ξ vector field or differential form (when required) in UF,  in short:  (Ψ∗T)(ξ) = T(Ψ∗ξ),  written  Ψ∗T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = T(Ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗T)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ξ(pF) = T(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ψ∗ξ(pE) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Similarly:  in short:  (Ψ∗T)(ξ) = T(Ψ∗ξ),  written  Ψ∗T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = T(Ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗T)(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ξ(pE) = T(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ψ∗ξ(pF) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' • Case T = αE ∈ Ω1(UE) (differential form = a  �0  1  �  tensor), then here ξ = ⃗wF ∈ Γ(UF)  and we have to check:  (Ψ∗αE)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ψ∗ ⃗wF(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF) =  αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (dΨ−1(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wF(pF)): True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Case T = ⃗wE ∈ Γ(UE) (vector field ≃ a  �1  0  �  tensor), then here ξ = αF ∈ Ω1(UF) we have to check:  (Ψ∗ ⃗wE)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='αF(pF) = ⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ψ∗(αF)(pE), where we implicitly use to the natural canonical isomorphism  J :  � E → E∗∗  ⃗w → w noted  =  ⃗w  �  defined by w(ℓ) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w for all ℓ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So we have to check: αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗ ⃗wE)(pF) =  Ψ∗(αF)(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE(pE)) = (αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wE)(pE) : True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2), use Ψ−1 instead of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Push-forward and pull-back of order 2 tensors  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Let T be an order 2 tensor in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its push-forward by Ψ is the order 2 tensor Ψ∗T in UF  defined by, for all ξ1, ξ2 vector field or differential form (when required) in UF,  in short:  Ψ∗T(ξ1, ξ2) := T(Ψ∗ξ1, Ψ∗ξ2)  written  Ψ∗T(·, ·) := T(Ψ∗·, Ψ∗·),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ψ∗T(pF)(ξ1(pF), ξ2(pF)) := T(pE)(Ψ∗ξ1(pE), Ψ∗ξ2(pE)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let T be an order 2 tensor in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its pull-back by Ψ is the order 2 tensor Ψ∗T in UE defined by, for  all ξ1, ξ2 vector field or differential form (when required) in UE,  in short:  Ψ∗T(ξ1, ξ2) := T(Ψ∗ξ1, Ψ∗ξ2)  written  Ψ∗T(·, ·) := T(Ψ∗·, Ψ∗·),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Ψ∗T(pE)(ξ1(pE), ξ2(pE)) := T(pF)(Ψ∗ξ1(pF), Ψ∗ξ2(pF)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 If T ∈ T 0  2 (UE) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a metric) then, for all vector fields ⃗w1, ⃗w2 in UF,  T∗(⃗w1, ⃗w2)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  = T(⃗w1  ∗, ⃗w2  ∗) = T(dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', T∗(pF)(⃗w1(pF), ��w2(pF)) = T(pE)(dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1(pF), dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2(pF)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Expression with bases (⃗ai) in E and (⃗bi) in F: In short we have (T∗)ij = T∗(⃗bi,⃗bj) = T(⃗bi∗,⃗bj∗) =  [⃗b∗  i ]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗b∗  j]|⃗a = ([⃗bi]T  |⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ]−T  |⃗a,⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([dΨ]−1  |⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗bj]|⃗b) = ([dΨ]−T  |⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ]−1  |⃗a,⃗b)ij, thus  [T∗]|⃗b = [dΨ]−T  |⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ]−1  |⃗a,⃗b,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  which means [(Ψ∗T)(pF)]|⃗b = ([dΨ(pE)]|⃗a,⃗b)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [T(pE)]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([dΨ(pE)]|⃗a,⃗b)−1 when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Particular case of an elementary tensor T = α1 ⊗ α2 ∈ T 0  2 (UE), where α1, α2 ∈ Ω1(UE), so T(⃗u1, ⃗u2) =  (α1 ⊗ α2)(⃗u1, ⃗u2) = (α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1)(α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2): For all ⃗w1, ⃗w2 ∈ Γ(UF),  (α1 ⊗ α2)∗(⃗w1, ⃗w2)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  = (α1 ⊗ α2)(⃗w∗  1, ⃗w∗  2) = (α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w∗  1)(α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w∗  2)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  = (α1∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1)(α2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  thus  (α1 ⊗ α2)∗ = α1∗ ⊗ α2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  (And any tensor is a finite sum of elementary tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  48  49  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Push-forward and pull-back of endomorphisms  And for the pull-back: For all vector fields ⃗u1, ⃗u2 in UE,  T ∗(⃗u1, ⃗u2)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  = T(⃗u1∗, ⃗u2∗) = T(dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1, dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 If T ∈ T 1  1 (UE) then for all vector fields ⃗w ∈ Γ(UF) and differential forms β ∈ Ω1(UF),  T∗(β, ⃗w) = T(β∗, ⃗w∗) = T(β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', T∗(pF)(β(pF), ⃗w(pF)) = T(pE)(β(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE), dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(pF)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For the elementary tensor T = ⃗u ⊗ α ∈ T 1  1 (UE), made of the vector field ⃗u ∈ Γ(UE) and of the  differential form α ∈ Ω1(UE): For all β, ⃗w ∈ Ω1(UF) × Γ(UF), in short,  (⃗u ⊗ α)∗(β, ⃗w)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  = (⃗u ⊗ α)(β∗, ⃗w∗) = (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='β∗)(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w∗)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  = (⃗u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='β)(α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = (⃗u∗ ⊗ α∗)(β, ⃗w),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  thus  (⃗u ⊗ α)∗ = ⃗u∗ ⊗ α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Expression with bases (⃗ai) in E and (⃗bi) in F:  In short we have (T∗)ij  =  T∗(bi,⃗bj)  =  T(Ψ∗(bi), Ψ∗(⃗bj)) = [Ψ∗(bi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ψ∗(⃗bj)] = [bi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[dΨ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[dΨ−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗bj] = ([dΨ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ−1])ij, thus  [T∗]|⃗b = [dΨ]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ]−1  |⃗a,⃗b,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  which means [(Ψ∗T)(pF)]|⃗b = [dΨ(pE)]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T(pE)]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ(pE)]−1  |⃗a,⃗b when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Push-forward and pull-back of endomorphisms  We have the natural canonical isomorphism  J2 :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) → L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  L → TL = J2(L)  where  TL(α, ⃗u) := α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  ∀(α, ⃗u) ∈ E∗ × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  Thus Ψ∗TL(m, ⃗w) = TL(Ψ∗m, Ψ∗ ⃗w) = (Ψ∗m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Ψ∗ ⃗w) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, thus:  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The push-forward by Ψ of a field of endomorphisms L on UE is the field of endomorphisms  Ψ∗L = L∗ on UF defined by  in short:  Ψ∗L = L∗ = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1 ,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L∗(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus with bases we get [L∗]|⃗b = [dΨ]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dΨ]−1  |⃗a,⃗b, “as in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 Elementary field of endomorphisms L = (J2)−1(⃗u ⊗ α), where ⃗u ∈ Γ(E) and α ∈ Ω1(E):  So TL = ⃗u ⊗ α and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2 = (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2)⃗u for all ⃗u2 ∈ Γ(UE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2 = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2 = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2∗ =  (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2∗)dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)⃗u∗ for all ⃗w2 ∈ Γ(E), thus (TL)∗ = ⃗u∗ ⊗ α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 Let L be a field of endomorphisms on UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its pull-back by Ψ is the field of endomorphisms  Ψ∗L = L∗ on UE defined by  in short:  Ψ∗L = L∗ = dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ ,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L∗(pE) = dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Application to derivatives of vector fields  ⃗u ∈ Γ(UE) is a C1 vector field in UE), pE ∈ UE, so d⃗u : UE → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) (given by d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(pE) =  limh→0  ⃗u(pE+h⃗w(pE))−⃗u(pE)  h  for all ⃗w ∈ Γ(UE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus its push-forward:  ((d⃗u)∗ =)  Ψ∗(d⃗u) = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗u)∗(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(pE)−1 when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  49  50  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity  Here Ψ is C2, ⃗u ∈ Γ(UE), pE ∈ UE, pF = Ψ(pE), so Ψ∗⃗u(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(pE) = (dΨ(Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u(Ψ−1(pF)),  and, for all ⃗w ∈ Γ(UF),  d(Ψ∗⃗u)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(pF) = (d2Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(pF))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(pE) + dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(pF),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  with Ψ∗(d⃗u)(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF), thus, in short,  d(Ψ∗⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = Ψ∗(d⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d2Ψ(Ψ∗ ⃗w, ⃗u) ̸= Ψ∗(d⃗u)  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  So the differentiation d and the push-forward ∗ do not commute (d(Ψ∗⃗u) = Ψ∗(d⃗u) iff Ψ is affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Application to derivative of differential forms  Let α ∈ Ω1(UE) (a differential form on UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its derivative dα : UE → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) is given by dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(pE) =  limh→0  α(pE+h⃗u(pE))−α(pE)  h  ∈ E∗, for all ⃗u ∈ Γ(UE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all ⃗u1, ⃗u2 ∈ Γ(UE),  (dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2(pE) = lim  h→0  α(pE + h⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2(pE) − (α(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2(pE)  h  ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  With the natural canonical isomorphism L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) with E∗∗ ≃ E, we can write  dα(pE)(⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2(pE) = dα(pE)(⃗u1(pE), ⃗u2(pE)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dα(⃗u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2 = dα(⃗u1, ⃗u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Thus the push-forward Ψ∗(dα) =noted (dα)∗ of dα, is given by, for all ⃗w1, ⃗w2 ∈ Γ(UF), in short,  (dα)∗(⃗w1, ⃗w2) = dα(⃗w∗  1, ⃗w∗  2),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', with pF = Ψ(pE), (dα)∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2(pF) = (dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular, (d2f)∗(⃗w1, ⃗w2) = d2f(dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2) (= d2f(⃗w∗  1, ⃗w∗  2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Ψ∗(dα) versus d(Ψ∗α): No commutativity  Here Ψ is C2, ⃗u ∈ Γ(UE), pE ∈ UE and pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We have Ψ∗α(pF) = α(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF) =  α(Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF), thus, for all ⃗w1 ∈ Γ(UF),  d(ψ∗α)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1(pF) = (dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ−1(pF) + α(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d2Ψ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1(pF) ∈ F ∗,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  thus, for all ⃗w1, ⃗w2 ∈ Γ(UF), in short  d(ψ∗α)(⃗w1, ⃗w2) = dα(dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2) + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d2Ψ−1(⃗w1, ⃗w2) ̸= dα(⃗w∗  1, ⃗w∗  2)  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  So the differentiation d and the push-forward ∗ do not commute (d(Ψ∗α) = Ψ∗(dα) iff Ψ is affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  50  51  Part III  Lie derivative  9  Lie derivative  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='0  Purpose and first results  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Purpose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cauchy’s approach may be insufficient, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' :  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' - Cauchy’s approach aims to compare two vectors deformed by a motion, thanks to a Euclidean dot  product and the deformation gradient F, with the deformation tensor C defined by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • ⃗W2 :=  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is a quantitative approach (needs a chosen Euclidean dot product: foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cauchy’s approach is a first order method (dedicated to linear material): Only the first order Taylor  expansion of the motion is used: Only dΦ = F is used (the “slope”), not d2Φ = dF (the “curvature”)  or higher derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' - Lie’s approach aims to build qualitative “covariant objective constitutive laws” (some will be discred-  ited afterward, because of invariance or thermodynamical requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lie’s approach “naturally” applies to non-linear materials thanks to second order Lie derivatives which  uses the second order Taylor expansion of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a non planar surface S, you need the Lie derivative if you want to derive along a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a Galilean Euclidean framework (quantification), the first order Lie derivatives approach give the  same results than Cauchy’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Cauchy died in 1857, and Lie was born in 1842: Unfortunately Cauchy could not use the Lie derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Basic results  The Eulerian velocity of the motion is ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With the material derivative is DEul  Dt := ∂Eul  ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lie derivative L⃗vf of a Eulerian scalar valued function f is the material derivative  L⃗vf = Df  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lie derivative L⃗v ⃗w of a (Eulerian) vector field ⃗w is more than just the material derivative D ⃗w  Dt :  L⃗v ⃗w = D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  L⃗v ⃗w gives the rate of stress on ⃗w due to a flow, and in particular the −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w term in L⃗v ⃗w tells that  the spatial variations of ⃗v (variations of the flow) act on the evolution of the stress (anticipated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)-(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) enable to define the Lie derivatives of tensors of any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Issue (ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �Φ is supposed to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗v(t, p(t)) = ∂�Φ  ∂t (t, PObj) is the Eulerian velocity at t at p(t) = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Recall: If Eul is a Eulerian function then its material time derivative is  DEul  Dt (t, p(t)) = lim  h→0  Eul(t+h, p(t+h)) − Eul(t, p(t))  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  Issue: The rate Eul(t+h,p(t+h))−Eul(t,p(t))  h  raises questions:  1- The difference Eul(t+h, p(t+h)) − Eul(t, p(t)) requires the time and space ubiquity gift to be cal-  culated by an observer, since it mixes two distinct times, t and t+h, and two distinct locations, p(t)  and p(t+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- The difference Eul(t+h, p(t+h)) − Eul(t, p(t)) can be impossible: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if Eul = ⃗w is a vector field  in a “non planar surface considered on its own” (manifold) then Eul(t+h, p(t+h)) and Eul(t, p(t)) don’t  belong to the same (tangent) vector space (so the difference ⃗w(t+h, p(t+h)) − ⃗w(t, p(t)) is meaningless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  51  52  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Toward a solution (without ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  To compare Eul(t+h, p(t+h)) and Eul(t, p(t)) (to get the evolution of Eul along a trajectory), you need  the duration h to get from t to t+h and to move from p(t) to p(t+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, you must:  take the value Eul(t, pt)) with you (for memory),  move along the considered trajectory, and doing so, the value Eul(t, pt) has possibly changed to,  with τ = t+h,  ((Φt  τ)∗Eult)(pτ) noted  =  Eult∗(τ, pτ)  (push-forward);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  And now, at (τ, pτ) where you are, you can compare the actual value Eul(τ, pτ) with the value  Eult∗(τ, pτ) you arrived with (the transported memory), thus the difference  Eul(τ, pτ) − Eult∗(τ, pτ)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  is meaningful for a human being since it is computed at a unique time τ and at a unique point pτ (no  gift of ubiquity required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1: To compute (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) with Eul = ⃗w a (Eulerian) vector field: At t define the vector field ⃗wt in Ωt  by ⃗wt(pt) := ⃗w(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The (spatial) curve ct : s → pt = ct(s) in Ωt is an integral curve of ⃗wt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' satisfies  ct′(s) = ⃗wt(ct(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ct is transformed by Φt  τ into the (spatial) curve cτ = Φt  τ ◦ct : s → pτ = cτ(s)=Φt  τ(ct(s))  in Ωτ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence cτ ′(s) = dΦt  τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='c′(s) = dΦt  τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt(pt) =noted ⃗wt∗(τ, pτ) is the tangent vector at cτ at pτ  (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the difference ⃗w(τ, pτ)− ⃗wt∗(τ, pτ) can be computed by a human being, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' without  ubiquity gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lie derivative, first definition  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The Lie derivative L⃗vEul along ⃗v of an Eulerian function Eul is the Eulerian function  L⃗vEul defined by, at t at pt = �Φ(t, PObj),  L⃗vEul(t, pt) := lim  h→0  Eul(t+h, p(t+h)) − (Φt  t+h)∗Eult(p(t+h))  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Interpretation: L⃗vEul measures the rate of change of Eul along a trajectory:  Eul(t+h, p(t+h)) is the value of Eul at t+h at p(t+h), see figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Eult∗(t+h, p(t+h)) = ((Φt  t+h)∗Eult)(t+h, p(t+h)) is exclusively strain related: It is the memory  transported along a flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the value Eul(t, pt) distorted by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, with g defined by g(τ) = ((Φt  τ)∗Eult)(pτ) (in particular g(t) = Eult(pt)):  L⃗vEul(t, pt) := g′(t) = lim  τ→t  g(τ) − g(t)  τ − t  also written  =  d((Φt  t+h)∗Eult)(p(t+h))  dt  |τ=t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  A more general definition  The rate in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) has to be slightly modified to be adequate in all situations:  Eul(t+h, p(t+h)) −  Eul∗(t+h, p(t+h)) is computed at (t+h, p(t+h)) which moves as h → 0, and on a “non-planar mani-  fold” this is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The “natural” definition is to arrive with the memory:  52  ex(E, Pe)  2  2  pz= 中(p)= Ex (b)  Pt=C (p)53  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lie derivative of a scalar function  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The Lie derivative L⃗vEul along ⃗v of an Eulerian function Eul is the Eulerian function  L⃗vEul defined by, at t at pt = �ΦPObj (t),  L⃗vEul(t, pt) := lim  h→0  Eul(t, pt) − (Φt−h  t  )∗Eult−h(pt)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with �g defined by �g(τ) = ((Φτ  t )∗Eulτ)(pt) (in particular �g(t) = Eul(t, pt)):  L⃗vEul(t, pt) := �g′(t) = lim  τ→t  �g(t) − �g(τ)  t − τ  = lim  τ→t  �g(τ) − �g(t)  τ − t  also written  =  d((Φτ  t )∗Eulτ)(pt)  dτ  |τ=t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Here the observer must:  At t−h at p(t−h) = �ΦPObj (t−h), take the value Eul(t−h, p(t−h)),  travel along the trajectory �ΦPObj ,  once at t at pt = �ΦPObj (t), this value has become ((Φt  t−h)∗Eult−h)(pt) (transported memory),  and then the comparison with Eul(t, pt) can be done in Ωt (no ubiquity gift required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Prove: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With (Φt  t+h)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Φt  t+h)∗  =  I,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) gives L⃗vEul(t, pt)  =  limh→0  (Φt  t+h)∗Eul(t,pt)−Eult(t,pt)  h  =  limh→0  (Φt  t−h)∗Eult−h)(pt)−Eult(pt)  −h  = limh→0  Eult(pt)−((Φt  t−h)∗Eult−h)(pt)  h  , and (Φt  t−h)∗ = (Φt−h  t  )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Equivalent definition (differential geometry)  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 The Lie derivative of a Eulerian function Eul along a flow of Eulerian velocity ⃗v is the  Eulerian function L⃗vEul defined at (t, pt) by  L⃗vEul(t, pt) := lim  h→0  ((Φt  t+h)∗Eult+h)(pt) − Eul(t, pt)  h  ,  rate in Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  In other words, if ˆg is defined by ˆg(τ) = ((Φt  τ)∗Eulτ)(pt) (in particular ˆg(t) = Eul(t, pt)), then  L⃗vEul(t, pt) := ˆg′(t) = lim  τ→t  ˆg(τ) − ˆg(t)  τ − t  also written  =  d((Φt  τ)∗Eulτ)(pt)  dτ  |τ=t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Prove: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) also reads L⃗vEul(t, pt) = limh→0  ((Φt  t−h)∗Eult−h)(pt)−Eult(pt)  −h  , and (Φt  t−h)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Φt−h  t  )∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 More precise definition, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3): E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10), the Lie derivative �L⃗vEul of a Eulerian  function �  Eul along a flow of Eulerian velocity ⃗v is the Eulerian function defined by, at t at pt = �Φ(t, PObj),  �L⃗vEul(t, pt) := ((t, pt), L⃗vEul(t, pt)  (pointed function at (t, pt)),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  And, to lighten the notation, �L⃗vEul(t, pt) =noted L⃗vEul(t, pt) (second component of �L⃗vEul(t, pt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Lie derivative of a scalar function  Let f be a C1 Eulerian scalar valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With (Φt−h  t  )∗ft−h(pt) = ft−h(p(t−h)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10), we get  L⃗vf(t, pt)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  =  lim  h→0  f(t, pt) − f(t−h, p(t−h))  h  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L⃗vf = Df  Dt  = ∂f  ∂t + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  So, for scalar functions, the Lie derivative is the material derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation: L⃗vf measures the rate of change of f along a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 L⃗vf = 0 iff f is constant along any trajectory (the real value is the memory value):  L⃗vf = 0  ⇐⇒  ∀t, τ ∈ [t0, T], (Φt  τ ∗)ft(pτ) = f(t, p(t)) when pτ = Φt  τ(pt),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' iff f(t, p(t)) = f(t0, pt0) when p(t) = Φt0(t, pt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' iff f let itself be carried by the flow (unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  53  54  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lie derivative of a vector field  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let p(t) = �Φ(t, PObj) = pt for all t, so p(τ) = �Φ(τ, PObj) = pτ = Φt  t+h(pt) = Φt(τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⇐: If fτ = (Φt  t+h)∗ft, then fτ(pτ) = ft(pt), thus limτ→t  f(τ,p(τ))−f(t,p(t))  τ−t  = 0, that is, Df  Dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⇒: If Df  Dt = 0 then f(t, p(t)) is a constant function on the trajectory t → �Φ(t, PObj), for any parti-  cle PObj, so f(τ, p(τ)) = f(t, pt) when p(τ) = Φt  t+h(pt), that is, f(τ, pτ) = (Φt  t+h)∗ft(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Prove: L⃗v(L⃗vf) = D2f  Dt2 = ∂2f  ∂t2 + 2d( ∂f  ∂t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + d2f(⃗v,⃗v) + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ( ∂⃗v  ∂t + d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Lie derivative of a vector field  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Formula  Let ⃗w be a C1 (Eulerian) vector field (interpreted as an “internal force field” in the following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  L⃗v ⃗w = D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ∂ ⃗w  ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  So the Lie derivative is not reduced to the material derivative D ⃗w  Dt (unless d⃗v = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' unless ⃗v is uniform):  The spatial variations d⃗v of ⃗v influences the rate of stress: ⃗v tries to bend ⃗w (which is expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗g : τ → ⃗g(τ) = (Φt∗  τ ⃗w)(t, p(t)) = dΦt  τ(pt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(τ, p(τ)) when p(τ) = Φt(τ, pt), so (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) reads  L⃗v ⃗w(t, pt) = ⃗g ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With ⃗z(τ) := ⃗w(τ, p(τ)) = dΦt(τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g(τ),  ⃗z ′(τ) = D ⃗w  Dτ (τ, p(τ)) = ∂(dΦt)  ∂τ  (τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g(τ) + dΦt(τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′(τ)  = (d⃗v(τ, p(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t(τ, pt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t(τ, pt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(τ, p(τ))) + F t  τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′(τ)  = d⃗v(τ, p(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(τ, p(τ)) + F t  τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  Thus D ⃗w  Dt (t, pt) = d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, pt) + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′(t), thus ⃗g ′(t) = D ⃗w  Dt (t, pt) − d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: Basis (⃗ei), ⃗v = �  i vi⃗ei, ⃗w = �  i wi⃗ei, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �  ij vi|j⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �  ij wi|j⃗ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  L⃗v ⃗w =  n  �  i=1  ∂wi  ∂t ⃗ei +  n  �  i,j=1  wi|jvj⃗ei −  n  �  i,j=1  vi|jwj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  So (column matrix), with [·] := [·]|⃗e,  [L⃗v ⃗w] = [D ⃗w  Dt ] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]  (= [∂ ⃗w  ∂t ] + [d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  (And [d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v] = [d⃗w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Duality notations: L⃗v ⃗w = �  i  ∂wi  ∂t ⃗ei + �  ij wi  |jvj⃗ei − �  ij vi  |jwj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Interpretation: Flow resistance measurement  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Φt0 is supposed to be a C2 motion and a C1 diffeomorphism in space, and ⃗w is a  vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L⃗v ⃗w = 0  ⇐⇒  ∀t ∈ [t0, T],  ⃗wt = (Φt0  t )∗ ⃗wt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', D ⃗w  Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w ⇔ the actual vector ⃗w(t, p(t)) is equal to F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0(pt0) = ⃗wt0∗(t, p(t)) the deformed  vector by the flow, see figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So: The Lie derivative L⃗v ⃗w vanishes iff ⃗w does not resist the flow (let  itself be deformed by the flow), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' iff ⃗w(t, pt) = ⃗wt0∗(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have L⃗v ⃗w = D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w and ∂F t0  ∂t (t, pt0) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⇐ (derivation): Suppose ⃗w(t, p(t)) = F t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t0, pt0) when p(t) = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then D ⃗w  Dt (t, p(t)) =  ∂F t0  ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t0, pt0) = (d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t0  t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t))) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)), thus  D ⃗w  Dt −  d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (See proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  ⇒ (integration):  Suppose  D ⃗w  Dt  =  d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let  ⃗f(t)  =  (F t0  t (pt0))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)) (= pull-back  (Φt0  t )∗ ⃗w(t0, pt0)) when p(t)  =  Φt0(t, pt0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So  ⃗w(t, p(t))  =  F t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f(t) and  D ⃗w  Dt (t, p(t))  =  ∂F t0  ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f(t) + F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f(t) + F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)) +  F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f ′(t) =hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w  Dt (t, p(t))+F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f ′(t) for all t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗f ′(t) = ⃗0, thus ⃗f ′(t) = ⃗0 (because  Φt0  t is a di���eomorphism), thus ⃗f(t) = ⃗f(t0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗wt = (Φt0  t )∗ ⃗wt0, for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  54  55  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Autonomous Lie derivative and Lie bracket  The Lie bracket of two vector fields ⃗v and ⃗w is  [⃗v, ⃗w] := d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w noted  =  L0  ⃗v ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  And L0  ⃗v ⃗w = [⃗v, ⃗w] is called the autonomous Lie derivative of ⃗w along ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  L⃗v ⃗w = ∂ ⃗w  ∂t + [⃗v, ⃗w] = ∂ ⃗w  ∂t + L0  ⃗v ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  NB: L0  ⃗v ⃗w is used when ⃗v et ⃗w are stationary vector fields, thus does not concern objectivity: A stationary  vector field in a referential is not necessary stationary in another (moving) referential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Examples  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Lie Derivative of a vector field along itself  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15) with ⃗w = ⃗v gives L⃗v⃗v = ∂⃗v  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, if ⃗v is a stationary vector field then L⃗v⃗v = ⃗0 (= [⃗v,⃗v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Lie derivative along a uniform flow  Here d⃗v = 0, thus  L⃗v ⃗w = D ⃗w  Dt = ∂ ⃗w  ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v  (when d⃗v = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Here the flow is rectilinear (d⃗v = 0): there is no curvature (of the flow) to influence the stress on ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Moreover, if ⃗w is stationary, that is ∂ ⃗w  ∂t = 0, then L⃗v ⃗w = d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = the directional derivative ∂ ⃗w  ∂⃗v of the  vector field ⃗w in the direction ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Lie derivative of a uniform vector field  Here d⃗w(t, p) = 0, thus  L⃗v ⃗w = ∂ ⃗w  ∂t − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  (when d⃗w = 0),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  thus the stress on ⃗w is due to the space variations of ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Moreover, is ⃗w is stationary then L⃗v ⃗w = −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Uniaxial stretch of an elastic material  Strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With [−−→  OP]|⃗e = [ ⃗X]|⃗e =  �  X  Y  �  , with ξ > 0, t ≥ t0, p(t) = Φt0(t, P) and [⃗x]|⃗e = [−−−→  Op(t)]|⃗e:  [⃗x]|⃗e =  �  x  y  �  =  �  X  Y  �  + ξ(t−t0)  �  X  0  �  =  �  X(1 + ξ(t−t0))  Y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Eulerian velocity ⃗v(t, p) =  �  ξX  0  �  =  �  ξ  1+ξ(t−t0)x  0  �  , d⃗v(t, p) =  �  ξ  1+ξ(t−t0)  0  0  0  �  (independent of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Deformation gradient (independent of P), with κt = ξ(t−t0):  Ft = dΦt0  t (P) =  �  1 + κt  0  0  1  �  = I + κt  �  1  0  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  Infinitesimal strain tensor, with F T  t = Ft here:  εt0  t (P) = Ft − I = κt  �  1  0  0  0  �  = εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Constitutive law = Linear isotropic elasticity:  σt(pt) = λTr(εt)I + 2µεt = κt  �  λ+2µ  0  0  λ  �  = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Cauchy stress vector ⃗T on a surface at p with normal ⃗nt(p) =  �  n1  n2  �  = ⃗n:  ⃗Tt(pt) = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n = κt  �  (λ+2µ)n1  λn2  �  = ξ(t−t0)  �  (λ+2µ)n1  λn2  �  = ⃗Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  Push-forwards: ⃗Tt0(pt0) = 0, thus F t0  t0+h(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Tt0(pt0) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  55  56  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  Lie derivative:  L⃗v ⃗T(t0, pt0) = lim  t→t0  ⃗Tt(pt) − F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Tt0(pt0)  t − t0  = ξ  �  (λ+2µ)n1  λn2  �  (rate of stress at (t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  Generic computation with L⃗v ⃗T  =  ∂ ⃗T  ∂t + d⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗T:  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28) gives  ∂ ⃗T  ∂t  = ξ  �  (λ+2µ) n1  λ n2  �  and  d⃗T = 0 and d⃗vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Tt =  �  ξ  1+ξ(t−t0)  0  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ξ(t−t0)  �  (λ+2µ) n1  λ n2  �  =  ξ2(t−t0)  1+ξ(t−t0)  �  (λ+2µ) n1  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particu-  lar, d⃗v(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗T(t0, pt0) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus L⃗v ⃗T(t0, pt0) = ξ  �  (λ+2µ) n1  λ n2  �  = rate of stress at the initial (t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Simple shear of an elastic material  Euclidean basis (⃗e1,⃗e2) in R2, the same basis at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Initial configuration Ωt0 = [0, L1] ⊗ [0, L2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Initial position [−−→  OP]⃗e = [−−→  Opt0]⃗e = [ ⃗X]⃗e =  �  X  Y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ξ ∈ R∗, pt = Φt0  t (pt0), [⃗x]|⃗e = [−−−→  Op(t)]|⃗e, and  [⃗x]⃗e =  �  x = ϕ1(t, X, Y ) = X  y = ϕ2(t, X, Y )  �  =  �  X + ξ(t−t0)Y  Y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  Eulerian velocity ⃗vt(pt) =  �  ξY  0  �  =  �  ξy  0  �  , thus d⃗vt(pt) =  �  0  ξ  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With κt = ξ(t−t0), deformation gradient (independent of P):  dΦt0  t (P) =  �  1  κt  0  1  �  = F t0  t ,  thus  F t0  t  − I = κt  �  0  1  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  Infinitesimal strain tensor:  εt0  t (P) = F t0  t (P)−I + (F t0  t (P)−I)T  2  = κt  2  �  0  1  1  0  �  = εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  Stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Constitutive law, usual linear isotropic elasticity (requires a Euclidean dot product):  σ(t, pt) = λTr(εt)I + 2µεt = µκt  �  0  1  1  0  �  = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Cauchy stress vector ⃗T(t, pt) (at t at pt) on a surface at p with normal ⃗nt(p) =  �  n1  n2  �  = ⃗n:  ⃗Tt = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n = µκt  �  n2  n1  �  = µξ(t−t0)  �  n2  n1  �  = ⃗T(t)  (stress independent of pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  Lie derivative, with ⃗Tt0 = ⃗0:  L⃗v ⃗T(t0, pt0) = lim  t→t0  ⃗Tt(pt) − F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Tt0(pt0)  t − t0  = µξ  �  n2  n1  �  (rate of stress at (t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Generic computation: L⃗v ⃗T = ∂ ⃗T  ∂t + d⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34) gives ∂ ⃗T  ∂t (t, p) = µξ  �  n2  n1  �  and d⃗T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With  d⃗vt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Tt0 = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus L⃗v ⃗T(t0, pt0) = µξ  �  n2  n1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Shear flow  Stationary shear field, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11) with α = 0 and t0 = 0:  ⃗v(x, y) =  �  v1(x, y) = λy,  v2(x, y) = 0,  d⃗v(x, y) =  �  0  λ  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  Let ⃗w(t, p) =  �  0  b  �  = ⃗w(t0, pt0) (constant in time and uniform in space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then L⃗v ⃗w = −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w =  �  −λb  0  �  measures “the resistance to deformation due to the flow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2, the virtual vector ⃗w∗(t, p) =  dΦ(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t0, pt0) being the vector that would have let itself be carried by the flow (the push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  56  57  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lie derivative of a differential form  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2: Shear flow, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36), with ⃗w constant and uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L⃗v ⃗w measures the resistance to the  deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Spin  Rotating flow: Continuing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14):  ⃗v(x, y) = ω  �  0  −1  1  0  � �  x  y  �  ,  d⃗v(x, y) = ω  �  0  −1  1  0  �  = ω Rot(π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  In particular d2⃗v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With ⃗w = ⃗w0 constant and uniform we get  L⃗v ⃗w0 = −d⃗v(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w0 = −ω Rot(π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w0  (⊥  �  a  b  �  = ⃗w0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  gives “the force at which ⃗w refuses to turn with the flow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Second order Lie derivative  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Let ⃗v, ⃗w be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  L⃗v(L⃗v ⃗w) = D2 ⃗w  Dt2 − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt − D(d⃗v)  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w,  = ∂2 ⃗w  ∂t2 + 2d∂ ⃗w  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂ ⃗w  ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t − d∂⃗v  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  + (d2 ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − (d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L⃗v(L⃗v ⃗w) = D(L⃗v ⃗w)  Dt  − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L⃗v ⃗w) = D( D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  Dt  − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  = D2 ⃗w  Dt2 − D(d⃗v)  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w,  thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)1, thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Lie derivative of a differential form  When the Lie derivative of a vector field ⃗w cannot be obtained by direct measurements, you need to use  a “measuring device” (Germain: To know the weight of a suitcase you have to lift it: You use work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here we consider a measuring device which is a differential form α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, if ⃗w is a vector field then  f = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v is a scalar function, and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) gives L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = D(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)  Dt  = Dα  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w  Dt , thus  L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = Dα  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  �  ��  �  →(L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  �  ��  �  =α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗v ⃗w  :  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Let α be a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lie derivative of α along ⃗v is the differential form  L⃗vα := Dα  Dt + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = ∂α  ∂t + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  (An equivalent definition is given at (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all vector field ⃗w,  L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w := Dα  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  (= ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  57  (B ) A  q=c  (t  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='%  w(t)  w(t,/p)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(t,p)  od  T  V(B)  (t)58  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lie derivative of a differential form  The definition of L⃗vα, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41), immediately gives the “derivation property”  L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = (L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L⃗v ⃗w)  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L⃗v is a derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  Quantification: Relative to a basis (⃗ei) and with [·] := [·]|⃗e,  [L⃗vα] = [Dα  Dt ] + [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v]  (row matrix) = [∂α  ∂t ] + [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v] + [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  Thus  [L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w] = [L⃗vα].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] = [∂α  ∂t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] + [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] + [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 Prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And prove [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v] = [⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dα]T (row matrix), thus  [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] = [⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dα]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] = [⃗w]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dα].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Basis (⃗ei), dual basis (πei), thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41) gives [L⃗vα] = [ Dα  Dt ] + [α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let α = �  i αiπei,  ⃗v = �  i vi⃗ei, d⃗v = �  ij vi|j⃗ei ⊗ πej (tensorial writing convenient for calculations), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v]|⃗e = [vi|j], thus  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = �  ij αivi|jπej, thus [α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v]|πe = [α]|πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v]|⃗e (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And dα = �  ij αi|jπei ⊗ πej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dα]|πe = [αi|j],  gives dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = �  ij αi|jvjπei = �  ij viαj|iπej, and [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v]|πe is a row matrix (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v is a differential form), thus  [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v]|πe = [⃗v]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dα]T  |πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Or compute (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = �  ij αi|jvjwi = [⃗w]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[dα]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗v]|⃗e = [⃗v]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [dα]T  |πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗w]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 Let α be a differential form, and let αt(p) := α(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove, when Φt0  t is a diffeomorphism,  L⃗vα = 0  ⇐⇒  ∀t ∈ [t0, T], αt = (Φt0  t )∗αt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' :  Dα  Dt = −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v ⇐⇒ αt(pt) = αt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0)−1 for all t, when pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⇐: If αt(p(t)) = αt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0)−1, then α(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0(t, pt0) = αt0(pt0), thus Dα  Dt (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0) +  αt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂F t0  ∂t (t, pt0) = 0, thus  Dα  Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0) + αt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0) = 0, thus L⃗vα = 0, since Φt0  t  is a  diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⇒: If β(t) := (Φt0  t )∗αt0(pt0) = αt(p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0) (pull-back at (t0, pt0)), then β(t) = α(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0(t, pt0), thus  β′(t) = Dα  Dt (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0) + α(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (pt0) = 0 (hypothesis L⃗vα = 0), thus β(t) = β(t0) = αt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 A definition equivalent to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41) is, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10),  L⃗vα(t, pt) := lim  τ→t  (Φt  τ)∗ατ(pt) − αt(pt)  τ − t  (= lim  τ→t  ατ(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt  τ(pt) − αt(pt)  τ − t  )  noted  =  D(Φt∗  τ ατ(pt))  Dτ  |τ=t  noted  =  D(α∗  τ(pt))  Dτ  |τ=t  (= D(ατ(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt  τ(pt))  Dτ  |τ=t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  Indeed, if β(τ) = (Φt  τ)∗ατ(pt) = ατ(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt  τ(pt), then β′(τ) and then τ = t give (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 ⃗v and α being C2, prove:  L⃗v(L⃗vα) = ∂2α  ∂t2 + 2d∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + 2∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂d⃗v  ∂t  + (d2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + 2(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41) gives  L⃗v(L⃗vα) = L⃗v(∂α  ∂t ) + L⃗v(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)  = ∂2α  ∂t2 + d∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + ∂(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)  ∂t  + d(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + ∂(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)  ∂t  + d(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  = ∂2α  ∂t2 + d∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + ∂dα  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t + (d2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  + ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂d⃗v  ∂t + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  = ∂2α  ∂t2 + 2d∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + 2∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗v  ∂t + (d2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + 2(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂d⃗v  ∂t  + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  58  59  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Incompatibility with Riesz representation vectors  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Incompatibility with Riesz representation vectors  The Lie derivative has nothing to do with any inner dot product (the Lie derivative does not compare  two vectors, contrary to a Cauchy type approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here we introduce a Euclidean dot product (·, ·)g and show that the Lie derivative of a linear form α  is not trivially deduced from the Lie derivative of a Riesz representation vector of α (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Same  issue as at § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let α be a Eulerian differential form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then let ⃗ag(t, p) ∈ ⃗Rn be the (·, ·)g-Riesz representation vector  of the linear form α(t, p) ∈ Rn∗: So, for all Eulerian vector field ⃗w,  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (⃗ag, ⃗w)g  (= ⃗ag •g ⃗w),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  which means α(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p) = (⃗ag(t, p), ⃗w(t, p))g at all admissible (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines the Eulerian vector  field ⃗ag (not intrinsic to α: ⃗ag depends on the choice of (·, ·)g, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 For all ⃗v, ⃗w ∈ ⃗Rn,  ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (∂⃗ag  ∂t , ⃗w)g,  (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (d⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v, ⃗w)g,  Dα  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (D⃗ag  Dt , ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)  Thus  L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (L⃗v⃗ag, ⃗w)g + (⃗ag, (d⃗v+d⃗vT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)g,  and  L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w ̸= (L⃗v⃗ag, ⃗w)g  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51)  So L⃗v⃗ag is not the Riesz representation vector of L⃗vα (but for solid body motions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Expected: A Lie  derivative is covariant objective, see § 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4, and the use of an inner dot product ruins this objectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A Euclidean dot product g(·, ·) is bilinear constant and uniform, thus:  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (⃗ag, ⃗w)g gives ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂ ⃗w  ∂t = ( ∂⃗ag  ∂t , ⃗w)g + (⃗ag, ∂ ⃗w  ∂t )g, with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂ ⃗w  ∂t = (⃗ag, ∂ ⃗w  ∂t )g, thus we are left  with ∂α  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ( ∂⃗ag  ∂t , ⃗w)g, for all ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (⃗ag, ⃗w)g gives d(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = d(⃗ag, ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v for all ⃗v, ⃗w, thus (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) = (d⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v, ⃗w)g +  (⃗ag, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)g, with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) = (⃗ag, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)g, thus we are left with (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (d⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v, ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus Dα  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ( D⃗ag  Dt , ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus (L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = Dα  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ( D⃗ag  Dt , ⃗w)g + (⃗ag, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)g = (L⃗v⃗ag + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ag, ⃗w)g + (d⃗vT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ag, ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 Chorus: a “differential form” (measuring instrument, covariant) should not be confused  with a “vector field” (object to be measured, contravariant);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, the use of a dot product (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  and the Riesz representation theorem should be restricted for computational purposes, after an objective  equation has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See also remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Lie derivative of a tensor  The Lie derivative of any tensor of order ≥ 2 is defined thanks to  L⃗v(T ⊗ S) = (L⃗vT) ⊗ S + T ⊗ (L⃗vS)  (derivation formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52)  (Or direct definition: L⃗vT(t0, pt0) = D((Φt0  t )∗Tt)(pt0)  Dt  |t=t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Lie derivative of a mixed tensor  Let Tm ∈ T 1  1 (Ω), and Tm is called a mixed tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its Lie derivative, called the Jaumann derivative, is  given by  L⃗vTm = DTm  Dt  − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Tm + Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = ∂Tm  ∂t  + dTm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Tm + Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53)  Can be checked with an elementary tensor T = ⃗w ⊗α: we have d(⃗w ⊗α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)⊗α+ ⃗w ⊗(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) and  (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)⊗α = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗w⊗α), and ⃗w⊗(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v) = (⃗w⊗α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v , thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52) gives L⃗v(⃗w⊗α) = (L⃗v ⃗w)⊗α+ ⃗w⊗(L⃗vα)  = ∂ ⃗w  ∂t ⊗ α + (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) ⊗ α − (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) ⊗ α + ⃗w ⊗ ∂α  ∂t + ⃗w ⊗ (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v) + ⃗w ⊗ (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)  = ∂ ⃗w⊗α  ∂t  + d(⃗w ⊗ α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗w ⊗ α) + (⃗w ⊗ α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  59  60  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lie derivative of a tensor  Quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Relative to a basis (⃗ei):  [L⃗vTm] = [DTm  Dt ] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Tm] + [Tm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v]  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  (the signs ∓ are mixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' “Mixed” also refers to positions of indices (up and down with duality notations):  Tm = �n  i,j=1T ij⃗ei ⊗ ej with the dual basis (ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Tm]|⃗e = [T ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19 With components, prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂Tm  ∂t  = �  ij  ∂T ij  ∂t ⃗ei ⊗ ej, dTm = �  ijk T i  j|k⃗ei ⊗ ej ⊗ ek, ⃗v = �  i vi⃗ei, d⃗v = �  ij vi  |j⃗ei ⊗ ej, thus  dTm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = �  ijk T i  j|kvk⃗ei ⊗ ej, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Tm = �  ijk vi  |kT k  j⃗ei ⊗ ej, Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = �  ijk T i  kvk  |j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Lie derivative of a up-tensor  Recall: If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) (a linear map) then its adjoint L∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) is defined by, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12,  ∀m ∈ F ∗,  L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  ∀m, ⃗u ∈ (F ∗ × E),  (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55)  (There is no inner dot product involved here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  In particular, d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v for all m ∈ ⃗Rn∗  t , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) for all m ∈ ⃗Rn∗  t  and all ⃗u ∈ ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let Tu ∈ T 2  0 (Ω), and Tu is called a up tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its Lie derivative is called the upper-convected (Maxwell)  derivative or the Oldroyd derivative and is given by  L⃗vTu = DTu  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Tu − Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v∗ = ��Tu  ∂t + dTu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Tu − Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56)  Can be checked with an elementary tensor T = ⃗u ⊗ ⃗w and L⃗v(⃗u ⊗ ⃗w) = (L⃗v⃗u) ⊗ ⃗w + ⃗u ⊗ (L⃗v ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Relative to a basis (⃗ei):  [L⃗vTu] = [DTu  Dt ] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Tu] − [Tu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='57)  “up” also refers to positions of indices (with duality notations): Tu = �n  i,j=1T ij⃗ei ⊗ ⃗ej with the dual  basis (ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Tu]|⃗e = [T ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 With components, prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂Tu  ∂t = �  ij  ∂T ij  ∂t ⃗ei⊗⃗ej, dTu = �  ijk T ij  |k⃗ei⊗⃗ej⊗ek, ⃗v = �  i vi⃗ei, d⃗v = �  ij vi  |j⃗ei⊗ej, d⃗v∗ = �  ij vj  |iei⊗⃗ej,  thus dTu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = �  ijk T ij  |k vk⃗ei ⊗ ej, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Tu = �  ijk vi  |kT kj⃗ei ⊗ ⃗ej, Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v∗ = �  ijk T ikvj  |kei ⊗ ⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Lie derivative of a down-tensor  Let Td ∈ T 0  2 (Ω), and Td is called a down tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lie derivative is called the lower-convected Maxwell  derivative and is given by  L⃗vTd = DTd  Dt + Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Td = ∂Td  ∂t + dTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58)  Can be checked with an elementary tensor T = ℓ ⊗ m and L⃗v(ℓ ⊗ m) = (L⃗vℓ) ⊗ m + ℓ ⊗ (L⃗vm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Relative to a basis (⃗ei):  [L⃗vTd] = [DTd  Dt ] + [Td].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v] + [d⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Td].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)  “down” also refers to positions of indices (with duality notations): Td = �n  i,j=1Tijei ⊗ ej with the dual  basis (ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Td]|⃗e = [Tij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 With components, prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂Td  ∂t = �  ij  ∂Tij  ∂t ei⊗ej, dTd = �  ijk Tij|kei⊗ej⊗ek, ⃗v = �  i vi⃗ei, d⃗v = �  ij vi  |j⃗ei⊗ej, d⃗v∗ = �  ij vj  |iei⊗⃗ej,  thus dTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = �  ijk Tij|kvkei ⊗ ej, Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = �  ijk Tikvk  |jei ⊗ ⃗ej, d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Td = �  ijk vk  |iTkjei ⊗ ⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 Let g = (·, ·)g ∈ T 0  2 (Ω) be a constant and uniform metric (a unique inner dot product  for all t, p, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a Euclidean dot product at all t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then Dg  Dt = 0, thus L⃗vg = 0 + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g, thus  [L⃗vg] = [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v] + [d⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  60  61  Part IV  Velocity-addition formula  10  Change of referential and velocity-addition formula  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='0  Issue and result (summary)  The velocity-addition formula is (in classical mechanics)  ⃗vA = ⃗vB + ⃗vD,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  where ⃗vA, ⃗vB and ⃗vD are the absolute, relative and drive velocity, ⃗vA and ⃗vD being velocities described by  an observer A with his referential RA = (OA, ( ⃗Ai)) and ⃗vB being a velocity described by an observer B  with his referential RB = (OB, ( ⃗Bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) is problematic (inconsistent):  The velocities ⃗vA and ⃗vD are quantified in RA, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' expressed in foot/s by the absolute observer,  The velocity ⃗vB is a quantified in RB, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' expressed in metre/s by the relative observer,  Thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) with ⃗vB + ⃗vD tells that you add metre/s and foot/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So:  Question: What are we missing (and what does (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) really mean)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: We miss a functional link: The translator between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Summary:  Call �Φ the motion of a observed object Obj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �Φ is quantified by A in his referential RA = (OA, ( ⃗Ai))  as the “motion” ⃗ϕA = [  −−→  OA�Φ]| ⃗A, and is quantified by B in his referential RB = (OB, ( ⃗Bi)) as the “motion”  ⃗ϕB = [  −−→  OB �Φ]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' At t, the translator Θ connects these numerical values: ⃗ϕA(t, PObj) = Θ(t, ⃗ϕB(t, PObj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus ∂ ⃗ϕA  ∂t (t, PObj) = ∂Θ  ∂t (t, ⃗xBt) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂ ⃗ϕB  ∂t (t, PObj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗vA(t, ⃗xAt) = ∂Θ  ∂t (t, ⃗xBt) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vB(t, ⃗xBt)  where ⃗xAt = ⃗ϕA(t, PObj) and ⃗xBt = ⃗ϕB(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then call  dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) = ⃗vBt∗(⃗xAt) = “the translated relative velocity at t from B to A”,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  thus, with ∂Θ  ∂t (t, ⃗xBt) = ⃗vD(t, ⃗xAt) the drive velocity, which gives ⃗vA(t, ⃗xAt) = ⃗vB∗(t, ⃗xAt) + ⃗vD(t, ⃗xAt): so  ⃗vA = ⃗vB∗ + ⃗vD  =  the velocity addition formula in RA,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : (Absolute velocity) = (Translated relative velocity) + (Drive velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In other words, with ⃗v the velocity of Obj and with ⃗vRB the velocity of RB in RA: For all pt = �Φ(t, PObj),  [⃗vt(pt)]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗vt(pt)] ⃗B + [⃗vRBt(pt)]| ⃗A,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  relation between the numerical values of the velocities stored by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 Translation motion of RB in RA, so [⃗vRBt(pt)]| ⃗A = [⃗vRBt]| ⃗A is independent of pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with ( ⃗Bit) = λ( ⃗Ait) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ai in foot and ⃗Bi in meter give λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28), dΘt = λI, hence [⃗vt(pt)]| ⃗A =  λ[⃗vt(pt)] ⃗B + [⃗vRBt]| ⃗A, which is the expected relation (“sum of the velocities with the good units”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Motion of the Earth around the Sun: See § 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Referentials and “matrix motions”  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Absolute and relative referentials  Classical mechanics framework: Time and space are decoupled, all the observers share the same time  unit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the second) and live in “our” Universe modeled as R3 (affine space) with its usual associated  vector space ⃗R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In the following, the affine space is Rn associated to the vector space ⃗Rn, n ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  An observer A, which we will call the absolute observer, chooses a (rigid body) object ObjRA in the  Universe, chooses one particle in ObjRA, calls OAt its position at t, and chooses three more particles  in ObjRA, calls PAti their positions at t (in the Universe), such that the bi-point vectors ⃗Ait := −−−−−→  OAtPAti  make a basis in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' He has thus built his (Cartesian) referential RAt = (OAt, ( ⃗Ait)), called the absolute  referential, and written RA = (OA, ( ⃗Ai)) when used by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ObjRA is the “Sun extended to infinity”,  and at t, OAt is the position of the center of the Sun in the Universe, ( ⃗Ait) is a Euclidean basis in foot  fixed relative to stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  61  62  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Referentials and “matrix motions”  An observer B, which we will call the relative observer, proceeds similarly: He chooses a (rigid body)  object ObjRB in the Universe, builds his Cartesian referential RBt = (OBt, ( ⃗Bit)), called the relative  referential, written RB = (OB, ( ⃗Bi)) when used by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ObjRB is the “Earth extended to infinity”, and  at t, OBt is the position of the center of the Earth and ( ⃗Bit) is a Euclidean basis in metre fixed relative  to the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Mn1 is the vectorial space of n ∗ 1 matrices (column matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A and B call Mn1(A) and Mn1(B) the  affine spaces of n ∗ 1 matrices made of the “matrix positions” [−−−→  OAtpt]| ⃗A and [−−−→  OBtpt]| ⃗B where pt is the  position at t of a particle in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If a function ϕ is given as ϕ(t, x), then ϕt(x) := ϕ(t, x), and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Motion of a material object Obj  An object Obj is considered by all observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its motion in the Universe is  �Φ :  �  [t1, t2] × Obj → Rn  (t, PObj) → pt = �Φ(t, PObj) = position of the particle PObj at t in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  At t at pt = �Φ(t, PObj), the Eulerian velocities and accelerations of PObj are  ⃗v(t, pt) = ∂�Φ  ∂t (t, PObj)  and  ⃗γ(t, pt) = ∂2�Φ  ∂2t (t, PObj)  (∈ ⃗Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Quantification: Absolute and relative “motion” of Obj  At t, the position pt = �Φ(t, PObj) of a particle PObj ∈ Obj is spotted by A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B, with the bi-point  vectors −−−→  OAtpt, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' −−−→  OBtpt in ⃗Rn, which components is stored by A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B, in his referentials: With  −−−→  OAtpt =  n  �  i=1  xAti ⃗Ait  and  −−−→  OBtpt =  n  �  i=1  xBti ⃗Bit,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  and with ( ⃗Ei) the canonical basis in Mn1, the n ∗ 1 matrices  ⃗xAt := [−−−→  OApt]| ⃗A =  �  �  xAt1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xAtn  �  � =  n  �  i=1  xAti ⃗Ei,  and  ⃗xBt := [−−−→  OBpt]| ⃗B =  �  �  xBt1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xBtn  �  � =  n  �  i=1  xBti ⃗Ei,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  are stored by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Initial notation: ⃗xAt := [−−−→  OAtpt]|( ⃗Ait), but here OAt and ( ⃗Ait) are fixed in RA,  idem for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Mind the notations: pt is a point, −−−→  OAtpt is a vector, ⃗xAt is a column matrix (components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  This defines the “absolute motion” ⃗ϕA and “relative motion” ⃗ϕB of Obj (matrix valued):  ⃗ϕA :  �  �  �  �  �  [t1, t2]×Obj → Mn1(A)  (t, PObj) → ⃗ϕA(t, PObj) := [  −−−−−−−−→  OA�Φ(t, PObj)]| ⃗A =  n  �  i=1  xAi(t) ⃗Ei  noted  =  ⃗xA(t) = [−−−−→  OAp(t)]| ⃗A,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  ⃗ϕB :  �  �  �  �  �  [t1, t2]×Obj → Mn1(B)  (t, PObj) → ⃗ϕB(t, PObj) := [  −−−−−−−−→  OB �Φ(t, PObj)]| ⃗B =  n  �  i=1  xBi(t) ⃗Ei  noted  =  ⃗xB(t) = [−−−−→  OBp(t)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  And the “absolute” and “relative” velocities and accelerations of PObj are (matrix valued in Mn1):  ⃗vA(t, ⃗xAt) := [⃗v(t, pt)]| ⃗A  and  ⃗γA(t, ⃗xAt) := [⃗γ(t, pt)]| ⃗A,  when  ⃗xAt := [−−−→  OApt]| ⃗A,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  ⃗vB(t, ⃗xBt) := [⃗v(t, pt)]| ⃗B  and  ⃗γB(t, ⃗xBt) := [⃗γ(t, pt)]| ⃗B,  when  ⃗xBt := [−−−→  OBpt]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  62  63  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Referentials and “matrix motions”  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Prove: ⃗vA(t, ⃗xAt) = ∂ ⃗ϕA  ∂t (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  −−−−−−−→  OAt�ΦPObj (t) = �  i xAi(t) ⃗Ai(t) gives ⃗v(t, pt) = �  i xAi  ′(t) ⃗Ai(t) + xAi(t) ⃗Ai  ′(t), thus [⃗v(t, pt)]| ⃗  A =  �  i xAi  ′(t)[ ⃗Ai(t)]| ⃗  A + xAi(t)[ ⃗Ai  ′(t)]| ⃗  A (since Mn1 is a vector space) = �  i xAi  ′(t) ⃗Ei + [⃗0] (the ⃗Ai(t) are static  in RA:  [ ⃗Ai  ′(t)]| ⃗  A = [limh→0  ⃗  Ai(t+h)− ⃗  Ai(t)  h  ]| ⃗  A = limh→0  [ ⃗  Ai(t+h)]| ⃗  A−[ ⃗  Ai(t)]| ⃗  A  h  = limh→0  ⃗  Ei− ⃗  Ei  h  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And  ⃗ϕAPObj (t) = [  −−−−−−−→  OA�ΦPObj (t)]| ⃗  A = �  i xAi(t)[ ⃗Ai(t)]| ⃗  A = �  i xAi(t) ⃗Ei, thus ⃗ϕAPObj  ′(t) = �  i xAi  ′(t) ⃗Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 ⃗u is a C1 vector field, p is a point, ⃗xA := [−−→  OAp]| ⃗A and ⃗uA(⃗xA) := [⃗u(p)]| ⃗A (matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Prove: d⃗uA(⃗xA) = [d⃗u(p)]| ⃗A (endomorphism in Mn1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d⃗uA(⃗xA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]| ⃗A = [d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w]| ⃗A for all ⃗w ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The point p+h⃗w ∈ Rn is referenced by A as [−−→  OAp + h⃗w]| ⃗  A = [−−→  OAp]| ⃗  A + h[⃗w]| ⃗  A = ⃗xA + h[⃗w]| ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus d⃗uA(⃗xA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]| ⃗  A = limh→0  ⃗uA(⃗xA+h[ ⃗w]| ⃗  A)−⃗uA(⃗xA)  h  = limh→0  [⃗u(p+h ⃗w)]| ⃗  A−[⃗u(p)]| ⃗  A  h  = limh→0  [⃗u(p+h ⃗w)− ⃗w(p)]| ⃗  A  h  =  [limh→0  ⃗u(p+h ⃗w)− ⃗w(p)  h  ]| ⃗  A = [d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w]| ⃗  A = [d⃗u(p)]| ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]| ⃗  A, true for all ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Call Qt the transition matrix from ( ⃗Ait) to ( ⃗Bit) at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove ⃗xAt = [−−−−→  OAOBt]| ⃗A + Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗xBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗xAt = [−−−→  OApt]| ⃗  A = [−−−−→  OAOBt + −−−→  OBtpt]| ⃗  A = [−−−−→  OAOBt]| ⃗  A + [−−−→  OBtpt]| ⃗  A, and the change of basis formula gives  [−−−→  OBtpt]| ⃗  B = Q−1  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [−−−→  OBtpt]| ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Motion of RB  Particular case Obj = ObjRB: Its motion in the Universe, also called the motion of RB, is noted  �ΦRB :  �  [t1, t2] × ObjRB → Rn  (t, QRB) → qt = �ΦRB(t, QRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  At t at qt = �ΦRB(t, QRB), the Eulerian velocities and accelerations of QRB are  ⃗vRB(t, qt) = ∂�ΦRB  ∂t (t, QRB)  and  ⃗γRB(t, qt) = ∂2�ΦRB  ∂2t (t, QRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Quantification: Drive and static “motion” of RB  The “drive motion” ⃗ϕD, also called the motion of RB in RA, and the “static motion” ⃗ϕS is the quantification  of �ΦRB by A and by B:  ⃗ϕD :  �  �  �  [t1, t2] × ObjRB → Mn1(A)  (t, QRB) → ⃗ϕD(t, QRB) := [  −−−−−−−−−−→  OA�ΦRB(t, QRB)]| ⃗A  noted  =  ⃗yD(t) = [−−−−→  OAq(t)]| ⃗A,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  ⃗ϕS :  �  �  �  ObjRB → Mn1(B)  QRB → ⃗ϕS(QRB) := [  −−−−−−−−−−→  OB �ΦRB(t, QRB)]| ⃗B  noted  =  ⃗yS = [−−−−→  OBq(t)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  (⃗ϕS is independent of t since ObjRB is fixed in RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  The drive velocity, also called the velocity of RB in RA, and static velocity of QRB are  ⃗vD(t, ⃗yDt) := [⃗vRB(t, qt)]| ⃗A  when  ⃗yDt := [−−→  OAqt]| ⃗A,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  ⃗vS(t, ⃗yS) := [⃗vRB(t, qt)]| ⃗B = [⃗0] noted  =  ⃗0 (null matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  And the drive and static accelerations are ⃗γD(t, ⃗yDt) = [⃗γRB(t, qt)]| ⃗A and ⃗γS(t, ⃗yS) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 Why introduce ⃗ϕS (static)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' You can’t confuse a particle QRB with its stored positions ⃗yS or ⃗yDt at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And see (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  63  64  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The translator Θt  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The translator Θt  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of Θt  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 At t, the translator Θt : Mn1(B) → Mn1(A) is defined with (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)-(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) by:  �  �  �  [−−−−→  OBq(t)]| ⃗B = ⃗ϕS(QRB) = ⃗yS position of QRB in RB (static), and  [−−−−→  OAq(t)]| ⃗A = ⃗ϕDt(QRB) = ⃗yDt position of QRB at t in RA (moving)  �  �  � =⇒ ⃗yDt = Θt(⃗yS),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Θt is the “inter-referential function at t” which translates the “matrix position” ⃗yS = ⃗ϕS(QRB) =  [  −−−−−−−−−−→  OB �ΦRB(t, QRB)]| ⃗B ∈ Mn1(B) (position of QRB as stored by B) to the “matrix position” ⃗yDt = ⃗ϕD(t, QRB) =  [  −−−−−−−−−−→  OA�ΦRB(t, QRB)]| ⃗A ∈ Mn1(A) (position of QRB as stored by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So Θt is defined by  ⃗ϕDt = Θt ◦ ⃗ϕS :  �  ObjRB → Mn1(A)  QRB → ⃗ϕDt(QRB) := Θt(⃗ϕS(QRB)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' defined by  Θt := ⃗ϕDt ◦ ⃗ϕ −1  S  :  �  Mn1(B) → Mn1(A)  ⃗yS → ⃗yDt = Θt(⃗yS) := ⃗ϕDt(⃗ϕ −1  S  (⃗yS))  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  (stored position by B to stored position by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for QOB the particle in ObjRB at t at OBt (chosen by B to locate its origin), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) gives, with  ⃗0 the null matrix in Mn1,  [−−−−→  OAOBt]| ⃗A = Θt(⃗0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  So, Θt is defined such that the following diagram commutes:  ⃗yS = ⃗ϕS(QRB) = localization of QRB by B  Θt  �  QRB ∈ ObjRB  ⃗ϕS  �  ⃗ϕDt  �  ⃗yDt = ⃗ϕDt(QRB) = Θt(⃗yS) = localization at t of QRB by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Translation at t for the motion �Φ  t is fixed, the position pt = �Φ(t, PObj) of a particle PObj ∈ Obj is also the position qt = �ΦRB(t, QRB) of  a particle QRB ∈ ObjRB, so ⃗ϕAt(PObj) = [−−−→  OApt]| ⃗A = [−−→  OAqt]| ⃗A = ⃗ϕDt(QRB), and ⃗ϕBt(PObj) = [−−−→  OBpt]| ⃗B =  [−−−→  OBqt]| ⃗B = ⃗ϕS(QRB), thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) gives  ⃗ϕAt = Θt ◦ ⃗ϕBt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  dΘt  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Push-forward  If ⃗yS ∈ Mn1(B), ⃗wS ∈ Mn1 and ⃗yDt = Θt(⃗yS) , then  ⃗wSt∗(⃗yDt) := dΘt(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wS  (= lim  h→0  Θt(⃗yS + h⃗wB) − Θt(⃗yS)  h  )  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  is the push-forward of the matrix ⃗wS ∈ Mn1 by Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, ⃗wSt∗([−−→  OAqt]| ⃗A) = dΘt([−−−→  OBqt]| ⃗B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗wt(qt)]| ⃗B for all qt ∈ Rn and all ⃗wt : Rn → ⃗Rn (vector field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  64  65  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Translated velocities  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Θt is affine in classical mechanics  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 In R3 (in classical mechanics), Θt is affine: For all QB0, QB1 ∈ ObjRB and all t, u ∈ R,  with qti = �ΦRB(t, QBi) ∈ Rn (positions at t in our Universe),  Θt([−−−→  OBqt0]| ⃗B + u [−−−→  qt0qt1]| ⃗B) = [−−−→  OAqt0]| ⃗A + u [−−−→  qt0qt1]| ⃗A,  and  [−−−→  qt0qt1]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [−−−→  qt0qt1]| ⃗B  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  the differential dΘt(⃗yS0) =noted dΘt being independent of ⃗yS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular [ ⃗Bit] ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Bi]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other  words, for all ⃗yS0, ⃗yS1 ∈ Mn1(B) and all t, u ∈ R,  Θt((1−u)⃗yS0 + u ⃗yS1) = (1−u)Θt(⃗yS0) + u Θt(⃗yS1),  and  Θt(⃗yS1) = Θt(⃗yS0) + dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(⃗yS1−⃗yS0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the straight line (possible in classical mechanics in R3) qt : u → qt(u) = qt0 +u −−−→  qt0qt1 ∈  Rn (fixed in RB), in particular, qt(0) = qt0 and qt(1) = qt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗yS(u) = [−−−−−→  OBqt(u)]| ⃗B (positions stored  by B), so ⃗yS(u) = [−−−→  OBqt0 + u −−−→  qt0qt1]| ⃗B = [(1−u)−−−→  OBqt0 + u −−−→  OBqt1]| ⃗B = (1−u)[−−−→  OBqt0]| ⃗B + u [−−−→  OBqt1]| ⃗B =  (1−u)⃗yS0 + u ⃗yS1, where ⃗yS0 = [−−−→  OBqt0]| ⃗B = ⃗yS(0) and ⃗yS1 = [−−−→  OBqt1]| ⃗B = ⃗yS(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for A: ⃗yDt(u) =  [−−−−−→  OAqt(u)]| ⃗A = (1−u)⃗yDt0 + u⃗yDt1 (positions stored by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  (1−u)Θt(⃗yS0) + uΘt(⃗yS1)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  =  (1−u)⃗yDt0 + u⃗yDt1 = ⃗yDt(u)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  =  Θt(⃗yS(u)) = Θt((1−u)⃗yS0 + u⃗yS1),  thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)1, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence (derivation in u): −Θt(⃗yS0) + Θt(⃗yS1) = dΘt(⃗yS(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (−⃗yS0 + ⃗yS1),  true for all u, thus dΘt(⃗yS(u)) is independent of u, dΘt(⃗yS(u)) = dΘt(⃗yS0), true for all ⃗yS0, so  dΘt(⃗yS0) =noted dΘt, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)2, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus [−−−−−→  OBtPBti]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [−−−−−→  OBtPBti]| ⃗B where PBti  is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Bit = −−−−−→  OBtPBti, thus [ ⃗Bit] ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Bi]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Call Qt = [Qt,ij] the transition matrix from ( ⃗Ait) to ( ⃗Bit) in ⃗Rn, and ( ⃗Ei) the canonical  basis in Mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove  [dΘt]| ⃗E = Qt,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej =  n  �  i=1  Qt,ij ⃗Ei, ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Bjt = �n  i=1Qt,ij ⃗Ait gives [ ⃗Bjt]| ⃗  A = �n  i=1Qt,ij ⃗Ei, and [ ⃗Bjt]| ⃗  A =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Bjt]| ⃗  B = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Translated velocities  t is fixed, ⃗vt(pt) =  ∂�Φ  ∂t (t, PObj) is the velocity of a particle PObj ∈ Obj at t at pt, ⃗xAt := [−−−→  OAtpt]| ⃗A,  ⃗xBt := [−−−→  OBtpt]| ⃗B, and Θt affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 The translated relative velocity and acceleration from B to A at t at pt are the matrices  ⃗vBt∗(⃗xAt) := dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt)  and  ⃗γBt∗(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗��Bt(⃗xBt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗vBt∗(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗vt(pt)] ⃗B and ⃗γBt∗(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗γt(pt)] ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation: Let qt0 and qt1 be particles in ObjRB s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗vBt(⃗xBt) = [−−−→  qt0qt1]| ⃗B where ⃗xBt = [−−−→  OBqt0]| ⃗B  (here −−−→  qt0qt1 is a tangent vector at qt0 to the curve qt : u → qt(u) = qt0 +u −−−→  qt0qt1 in the proof of prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then [−−−→  qt0qt1]| ⃗A =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [−−−→  qt0qt1]| ⃗B gives [−−−→  qt0qt1]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) = ⃗vBt∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Similarly for ⃗γBt∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 ( ⃗Ai) and ( ⃗Bi) are Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in foot and metre), (·, ·)A and (·, ·)B are the  associated Euclidean dot products, λ = || ⃗Bi||A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28), ( ⃗Ei) is the canonical basis in Mn1, and  (·, ·)M is the canonical inner dot product in Mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Call ⃗Eit∗ := dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ei and prove:  ∀i, j, ( ⃗Eit∗, ⃗Ejt∗)M = λ2δij,  and  dΘt  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt = λ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ( ⃗Bit) is a Euclidean basis for B, thus is a Euclidean orthogonal basis for all observers, in particular  for A, with || ⃗Bit||A = λ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗Eit∗ = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Bjt]| ⃗  B =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) [ ⃗Bit]| ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ( ⃗Eit∗, ⃗Ejt∗)M = [ ⃗Eit∗]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Ejt∗] =  [ ⃗Bit]T  | ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Bjt]| ⃗  A = ( ⃗Bit, ⃗Bjt)A = λ2( ⃗Bit, ⃗Bjt)B = λ2δij, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then λ2δij = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ei, dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej)M =  (dΘt  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ei, ⃗Ej)M, true for all i, j, thus dΘt  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt = λ2I, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  65  66  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition of Θ  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Definition of Θ  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 The translator from B to A is the function Θ defined with (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) by  Θ :  �  �  �  �  �  �  t∈[t1,t2]  ({t} × Mn1(B)) → Mn1(A)  (t, ⃗yS) → Θ(t, ⃗yS) := Θt(⃗yS) ,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all QRB ∈ ObjRB and all t,  Θ(t, [  −−−−−−−−−−→  OB �ΦRB(t, QRB)]| ⃗B) = [  −−−−−−−−−−→  OA�ΦRB(t, QRB)]| ⃗A,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Θ(t, ⃗ϕS(QRB)) = ⃗ϕD(t, QRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22) gives Θ(t,⃗0) = [−−−−−→  OAOB(t)]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 The translator Θ looks like a motion, but is not: A “usual” motion is defined by one  observer and connects one particle to its position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While Θ connects two “matrix positions” of one particle  relative to two referentials: Θ is an “inter-referential” function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  The “Θ-velocity” is the drive velocity  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 The “Θ-velocity” and “Θ-acceleration” are defined by (Eulerian type definition)  with  ⃗yDt = Θ(t, ⃗yS),  �  �  �  �  �  ⃗vΘ(t, ⃗yDt) := ∂Θ  ∂t (t, ⃗yS) (∈ Mn1),  ⃗γΘ(t, ⃗yDt) = ∂2Θ  ∂t2 (t, ⃗yS) (∈ Mn1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15  ⃗vΘ = ⃗vD  and  ⃗γΘ = ⃗γD ,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗vΘ(t, ⃗y) = ⃗vD(t, ⃗y) and ⃗γΘ(t, ⃗y) = ⃗γD(t, ⃗y) in Mn1, for all t and all ⃗y ∈ Mn1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Θ(t, ⃗ϕS(QRB)) = ⃗ϕD(t, QRB) gives  ∂Θ  ∂t (t, ⃗ϕS(QRB)) = ∂⃗ϕD  ∂t (t, QRB),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗vΘ(t, Θ(t, ⃗ϕS(QRB))) = ⃗vD(t, ⃗ϕD(t, QRB)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  thus ⃗vΘ(t, ⃗ϕD(t, QRB)) = ⃗vD(t, ⃗ϕD(t, QRB)), thus ⃗vΘ(t, ⃗y) = ⃗vD(t, ⃗y) for all ⃗y ∈ Mn1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem with  ∂2  ∂t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  The velocity-addition formula  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24) gives  ⃗ϕA(t, PObj) = Θ(t, ⃗ϕB(t, PObj)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  thus  ∂⃗ϕA  ∂t (t, PObj) = ∂Θ  ∂t (t, ⃗ϕB(t, PObj)) + dΘ(t, ⃗ϕB(t, PObj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗ϕB  ∂t (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Thus  ⃗vA(t, ⃗xAt) = ⃗vΘ(t, ⃗xAt) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vB(t, ⃗xBt),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  where ⃗xBt = ⃗ϕB(t, PObj) and ⃗xAt = ⃗ϕA(t, PObj) = Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, with ��vΘ =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34) ⃗vD,  ⃗vAt = ⃗vBt∗ + ⃗vDt  where  ⃗vBt∗(⃗xAt) := dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  which is the velocity-addition formula in RA:  ⃗vAt the absolute velocity = ⃗vBt∗ the translated relative velocity from B to A  + ⃗vDt the drive velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  In other words (relation between the numerical values of the velocities stored by A and B),  [⃗vt(pt)]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗vt(pt)] ⃗B + [⃗vRBt(pt)]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  66  67  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Coriolis acceleration, and the acceleration-addition formula  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Coriolis acceleration, and the acceleration-addition formula  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37) gives  ∂2⃗ϕA  ∂t2  (t, PObj) = ∂2Θ  ∂t2 (t, ⃗xBt) + d∂Θ  ∂t (t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗ϕB  ∂t (t, PObj)  +  �∂(dΘ)  ∂t  (t, ⃗xBt) + d2Θ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗ϕB  ∂t (t, PObj)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗ϕB  ∂t (t, PObj) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂2⃗ϕB  ∂t2 (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  And d2Θt = 0 in our classical framework (Θt is affine);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ∂Θ  ∂t (t, ⃗yS) = ⃗vΘt(Θt(⃗yS)) gives ∂(dΘ)  ∂t (t, ⃗yS) =  d( ∂Θ  ∂t )(t, ⃗yS) = d⃗vΘt(Θt(⃗yS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗yS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ∂ ⃗ϕB  ∂t (t, PObj) = ⃗vBt(⃗xBt) where ⃗xBt = ⃗ϕB(t, PObj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  ⃗γAt(⃗xAt) = ⃗γΘt(⃗xAt) + 2d⃗vΘt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γBt(⃗xBt)  = ⃗γDt(⃗xAt) + 2d⃗vDt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt∗(⃗xAt) + ⃗γBt∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 At t, the Coriolis acceleration ⃗γCt at ⃗xAt is  ⃗γCt(⃗xAt) = 2d⃗vDt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt∗(⃗xAt),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗γCt = 2d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  And the Coriolis acceleration ⃗γC at t at ⃗xAt is ⃗γC(t, ⃗xAt) := ⃗γCt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43) gives the acceleration-addition formula in RA:  ⃗γAt = ⃗γBt∗ + ⃗γDt + ⃗γCt ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  ⃗γAt the absolute acceleration = ⃗γBt∗ the translated relative acceleration from B to A  + ⃗γDt the drive acceleration + ⃗γCt the Coriolis acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  In other words (relation between the numerical values of the acceletations stored by A and B),  [⃗γt(pt)]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗γt(pt)] ⃗B + [⃗γRBt(pt)]| ⃗A + 2[d⃗vRBt]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗vt(pt)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  With an initial time  Let t0, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the Lagrangian associated function Φt0  t with the motion �Φ of Obj:  Φt0  t :  �  Ωt0 → Ωt  pt0=�Φ(t0, PObj) → pt = Φt0  t (pt0) := �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  And, with ⃗xAt = ⃗ϕA(t, PObj) = [−−−→  OApt]| ⃗A and ⃗xBt = ⃗ϕB(t, PObj) = [−−−→  OBpt]| ⃗B, define the “matrix motions”  ⃗ϕt0  At : Mn1(A) → Mn1(A) and ⃗ϕt0  Bt : Mn1(B) → Mn1(B) by  �  �  �  ⃗ϕt0  At(⃗xAt0) := ⃗xAt  (= [  −−−−−−−−→  OA�Φ(t, PObj)]| ⃗A = [−−−−−−−→  OAΦt0  t (pt0)]| ⃗A = ⃗ϕAt(PObj)),  ⃗ϕt0  Bt(⃗xBt0) := ⃗xBt  (= [  −−−−−−−−→  OB �Φ(t, PObj)]| ⃗B = [−−−−−−−→  OBΦt0  t (pt0)]| ⃗B = ⃗ϕBt(PObj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  And Θt(⃗xBt) = ⃗xAt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Θt(⃗ϕt0  Bt(⃗xBt0)) = ⃗ϕt0  At(⃗xAt0) with ⃗xAt0 = Θt0(⃗xBt0), thus  Θt ◦ ⃗ϕt0  Bt = ⃗ϕt0  At ◦ Θt0 : Mn1(B) → Mn1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)  In other words, the following diagram commutes:  ⃗xBt0 = ⃗ϕB(t0, PObj)  Θt0  �  ⃗ϕt0  Bt  � ⃗xBt = ⃗ϕt0  Bt(⃗xBt0)  Θt  �  PObj ∈ Obj  ⃗ϕBt0  �  ⃗ϕAt0  �  ⃗xAt0 = ⃗ϕA(t0, PObj) = Θt0(⃗xBt0)  ⃗ϕt0  At  � ⃗xAt = ⃗ϕt0  At(⃗xAt0) = Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51)  Thus, for any vector field ⃗uBt0 in RB,  dΘt(⃗xBt)  �  ��  �  (translation at t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d⃗ϕt0  Bt(⃗xBt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uBt0(⃗xBt0)  �  ��  �  (deformation from t0 to t)  =  d⃗ϕt0  At(⃗xAt0)  �  ��  �  (deformation from t0 to t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' dΘt0(⃗xBt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uBt0(⃗xBt0)  �  ��  �  (translation at t0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52)  67  68  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Drive and Coriolis forces  Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 Redo the above steps with ObjRB instead of Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the Lagrangian associated function Φt0  RBt with the motion �ΦRB of ObjRB:  Φt0  RBt :  �  ΩRBt0 = Rn → ΩRBt = Rn  qt0 = �ΦRB(t0, QRB) → qt = Φt0  RBt(qt0) := �ΦRB(t, QRB),  �  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53)  then define the “matrix motions” ⃗ϕt0  Dt : Mn1(A) → Mn1(A) and ⃗ϕt0  St : Mn1(B) → Mn1(B) by  �  �  �  ⃗ϕt0  Dt(⃗yDt0) := ⃗yDt  (= [  −−−−−−−−−−→  OA�ΦRB(t, QRB)]| ⃗  A = [−−−−−−−−→  OAΦt0  RBt(pt0)]| ⃗  A = ⃗ϕDt(QRB)),  ⃗ϕt0  St(⃗yS) := ⃗yS  (= [  −−−−−−−−−−→  OB �ΦRB(t, QRB)]| ⃗  B = [−−−−−−−−→  OBΦt0  RBt(qt0)]| ⃗  B = ⃗ϕS(QRB)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  Thus ⃗ϕS is a time-shift, which is also abusively noted ⃗ϕt0  St = I (algebraic identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So with Θt(⃗yS) = ⃗yDt we get  Θt(⃗ϕt0  Dt(⃗yS)) = ⃗ϕt0  Dt(⃗yDt0), with ⃗yDt0 = Θt0(⃗yS), thus  Θt ◦ ⃗ϕt0  St = ⃗ϕt0  Dt ◦ Θt0 : Mn1(B) → Mn1(A)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55)  (also abusively written Θt = ⃗ϕt0  Dt ◦ Θt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other words, the following diagram commutes:  ⃗yS = ⃗ϕS(QRB)  Θt0  �  ⃗ϕt0  St = time shift  � ⃗yS = ⃗ϕS(QRB)  Θt  �  QRB ∈ ObjRB  ⃗ϕS  �  ⃗ϕt0  D  �  ⃗yDt0 = ⃗ϕDt0(QRB) = Θt0(⃗yS)  ⃗ϕt0  Dt � ⃗yDt = ⃗ϕDt(QRB) = ⃗ϕt0  Dt(⃗yDt0) = Θt(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56)  And (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55) gives, for any ⃗yS = ⃗ϕS(QRB) and all vector field ⃗uS (static in RB), with ⃗yDt0 = Θt0(⃗yS),  dΘt(⃗yS)  �  ��  �  (translation at t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  d⃗ϕt0  St(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uS(⃗yS)  �  ��  �  (time shift from t0 to t)  =  d⃗ϕt0  Dt(⃗yDt0)  �  ��  �  (Drive motion from t0 to t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' dΘt0(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗uS(⃗yS)  �  ��  �  (translation at t0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='57)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10  Drive and Coriolis forces  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Fundamental principal: requires a Galilean referential  Second Newton’s law of motion (fundamental principle of dynamics): In a Galilean referential, the sum  of the external forces ⃗f on an object is equal to its mass multiplied by its acceleration:  �  external⃗f = m⃗γ  (in a Galilean referential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58)  Question: And in a Non Galilean referential?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: Then you have to add “observer dependent forces”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' you have to add “apparent forces”  due to the motion of the non Galilean observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Indeed, the motion of an object in our Universe does  not care about the observer motion (his accelerations and velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='v=_36MiCUS1ro for a carousel (a merry-go-round),  See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='v=aeY9tY9vKgs for tornadoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Drive + Coriolis forces = the inertial force  Consider ⃗f(t, pt) = the sum of the external forces acting on PObj at t at pt = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a Galilean referential RA, Newton laws (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58) means  [⃗ft(pt)]| ⃗A = m [⃗γt(pt)]| ⃗A,  written  ⃗fAt(⃗xAt) = m⃗γAt(⃗xAt)  ∈ Mn1,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)  with ⃗xAt := [−−−→  OApt]| ⃗A, ⃗fAt(⃗xAt) := [⃗ft(pt)]| ⃗A and ⃗γAt(⃗xAt) = [⃗γt(pt)]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With ⃗xAt = Θt(⃗xBt), the accelera-  tion addition formula gives ⃗fAt(⃗xAt) = m(dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γB(⃗xBt) + ⃗γDt(⃗xAt) + ⃗γCt(⃗xAt)) ∈ RA, thus, in RB,  dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗fAt(⃗xAt)  �  ��  �  ⃗fAt∗(⃗xBt)= ⃗fBt(⃗xBt)  = m⃗γB(⃗xBt) + m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γDt(⃗xAt)  �  ��  �  m ⃗γDt∗(⃗xBt)  + m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γCt(⃗xAt)  �  ��  �  m ⃗γCt∗(⃗xBt)  ,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60)  and dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ft(pt)]| ⃗A = dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗fAt(⃗xAt) =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) [⃗ft(pt)]| ⃗B =noted ⃗fBt(⃗xBt) is the external forces as quanti-  fied by B at t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) (with Θt supposed to be affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And with the pull-back notation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26):  68  69  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Summary for “Sun and Earth”  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 At t on pt, define  The drive force ⃗fBDt(⃗xBt) := −m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γDt(⃗xAt)  (= −m⃗γDt  ∗(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Coriolis force ⃗fBCt(⃗xBt) := −m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γCt(⃗xAt)  (= −m⃗γCt  ∗(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (The inertial, or fictitious, force := ⃗fBDt(⃗xBt) + ⃗fBCt(⃗xBt) = −m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗γDt + ⃗γCt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61)  Then (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60) gives the fundamental principle quantified in RB (non Galilean referential):  ⃗fBt(⃗xBt) + ⃗fBDt(⃗xBt) + ⃗fBCt(⃗xBt) = m⃗γB(⃗xBt) ,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='62)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', at t, in RB: The external force + the Drive and Coriolis forces = m times the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11  Summary for “Sun and Earth”  Illustation with a simplified (circular) motion of the Earth around the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Coriolis forces on the Earth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Referentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Relative referential RB = (OB, ( ⃗B1, ⃗B2, ⃗B3)) chosen by the observer B fixed on the Earth, where OBt =  �ΦRB(t, QOB) is the position of the particle QOB at the center of the Earth, written OB by B (fixed for B),  and ( ⃗B1t, ⃗B2t, ⃗B3t) is a Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' built with the metre) fixed in the Earth, written ( ⃗B1, ⃗B2, ⃗B3)  by B (fixed for B), with ⃗B3 chosen to be along the rotation axis of the Earth and oriented from the south  pole to the north pole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (·, ·)B is the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, a fixed particle QRB in  the Earth at longitude θQRB ∈] − π, π] and latitude ϕQRB ∈ [− π  2 , π  2 ] is referenced by observer B as the  matrix ⃗yS = ⃗ϕS(QRB) = [  −−−−−−−−−−→  OB �ΦRB(t, QRB)]| ⃗B = RB  �  �  cos(θQRB ) cos(ϕQRB )  sin(θQRB ) cos(ϕQRB )  sin(ϕQRB )  �  � where RB = ||  −−−−−−−−−−→  OB �ΦRB(t, QRB)||B  is the distance between QOB and QRB (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if QRB is on the surface of the Earth then RB ≃ 6371 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Initial Galilean referential RA0 = (OA0, ( ⃗A1, ⃗A2, ⃗A3)): OA0 is at the center of the Sun and ( ⃗A1, ⃗A2, ⃗A3) is  a Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' built with the foot) fixed relative to the stars, such that ⃗A3 = µ ⃗B3 with µ > 0  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3048 and λ = 1  µ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (·, ·)A is the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Deduced absolute Galilean referential RA = (OAt, ( ⃗A1, ⃗A2, ⃗A3)) chosen by observer A fixed on Earth,  where OAt = OBt, written OA by A (fixed for A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since it takes more that 365 days for QOB to complete a  rotation around the Sun, the motion of QOB will be considered to be rectilinear at constant velocity “in a  short interval of time” sufficient for the computation of the Coriolis acceleration with “sufficient accuracy”  (simplifies the calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (If A prefers to work with the initial Galilean referential RA0, then the absolute matrix motion  ⃗ϕA(t, PObj) = [  −−−−−−−−→  OA�Φ(t, PObj)]| ⃗A has to be replaced by ⃗ϕA(t, PObj) = [−−−−−−→  OA0OB(t)]| ⃗A + [  −−−−−−−−−−→  OB(t)�Φ(t, PObj)]| ⃗A,  idem for the drive motion ⃗ϕD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Drive motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The motion t → qt = �ΦRB(t, QRB) of a particle QRB in the Earth is stored by A as the drive motion ⃗ϕD  given by (matrix valued), with ω the angular velocity of the Earth in RA,  ⃗yD(t) = ⃗ϕD(t, QRB) = RA(QRB)  �  �  cos(ωt) cos ϕQRB  sin(ωt) cos ϕQRB  sin ϕQRB  �  � = [−−−−→  OAq(t)]| ⃗A =  �  �  yD1(t)  yD2(t)  yD3  �  � ,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='63)  where RA(QRB) = ||−−−−−→  QOBQRB||| ⃗A is the distance between QOB and QRB for A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' RA ≃ 20902231 foot if  QRB is on the surface of the Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (And (ωt) by replaced by (α0+ω(t−t0)) to be more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Drive velocity: With ⃗ωD := ω ⃗A3,  ⃗vD(t, ⃗yD(t)) = ⃗yD  ′(t) = ωRA  �  �  − sin(ωt) cos ϕQRB  cos(ωt) cos ϕQRB  0  �  � = ω  �  �  −y2(t)  y1(t)  0  �  � = ω  �  �  0  −1  0  1  0  0  0  0  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗yD(t) = ⃗ωD∧⃗yD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='64)  69  70  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Summary for “Sun and Earth”  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Drive acceleration:  ⃗γD(t, ⃗yDt) = ⃗yD  ′′(t) = ⃗ωD ∧ ⃗yD  ′(t) = ⃗ωD ∧ ⃗vD(t, ⃗yDt) = ⃗ωD ∧ (⃗ωD ∧ ⃗yD(t)) = −ω2  �  �  yD1(t)  yD2(t)  0  �  �  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65)  = the usual centrifugal acceleration (in a plane parallel to the equatorial plane, drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Differential of the drive velocity (time and space independent here): (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='64) gives  d⃗vD(t, ⃗yDt) = d⃗vD =  �  �  0  −ω  0  ω  0  0  0  0  0  �  � = ⃗ωD ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='66)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Here OAt = OBt, thus Θt(⃗0) = ⃗0 (with [⃗0] =noted ⃗0 = the null matrix), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Calculation of dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With Θt affine, dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗Bit]| ⃗B = [ ⃗Bit]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗B3 = λ ⃗A3 (hypothesis) and dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗B3]| ⃗B =  [ ⃗B3t]| ⃗A give dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E3 = λ ⃗E3 where ( ⃗Ei) is the canonical basis in Mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then let QBi ∈ ObjRB be the  Earth particle which position qti = �ΦRB(t, QBi) makes ⃗Bit := −−−−→  OBtqti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, ⃗B1 and ⃗B2 being in the  equatorial plane, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='63) gives dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1 = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗B1]| ⃗B = [ ⃗B1]| ⃗A = [−−−→  OAqt1]| ⃗A = λ  �  �  cos(ωt)  sin(ωt)  0  �  �, and dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E2 =  dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗B2]| ⃗B = [ ⃗B2]| ⃗A = [−−−→  OAqt2]| ⃗A = λ  �  �  − sin(ωt)  cos(ωt)  0  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus [dΘt]| ⃗E = λ  �  �  cos(ωt)  − sin(ωt)  0  sin(ωt)  cos(ωt)  0  0  0  1  �  � = the  expected rotation matrix expanded by λ (change of unit of measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Calculation of Θt (affine): Θt(⃗yS) = Θt(⃗0) + dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗yS, so, with OAt = OBt here,  ⃗yDt := Θt(⃗yS) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗yS  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Motions of Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B quantifies the motion �Φ of Obj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' he stores the relative motion ⃗ϕB of Obj, and the relative velocities  and accelerations ⃗vBt and ⃗γB (matrices), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)-(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Translations for A: With ⃗xAt = Θt(⃗xBt),  ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt)  and  ⃗γBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Drive force (apparent force in RB due to the motion of B):  ⃗fBDt(⃗xBt) = −m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γDt(⃗xAt)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65)  =  λmω2dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  xA1(t)  xA2(t)  0  �  � (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67)  =  λmω2  �  �  xB1(t)  xB2(t)  0  �  � ,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='69)  centrifugal force (in a “parallel plane” at latitude of PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Coriolis acceleration (apparent acceleration due to the motion of B):  ⃗γCt(⃗xAt) = 2 d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt)) = 2 dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='70)  because dΘt commutes with d⃗vDt (composition of “rotations along the same south-north axis” which reads  as eiωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ei π  2 = ei π  2 eiωt = ei( π  2 +ωt) in the equatorial plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Coriolis force (apparent force due to the motion of B):  ⃗fBCt(⃗xBt) = −m dΘt  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗γCt(⃗xAt) = −2m d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) = −2m⃗ω ∧ ⃗vBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='71)  70  71  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  “Isometric objectivity” and “Frame Invariance Principle”  11  Objectivities  Goal: To give an objective expression of the laws of mechanics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' As Maxwell [13] said: “The formula at  which we arrive must be such that a person of any nation, by substituting for the different symbols the  numerical value of the quantities as measured by his own national units, would arrive at a true result”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Generic notation: if a function z is given as z(t, x), then zt(x) := z(t, x), and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  “Isometric objectivity” and “Frame Invariance Principle”  This manuscript is not intended to describe “isometric objectivity”:  “Isometric objectivity” is the framework in which the “principle of material frame-indifference” (“frame  invariance principle”) is settled, principle which states that “Rigid body motions should not affect the  stress constitutive law of a material”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Truesdell–Noll [19] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 41:  « Constitutive equations must be invariant under changes of frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' »  Or Germain [9] :  « Axiom of power of internal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The virtual power of the "internal forces" acting on a  system S for a given virtual motion is an objective quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', it has the same value whatever be the  frame in which the motion is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' »  NB: Both of these affirmations are limited to “isometric changes of frame” (the same metric for all), as  Truesdell–Noll [19] page 42-43 explain: The “isometric objectivity” concern one observer who defines his  Euclidean dot product and consider only orthonormal change of bases to validate a constitutive law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If you want to interpret “isometric objectivity” in the “covariant objectivity” framework, then “isometric  objectivity” corresponds to a dictatorial management: One observer with his Euclidean referential (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  based on the English foot), imposes his unit of length to all other users (isometry hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Note:  The metre was not adopted by the scientific community until after 1875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Moreover, isometric objectivity leads to despise the difference between covariance and contravariance,  due to the uncontrolled use of the Riesz representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 Marsden and Hughes [12] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 8 use this isometric framework to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But, pages 22  and 163, they write that a “good modelization” has to be “covariant objective” (observer independent) to  begin with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And they propose a covariant modelization for elasticity at § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Definition and characterization of the covariant objectivity  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Framework of classical mechanics  Framework of classical mechanics to simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider two observers A and B and their referentials  RA = (OA, ( ⃗Ai)) and RB = (OB, ( ⃗Bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ( ⃗Ai) and ( ⃗Bi) are Euclidean bases in foot and metre, (·, ·)A  and (·, ·)B is their associated Euclidean dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And Θ is the translator, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Consider a regular motion �Φ of an object Obj, pt = �Φ(t, PObj) ∈ Rn the position at t of a particle in  our Universe, Ωt = �Φ(t, Obj) the configuration at t, and C = �  t∈[a,b]({t} × Ωt) the set of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And ⃗xAt := [−−−→  OApt]| ⃗A ∈ Mn1(A) and ⃗xBt := [−−−→  OBpt]| ⃗B ∈ Mn1(B) are the stored components of pt relative to  the chosen referentials, Mn1(A) and Mn1(B) being the spaces of n ∗ 1 matrices as referred to by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Covariant objectivity of a scalar function  Let f  :  �  C → R  (t, pt) → f(t, pt)  �  be a Eulerian scalar function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a temperature field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  f is  quantified by A and B as the functions fA  :  �  R×Mn1(A) → R  (t, ⃗xAt) → fA(t, ⃗xAt) := f(t, pt)  �  and fB  :  �  R×Mn1(B) → R  (t, ⃗xBt) → fB(t, ⃗xBt) := f(t, pt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 f is objective covariant iff, for all referentials RA and RB and for all t,  fAt(⃗xAt) = fBt(⃗xBt)  when  ⃗xAt = Θt(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' fAt = fBt∗ is the push-forward of fBt by Θt cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  71  72  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition and characterization of the covariant objectivity  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Covariant objectivity of a vector field  Let  ⃗w  :  �  C → ⃗Rn  (t, pt) → ⃗w(t, pt)  �  be a Eulerian vector field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  a force field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗w is quan-  tified by A and B as the functions  ⃗wA  :  �  R×Mn1(A) → Mn1(A)  (t, ⃗xAt) → ⃗wA(t, ⃗xAt) := [⃗w(t, pt)] ⃗A  �  and  ⃗wB  :  �  R×Mn1(B) → Mn1(B)  (t, ⃗xBt) → ⃗wB(t, ⃗xBt) := [⃗w(t, pt)] ⃗B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So ⃗wA(t, ⃗xAt) and ⃗wB(t, ⃗xBt) are the column matrices of the  components of ⃗w(t, pt) in RA and RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 ⃗w is objective covariant iff, for all referentials RA and RB and for all t,  ⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt)  when  ⃗xAt = Θt(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗wAt = ⃗wBt∗ is the push-forward of ⃗wBt by Θt cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Fundamental counter-example: A Eulerian velocity field is not objective, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39),  because of the drive velocity ⃗vD ̸= ⃗0 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Neither is a Eulerian acceleration field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The field of gravitational forces (external forces) is objective covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Covariant objectivity of differential forms  Let α :  �  C → Rn∗  (t, pt) → α(t, pt)  �  be a Eulerian differential form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a measuring device used to get the inter-  nal power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α is quantified by A and B as the functions αA :  �  R×Mn1(A) → Mn1(A)  (t, ⃗xAt) → αA(t, ⃗xAt) := [α(t, pt)] ⃗A  �  and αB :  �  R×Mn1(B) → Mn1(B)  (t, ⃗xBt) → αB(t, ⃗xBt) := [α(t, pt)] ⃗B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So αA(t, ⃗xAt) and αB(t, ⃗xBt) are the row matrices  of the components of α(t, pt) in RA and RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 α is objective covariant iff, for all referentials RA and RB and for all t,  αAt(⃗xAt) = αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)−1  when  ⃗xAt = Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' αAt = αBt∗ is the push-forward of αBt by Θt cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) are compatible: If ⃗w is an objective vector field and if α is an objective differential  form, then the scalar function α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w is objective:  αAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wAt(⃗xAt) = αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt)  (= (α(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, pt)),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  since αAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wAt(⃗xAt) = (αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt)) = αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Covariant objectivity of tensors  A tensor acts on both vector fields and differential forms, and its objectivity is deduced from the previous §.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, let T be a (Eulerian) tensor corresponding to a “physical quantity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The observers A and B  describe T as being the functions TA and TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 T is objective covariant iff, for all referentials RA and RB and for all t,  TAt(⃗xAt) = TBt∗(⃗xAt)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' TAt is the push-forward of TBt by Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Recall: TBt∗(⃗xAt)(α1(⃗xAt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗w1(⃗xAt)) := TBt(⃗xBt)(α1∗(⃗xBt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗w1∗(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  72  73  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Non objectivity of the velocities  Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 (Non covariant objectivity of a differential d⃗w) Let ⃗w be an objective vector field,  seen as ⃗wA by A and ⃗wB by B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So ⃗wAt(⃗xAt) =(11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), thus  d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wBt(⃗xBt) + (d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt)),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  hence  d⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)−1 + (d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)−1  ̸= dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)−1  when  d2Θt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Thus d⃗w is not covariant objective in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' However in classical mechanics for “change of Cartesian  referentials” Θt is affine, so d2Θt = 0, and in particular d⃗w is objective when ⃗w is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  (d2 ⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt) + d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d2Θt(⃗xBt)  = dΘt(���xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d2 ⃗wBt(⃗xBt) + 2 d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wBt(⃗xBt) + d3Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Thus d2 ⃗w is not covariant objective in general (but if Θt is affine then d2 ⃗w is objective if ⃗w is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Non objectivity of the velocities  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Eulerian velocity ⃗v : not covariant (and not isometric) objective  Velocity addition formala: With ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(⃗xBt) when ⃗xAt = Θt(⃗xBt), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39),  ⃗vAt(⃗xAt) = ⃗vBt∗(⃗xAt) + ⃗vDt(⃗xAt)  ̸= ⃗vBt∗(⃗xAt)  when  ⃗vDt(⃗xAt) ̸= ⃗0,  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  thus a Eulerian velocity field is not covariant objective (and not isometric objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  d⃗v is not objective  The velocity addition formula, (⃗vAt − ⃗vDt)(⃗xAt) = ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) when ⃗xAt = Θt(⃗xBt),  gives  d(⃗vAt − ⃗vDt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vBt(⃗xBt) + d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  thus d⃗v is neither covariant objective nor isometric objective because of d⃗vD:  d⃗vAt(⃗xAt) = d⃗vBt∗(⃗xAt) + d⃗vDt(⃗xAt) + d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt)−1 ̸= d⃗vBt∗(⃗xAt)  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Recall: “Isometric objective” implies  The use of the same Euclidean metric in RB and RA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (·, ·)A = (·, ·)B,  �ΦRB (motion of RB) is a solid body motion, and  Θt is affine (so d2Θt = 0 for all t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Prove, with Qt the (orthonormal) transition matrix from ( ⃗Ai) to ( ⃗Bi):  [d⃗vt]| ⃗B = Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗vt]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q−1  t  + Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q−1  t ,  written  [L]| ⃗B = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='QT + Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  (Used in classical mechanics courses, to prove that d⃗v isn’t “isometric objective” because of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' t0, t ∈ R, pt0 = �Φ(t0, PObj), pt = �Φ(t, PObj) = Φt0  t (pt0), ⃗v(t, pt) = ∂ �Φ  ∂t (t, PObj), and F t0  t (pt0) = dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So ⃗v(t, Φt0  t (pt0)) =  ∂Φt0  pt0  ∂t  (t, pt0), thus d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  pt0 (t) =  ∂F t0  pt0  ∂t  (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30), with F t0  pt0 =noted F, gives  [F(t)]|⃗at0 , ⃗  B = Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t)]|⃗at0 , ⃗  A, thus [F ′(t)]|⃗at0 , ⃗  B = Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t)]|⃗at0 , ⃗  A + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F ′(t)]|⃗at0 , ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus [d⃗v(t, pt)]| ⃗  B =  [F t0  pt0  ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  pt0 (t)]| ⃗  B  = [F t0  pt0  ′(t)]| ⃗  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F t0  pt0 (t)]| ⃗  B  = (Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t)]|⃗at0 , ⃗  A + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F ′(t)]|⃗at0 , ⃗  A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t)]−1  |⃗at0 , ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q(t)−1  =  Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q(t)−1 + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F ′(t)]|⃗at0 , ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t)]−1  |⃗at0 , ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q(t)−1 = Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q(t)−1 + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v(t, pt)]| ⃗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)-  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Prove that d2⃗v is “isometric objective” when �ΦRB is a rigid body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) with ⃗vA − ⃗vD instead of ⃗wA, and ⃗vB instead of ⃗wB give, in an “isometric objective” framework,  d2(⃗vAt − ⃗vDt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗uBt∗, ⃗wBt∗) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d2⃗vBt(⃗xBt)(⃗uB, ⃗wB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Here d2⃗vDt = 0 (rigid body motion), thus d2⃗v is “isometric objective”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  73  74  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Lie derivatives are covariant objective  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  d⃗v + d⃗vT is “isometric objective”  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 If �ΦRB is a rigid body motion then d⃗vt + d⃗vT  t is “isometric objective”  d⃗vAt + d⃗vT  At = (d⃗vBt + d⃗vT  Bt)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  (Isometric framework: The rate of deformation tensor is independent of an added added rigid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='QT = I gives Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='QT + ( Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='QT )T = 0, then apply (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 Prove that Ω = d⃗v−d⃗vT  2  is not isometric objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11) gives d⃗vT  At = d⃗vT  Bt∗ + d⃗vT  Dt, thus  d⃗vAt−d⃗vT  At  2  =  d⃗vBt∗−d⃗vT  Bt∗  2  +  d⃗vDt−d⃗vT  Dt  2  ̸=  d⃗vBt∗−d⃗vT  Bt∗  2  , even if  �ΦRB is a solid body motion (then  d⃗vDt−d⃗vT  Dt  2  = ⃗ω∧ is a rotation time a dilation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Lagrangian velocities  The Lagrangian velocities do not define a vector field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus asking about the objectivity of  Lagrangian velocities is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  The Lie derivatives are covariant objective  Framework of § 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular we have the velocity-addition formula ⃗vAt = ⃗vBt∗ + ⃗vDt in RA where  ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) and ⃗xBt = Θt(⃗xAt), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The objectivity under concern is the covariant objectivity (no inner dot product or basis required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Lie derivatives are also called “objective rates” because they are covariant objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Easy proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Scalar functions  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 If f be a covariant objective function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), then its Lie derivative L⃗vf is  covariant objective:  L⃗vAfA = Θ∗(L⃗vBfB),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L⃗vAfA(t, ⃗xAt) = L⃗vBfB(t, ⃗xBt)  when  ⃗xAt = Θt(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', DfA  Dt (t, ⃗xAt) = DfB  Dt (t, ⃗xBt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ( ∂fA  ∂t + dfA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vA)(t, ⃗xAt) = ( ∂fB  ∂t + dfB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vB)(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the motion t → p(t) = �Φ(tPObj) of a particle PObj, and ⃗xA(t) = [−−−−→  OAp(t)]| ⃗A and ⃗xB(t) =  [−−−−→  OBp(t)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With f objective, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) gives fB(t, ⃗xB(t)) = fA(t, Θ(t, ⃗xB(t))) (= fA(t, ⃗xA(t))), thus  DfB  Dt (t, ⃗xB(t)) = ∂fA  ∂t (t, ⃗xA(t)) + dfAt(⃗xA(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (∂Θ  ∂t (t, ⃗xB(t))  �  ��  �  ⃗vDt(⃗xAt)  + dΘt(⃗xB(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xB(t)))  �  ��  �  ⃗vBt∗(⃗xAt)  = ∂fA  ∂t (t, ⃗xAt) + dfAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vAt(⃗xAt) = DfA  Dt (t, ⃗xAt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  thanks to velocity addiction formula ⃗vAt = ⃗vBt∗ + ⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Vector fields  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 Let ⃗w be a covariant objective vector field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then its Lie derivative L⃗v ⃗w  is covariant objective:  L⃗vA ⃗wA = Θ∗(L⃗vB ⃗wB),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', when ⃗xAt = Θt(⃗xBt),  L⃗vA ⃗wA(t, ⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗vB ⃗wB(t, ⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  (D ⃗wA  Dt  − d⃗vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wA)(t, ⃗xAt) = dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (D ⃗wB  Dt  − d⃗vB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wB)(t, ⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  (∂ ⃗wA  ∂t  + d⃗wA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vA − d⃗vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wA)(t, ⃗xAt) = dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (∂ ⃗wB  ∂t  + d⃗wB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vB − d⃗vB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wB)(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  But the partial, convected, material, and Lie autonomous derivatives are not covariant objective (not  74  75  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Lie derivatives are covariant objective  even isometric objective because of the drive velocity ⃗vD): We have  (d⃗wAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗vAt−⃗vDt))(⃗xAt) = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗wBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt) + (d2Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt)(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  (d(⃗vAt−⃗vDt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wAt)(⃗xAt) = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗vBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt) + (d2Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt)(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  (d(⃗vAt−⃗vDt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗vAt−⃗vDt))(⃗xAt) = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗vBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt) + d2Θt(⃗vBt,⃗vBt))(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  L0  (⃗vAt−⃗vDt) ⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L0  ⃗vBt ⃗wBt(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  ∂ ⃗wA  ∂t (t, ⃗xAt) + L0  ⃗vD ⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂ ⃗wB  ∂t (t, ⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  D ⃗wA  Dt (t, ⃗xAt) − d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wAt(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗wB  Dt (t, ⃗xBt) + d2Θt(⃗vBt, ⃗wBt)(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  ∂(⃗vA−⃗vD)  ∂t  (t, ⃗xAt) + L0  ⃗vD(⃗vA−⃗vD)(t, ⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗vB  ∂t (t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' • ⃗wAt(Θt(⃗xBt)) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt) gives  d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt) = d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wB(⃗xBt),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  thus, with dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) = (⃗vAt−⃗vDt)(⃗xAt) = ⃗vBt∗(⃗xAt) (velocity-addition formula),  d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗vAt−⃗vDt)(⃗xAt) = (d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt),  hence (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vAt(⃗xAt) ̸= dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt)) (the vector field d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v is  not objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (⃗vAt−⃗vDt)(Θt(⃗xBt)) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) gives  d(⃗vAt−⃗vDt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xBt) = d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vBt(⃗xBt),  so, applied to ⃗wBt (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗vBt), we get (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If ⃗xAt = Θt(⃗xB), then ⃗wA(t, Θ(t, ⃗xB)) = dΘ(t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wB(t, ⃗xB), so, with ∂Θ  ∂t (t, ⃗xB) = ⃗vΘt(⃗xAt), we get  ∂ ⃗wA  ∂t (t, ⃗xAt) + d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vΘt(⃗xAt) = d∂Θ  ∂t (t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xB) + dΘt(⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂ ⃗wB  ∂t (t, ⃗xB)  = (d⃗vΘt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wBt(⃗xB) + dΘt(⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂ ⃗wB  ∂t (t, ⃗xB),  Thus (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25) since ⃗vΘ = ⃗vD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21) gives (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗vB∗(t, Θ(t, ⃗xB)) = dΘ(t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vB(t, ⃗xB) gives  ∂⃗vB∗  ∂t (t, ⃗xAt) + d⃗vB∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vΘ(t, ⃗xAt) =  ∂dΘ  ∂t (t, ⃗xB)  �  ��  �  d⃗vΘt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΘt(⃗xB)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vBt(⃗xB) + dΘ(t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂⃗vB  ∂t (t, ⃗xB, )  since ∂dΘ  ∂t (t, ⃗xB) = d( ∂Θ  ∂t )(t, ⃗xB) and ∂Θ  ∂t (t, ⃗xB) = ⃗vΘ(t, ⃗xAt) = ⃗vΘt(Θt(⃗xB));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' hence (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Tensors  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 It T is a covariant objective tensor, then its Lie derivatives are covariant objectives:  L⃗vATA = Θ∗(L⃗vBTB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Corollary of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) to get L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = (L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L⃗v ⃗w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then use L⃗v(t1 ⊗ t2) =  (L⃗vt1) ⊗ t2 + t1 ⊗ (L⃗vt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  75  76  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Taylor expansions and ubiquity gift  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Taylor expansions and ubiquity gift  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  In Rn with ubiquity  Generic formula:  f(t) = f(t0) + (t−t0) f ′(t0) + (t−t0)2  2  f ′(t0)2 + o((t−t0)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  In particular f(t) = ⃗w(t, p(t)) gives  ⃗w(t, p(t)) = ⃗w(t0, p(t0)) + (t−t0) D ⃗w  Dt (t0, p(t0)) + (t−t0)2  2  D ⃗w  Dt (t0, p(t0))2 + o((t−t0)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  Problem :  ⃗w(t, p(t)) is a vecteur at t at p(t) while ⃗w(t0, p(t0)) is a vecteur at t0 at p(t0), so (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  cannot be written  ⃗w(t, p(t)) −  �  ⃗w(t0, p(t0)) + (t−t0) D ⃗w  Dt (t0, p(t0)) + (t−t0)2  2  D ⃗w  Dt (t0, p(t0))2�  = o((t−t0)2),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  since the left-hand side supposes the ubiquity gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in a non-planar manifold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a surface in R3 considered on its own), ⃗w(t, pt) ∈ Tpt(Ωt) = the  linear tangent space at p(t) = pt, whereas ⃗w(t0, pt0) ∈ Tpt0(Ωt0) = the linear tangent space at p(t0) = pt0,  and the tangent spaces Tpt(Ωt) and Tpt0(Ωt0) are distinct at two distinct points in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the  left-hand side of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32) is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In R3 our affine space (our Universe), Tpt(Ωt) and Tpt0(Ωt0) are identified with ⃗R3, and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31) is well  defined, and very useful!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  General case  By definition, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10), with p(t) = Φt0(t, pt0) = Φt0  t (pt0),  L⃗v ⃗w(t0, pt0) = dΦt0  t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)) − ⃗w(t0, pt0)  t − t0  + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Thus,  dΦt0  t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)) = ⃗w(t0, pt0) + (t−t0) L⃗v ⃗w(t0, pt0) + o(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  Hence we get the first order Taylor expansion without ubiquity gift:  ⃗w(t, p(t)) = dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  ⃗w + (t−t0) L⃗v ⃗w  �  (t0, pt0) + o(t−t0),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  both side of the equality being in Tpt(Ωt) (meaningful in any manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 In Rn, with the gift of ubiquity, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35) gives (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' dΦt0(t0+h, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t0, pt0) =(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35) ⃗w(t0, pt0) + h d⃗v(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t0, pt0) + o(h), thus  ⃗w(t, pt)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  =  (I + h d⃗v(t0, pt0) + o(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  (⃗w + h D ⃗w  Dt − h d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)(t0, pt0) + o(h)  �  = (⃗w + h d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + h D ⃗w  Dt − h d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)(t0, pt0) + o(h) = (⃗w + h D ⃗w  Dt )(t0, pt0) + o(h),  which is (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 In Rn, at second order,  ⃗w(t, p(t)) = dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  (⃗w + hL⃗v ⃗w + h2  2 L⃗v(L⃗v ⃗w))(t0, pt0) + o(h2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  So if the values ⃗w(t0, pt0), L⃗v ⃗w(t0, pt0) and L⃗v(L⃗v ⃗w)(t0, pt0) are known, then ⃗w(t, p(t)) is estimated at  second order thanks to the push-forward of (⃗w + hL⃗v ⃗w + h2  2 L⃗v(L⃗v ⃗w))(t0, pt0) by Φt0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  76  77  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Taylor expansions and ubiquity gift  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let dΦt0(t, pt0) = F t0  pt0 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let ⃗g(t) = dΦt0(t, pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)) when p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So  L⃗v ⃗w(t0, pt0) = ⃗g ′(t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And D ⃗w  Du (u, p(u)) = d⃗v(u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(u, p(u)) + F t  u(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′(u), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  D2 ⃗w  Du2 (u, p(u)) =  D(d⃗v)  Du (u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(u, p(u)) + d⃗v(u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w  Du (u, p(u)) + d⃗v(u, p(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t  u(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′(u) +  F t  u(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g ′′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus D2 ⃗w  Dt2 (t, p(t)) = ( D(d⃗v)  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w  Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗v ⃗w)(t, p(t)) + ⃗g ′′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39) we get  ⃗g ′′(t) = L⃗v(L⃗v ⃗w)(t, p(t)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Alternate proof (calculation): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34) gives F t0  pt0 (t) = It0 + h d⃗v(t0, pt0) + h2  2 d⃗γ(t0, pt0) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus,  omitting the reference to (t0, pt0) to lighten the writing,  dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗w + hL⃗v ⃗w + h2  2 L⃗vL⃗v ⃗w + o(h2))  =  �  I + h d⃗v + h2  2 d(D⃗v  Dt ) + o(h2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  ⃗w + hL⃗v ⃗w + h2  2 L⃗vL⃗v ⃗w + o(h2)  �  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  The h0 term is I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The h term is L⃗v ⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = D ⃗w  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The h2 term is the sum of  1  2L⃗vL⃗v ⃗w = 1  2(D2 ⃗w  Dt2 − 2 d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt − D(d⃗v)  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39),  d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗v ⃗w = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = 1  2(2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D ⃗w  Dt − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w),  1  2d(D⃗v  Dt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = 1  2(D(d⃗v)  Dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And the sum gives D2 ⃗w  Dt2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  77  78  Part V  Appendix  In this appendix, we tried to give standard results useful in mechanics, results that are scattered in the  existing literature, and sometimes difficult to find except in math books (differential geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The definitions, notations and results are detailed, so that no ambiguity is possible (some notations  can be nightmarish when not understood, or misused, or come like a bull in a china-shop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  All the  results presented apply to solids, fluids, thermodynamics, general relativity, electromagnetism, quantum  mechanics, chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (the same math applies to all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' even applies to mechanical engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A  Classical and duality notations  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Contravariant vector and basis  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Contravariant vector  Let (E, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') =noted E be a real vector space (= a linear space on the field R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 An element ⃗x ∈ E is called a vector, or a “contravariant vector”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A vector is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So why this name contravariant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Historical answer: Because of the change of  basis formula [⃗x]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|old, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28), which uses P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, what is a covariant vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Answer: From the vector space E, you can build the vector space (an  overlay) L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) =noted E∗ = the space of linear forms on E (a linear form is a measuring instrument  that gives values to vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then an element ℓ ∈ E∗ will be called a covariant vector, because of the  change of basis formula [ℓ]|new = [ℓ]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Basis  Definitions: • n vectors ⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en ∈ E are linearly independent iff, for all λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λn ∈ R, the equality  �n  i=1λi⃗ei = ⃗0 implies λi = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  n vectors ⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en ∈ E span E iff, for all ⃗x ∈ E, ∃λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λn ∈ R such that ⃗x = �n  i=1λi⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A basis in E is a set {⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en} ⊂ E made of n linearly independent vectors which span E, in which  case the dimension of E is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Canonical basis  Consider the field R of reals and the Cartesian product ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × R, n times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The canonical basis is  ⃗e1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗en = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', 0, 1),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  with 0 = the addition identity element used n−1 times, and 1 = the multiplication identity element used  once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The 3-D geometric space we live in has no canonical basis: What would the identity  element 1 mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1 metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1 foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And there is no “intrinsic” preferred direction to define ⃗e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So the  Cartesian product ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × R and its canonical basis form an abstract mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Cartesian basis  (René Descartes 1596-1650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Let n = 1, 2, 3, let Rn be the usual affine space (space of points), and let  ⃗Rn = (⃗Rn, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') be the usual real vector space of bipoint vectors with its usual algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let p ∈ Rn, and let (⃗ei(p)) be a basis at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A Cartesian basis in ⃗Rn is a basis independent of p (the same at all p), and then (⃗ei(p)) =noted (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example of a non Cartesian basis: The polar basis, see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 (polar coordinate system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And a Euclidean basis is a particular Cartesian basis described in § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  More generally, a Cartesian basis refers to En = E ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='×E (n-times) where E is a dimension 1 vector  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  78  79  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Representation of a vector relative to a basis  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Representation of a vector relative to a basis  We give:  the classical notation (non ambiguous), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' used by Arnold [3] and Germain [8], and  the duality notation (can be ambiguous because of misuses), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' used by Marsden and Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Both classical and duality notation are equally good, but if you have any doubt, use the classical notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Let ⃗x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ei) be a basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The components of ⃗x relative to the basis (⃗ei) are  the n real numbers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn (classical notation) also named x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn (duality notation) such that  ⃗x = x1⃗e1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' + xn⃗en  �  ��  �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  = x1⃗e1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' + xn⃗en  �  ��  �  dual  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗x]|⃗e =  �  �  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xn  �  �  � �� �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  �  �  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xn  �  �  � �� �  dual  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  [⃗x]|⃗e being the column matrix representing ⃗x relative to the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Of course xi = xi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') And  the column matrix [⃗x]|⃗e is simply named [⃗x] if one chosen basis is imposed to all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With the sum sign:  ⃗x =  n  �  i=1  xi⃗ei  � �� �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  xi⃗ei  � �� �  dual  (=  n  �  J=1  xJ⃗eJ =  n  �  α=1  xα⃗eα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  (The index in a summation is a dummy index, even if you do not write the sum sign � as can be done  with Enstein’s convention: ⃗x = �n  j=1xj⃗ej =noted xj⃗ej = xi⃗ei = xJ⃗eJ = xα⃗eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 In ⃗R2 with ⃗x = 3⃗e1 + 4⃗e2 = �2  i=1 xi⃗ei = �2  i=1 xi⃗ei: We have x1=x1=3 and x2=x2=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And [⃗x]|⃗e = 3[⃗e1]|⃗e +4[⃗e2]|⃗e = �2  i=1 xi[⃗ei]|⃗e = �2  i=1 xi[⃗ei]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, with δi  j = δij :=  �  = 1 if i=j  = 0 if i̸=j  �  the Kronecker symbols,  ⃗ej =  n  �  i=1  δij⃗ei  � �� �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  δi  j⃗ei  � �� �  dual  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗e1]|⃗e =  �  �  �  �  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  0  �  �  �  � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', [⃗en]|⃗e =  �  �  �  �  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  0  1  �  �  �  � ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  that is, the components of ⃗ej are δij with classical notations, and δi  j with duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the  matrices [⃗ej]|⃗e mimic the use of theoretical Cartesian space ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × R and its canonical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The column matrix [⃗x]|⃗e is also called a “column vector”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' NB: A “column vector” is not a  vector, but just a matrix (a collection of real numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See the change of basis formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28) where  the same vector is represented by two “column vectors” (two column matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Dual basis  Recall: Let E and F be vector spaces and (F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') =noted F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) be the usual real vector space  of functions with the internal addition (f, g) → f +g defined by (f +g)(x) := f(x)+g(x) and the external  multiplication (λ, f) → λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='f defined by (λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='f)(x) := λ(f(x)), for all f, g ∈ F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), x ∈ E, λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='f =noted λf for all f ∈ F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) and λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Linear forms = “Covariant vectors”  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 The set E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) of linear scalar valued functions is called the dual of E:  E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = the dual of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  And a linear scalar valued function ℓ ∈ E∗ is called a linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  More precisely, E∗ as defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) is the algebraic dual of E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To define the topological dual usually  needed with L2 functions in mechanics, E needs to be a Banach space (a vector space equipped with  a norm with which E is complete), and E∗ is then the set of continuous linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (If E is finite  dimensional then any linear form is continuous relative to any norm since all norms are equivalent in  finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  79  80  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Dual basis  E∗ is a vector space: sub-space of (F(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') (trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation: It answers the question: What does a function E → R do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Answer: Like any function,  it gives values to vectors: ℓ(⃗u) = the value of ⃗u through ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' That is, a ℓ ∈ E∗ is a measuring tool for  vectors: If ���u ∈ E then ℓ(⃗u) = real value given by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation: If ℓ ∈ E∗ then  ∀⃗u ∈ E,  ℓ(⃗u) noted  =  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  also written ⟨ℓ, ⃗u⟩E∗,E where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⟩E∗,E is the duality bracket: The dot in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u is “the distributivity dot”  since linearity ℓ(⃗u + λ⃗v) = ℓ(⃗u) + λℓ(⃗v) = distributivity ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u + λ⃗v) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u + λℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: The dot in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u is not an inner dot product (since ℓ /∈ E while ⃗u ∈ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 A linear form ℓ in E∗ is also called a “covariant vector”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Co-variant refers to:  1- The action of a function on a vector, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) (co-variant calculation), and  2- The change of coordinate formula [ℓ]new = [ℓ]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28) (covariant formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: E∗ being a vector space, an element ℓ ∈ E∗ is indeed a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But E∗ has no existence if E has  not been specified first since E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ℓ ∈ E∗ can’t be confused with a vector ⃗u ∈ E since  there is no natural canonical isomorphism between E and E∗ (no “intrinsic representation”), see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Misner–Thorne–Wheeler [14], box 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1, insist: “Without it [the distinction between covari-  ance and contravariance], one cannot know whether a vector is meant or the very different object that is  a linear form.”  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Covariant dual basis (= the functions that give the components of a vector)  Notation: If ⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗uk are vectors in E, then Vect{⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗uk} := the vector space spanned by ⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let E be a finite dimensional vector space, and let (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n be a basis in E  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Let i ∈ [1, n]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The scalar projection on Vect{⃗ei} parallel to Vect{⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗ei−1,⃗ei+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en}  is the linear form named πei ∈ E∗ with the classical notation, named ei ∈ E∗ with the duality notation,  defined by, for all i, j,  �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' :  πei(⃗ej) = δij,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = δij,  dual not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' :  ei(⃗ej) = δi  j,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = δi  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Thus, πei = ei being linear, if ⃗x =clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �n  i=1xi⃗ei =dual �n  i=1xi⃗ei (classical or duality notations), then  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) gives  πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  = xi  =  xi dual  = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' πei = ei gives the i-th component of a vector ⃗x relative to the basis (⃗ei), see figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1:  Parallel projections: πe1(⃗x) = x1 and πe2(⃗x) = x2 (dual not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : e1(⃗x) = x1 and e2(⃗x) = x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: The dual basis (πei) is intrinsic to (⃗ei);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And there can’t be any notion of orthogonality in E here  since we can’t use a inner dot product: The functions πei = ei and vectors ⃗x do not belong to a same  vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 and definition of the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n = (ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n is a basis in E∗,  called the (covariant) dual basis of the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  80  2  f  1  1  2  1  er81  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Dual basis  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If �n  i=1λiπei = 0, then 0 = (�n  i=1λiπei)(⃗ej) = �n  i=1λiπei(⃗ej) = �n  i=1λiδij = λj for all j,  thus (πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n is a family of n independent vectors in E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then let ℓ ∈ E∗ and m = �  i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei)πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus m ∈ E∗ (since E∗ is a vector space), and m(⃗ej) = �  i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei)(πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej) = �  i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei)δij = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej), thus  m = ℓ, thus ℓ = �  i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei)πei, thus Vect{(πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n} span E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n is a basis in E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  dim E∗ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Use duality notations if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Following example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The size of a child is represented on a wall by a bipoint vector ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And English observer chooses the foot as unit of length, represented by a vertical bipoint vector which he  names ⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And then defines the linear form πe : ⃗R → R by πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus πe is a measuring instrument,  which gives s = πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = the size of the child in foot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗u = s⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Let (⃗ai) and (⃗bi) be bases and let (πai) and (πbi) be the dual bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  If ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, ⃗bi = λ⃗ai,  then  ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, πbi = 1  λ πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  (With duality notations, bi = 1  λ ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' πbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = δij = πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗bj  λ = 1  λ πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj for all j (since πai is linear), thus πbi = 1  λ πai, true for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Example: aeronautical units  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 International aeronautical units: Horizontal length = nautical mile (NM), altitude =  English foot (ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Application: An air traffic controller chooses the point O = the position of its control  tower, and a plane p is located thanks to the bipoint vector ⃗x = −→  Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the traffic controller chooses ⃗e1 =  the vector of length 1 NM oriented South (first runway), ⃗e2 = the vector of length 1 NM oriented Southwest  (second runway), ⃗e3 = the vertical vector of length 1 ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus his referential is R = (O, (⃗e1,⃗e2,⃗e3)), and  his dual basis (πe1, πe2, πe3) is defined by πei(⃗ej) = δij for all i, j, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' He writes ⃗x = �n  i=1xi⃗ei ∈ ⃗Rn,  so that x1 = πe1(⃗x) = the distance to the south in NM, x2 = πe2(⃗x) = the distance to the southwest  in NM, x3 = πe3(⃗x) = the altitude in ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here the basis (⃗ei) is not a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This non Euclidean basis (⃗ei) is however vital if you take  a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A Euclidean basis is not essential to life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See next remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 The metre is the international unit for NASA that launched the Mars Climate Orbiter  probe, and the foot is the international vertical unit for aviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And for the Mars Climate Orbiter landing  procedure, NASA (uses the metre) asked Lockheed Martin (uses the foot) to do the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Result?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Mars Climate Orbiter space probe burned in the Martian atmosphere because of λ ∼ 3 times too  high a speed during the landing procedure: One metre is λ ∼ 3 times one foot, and someone forgot it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Although NASA and Lockheed Martin used a Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But not the same (one based on  a metre, and one based on the foot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Objectivity and covariance can be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Matrix representation of a linear form  Let ℓ ∈ E∗, let (⃗ei) be a basis: The components of ℓ are the n reals  ℓi := ℓ(⃗ei) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei,  and  [ℓ]|⃗e = ( ℓ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓn )  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  is the row matrix of ℓ,  called the matrix of ℓ relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus,  if ⃗x  ∈  E  and  ⃗x =clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �n  i=1xi⃗ei =dual �n  i=1xi⃗ei, then  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  ℓixi  dual  =  n  �  i=1  ℓixi = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗e  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  with usual matrix computation rules (a 1 ∗ n matrix times a n ∗ 1 matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular for the dual basis (πei) = (ei) (classical and duality notations),  [πej]|⃗e = [ej]|⃗e = (0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 0  1  ����  jth position  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 0)  (= row matrix = [⃗ej]T  |⃗e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Thus we have, with classical and duality notations,  ℓ clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  ℓi πei  dual  =  n  �  i=1  ℓi ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  81  82  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Einstein convention  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 Relative to a basis, a vector is represented by a column matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2), and a linear  form by a row matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This enables:  The use of matrix calculation to compute ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗e, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11), not to be confused with an  inner dot product calculation ⃗x • ⃗y relative to an inner dot product in E for ⃗x, ⃗y ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Not to confuse the “nature of objects”: Relative to a basis, a (contravariant) vector is a mathematical  object represented by a column matrix, while a linear form (covariant vector) is a mathematical object  represented by a row matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Example: Thermodynamic  Consider the Cartesian space ⃗R2 = {(T, P) ∈ R×R} = {(temperature,pressure)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' There is no meaningful  inner dot product in this ⃗R2: What would  √  T 2+P 2 mean (Pythagoras: Can you add Kelvin degrees and  kg/(m·s2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, in thermodynamics, the (covariant) dual bases are the main ingredient for calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', in the Cartesian product ⃗R2 = R × R consider the basis ( ⃗E1=(1, 0), ⃗E2=(0, 1)) (after a choice  of temperature and pressure units);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗X ∈ ⃗R2, ⃗X = T ⃗E1 + P ⃗E2 =noted (T, P), and let (πE1, πE2) =  (E1, E2) =noted (dT, dP) be the (covariant) dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The first principle of thermodynamics tells that  the density α of internal energy is an exact differential form: ∃U ∈ C1( ⃗R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α = dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, at any  ⃗X0 = (T0, P0),  α( ⃗X0) = dU( ⃗X0) = ∂U  ∂T ( ⃗X0) dT + ∂U  ∂P ( ⃗X0) dP  and  [dU( ⃗X0)]| ⃗E =  � ∂U  ∂T ( ⃗X0)  ∂U  ∂P ( ⃗X0)  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And we have the first order Taylor expansion in the vicinity of ⃗X0,  U( ⃗X0 + δ ⃗X) = U( ⃗X0) + dU( ⃗X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δ ⃗X + o(δ ⃗X)  = U(T0, P0) + δT ∂U  ∂T (T0, P0) + δP ∂U  ∂T (T0, P0) + o((δT, δP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  Matrix computation: Column matrices for vectors, row matrices for linear forms:  [ ⃗E1]| ⃗E =  �  1  0  �  ,  [ ⃗E2]| ⃗E =  �  0  1  �  ,  [ ⃗X0]| ⃗E =  �  T0  P0  �  ,  [δ ⃗X]| ⃗E =  �  δT  δP  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  [E1]| ⃗E = [dT]| ⃗E = ( 1  0 ) ,  [E2]| ⃗E = [dP]| ⃗E = ( 0  1 ) ,  [dU]| ⃗E = ( ∂U  ∂T  ∂U  ∂P )  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  give  dU( ⃗X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δ ⃗X =  � ∂U  ∂T ( ⃗X0)  ∂U  ∂P ( ⃗X0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  δT  δP  �  = ∂U  ∂T ( ⃗X0)δT + ∂U  ∂P ( ⃗X0)δP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  This is a “covariant calculation” (in particular no inner dot product has been used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Einstein convention  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  When you work with components (after a choice of a basis), the goal is to visually differentiate a lin-  ear form from a vector (to visually differentiate covariance from contravariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Framework: a finite  dimension vector space E, dim E = n, and duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Einstein Convention:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A basis in E (contravariant) is written with bottom indices: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (⃗ei) is a basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A vector ⃗x ∈ E (contravariant) has its components relative to (⃗ei) (quantification) written with top  indices: ⃗x = �n  i=1xi⃗ei, and is represented by the column matrix [⃗x]|⃗e =  �  �  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  xn  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Classical notations:  ⃗x = �n  i=1xi⃗ei, and column matrix of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A basis in E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) (covariant) is written with top indices: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (ei) ∈ E∗n is the dual basis of  the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Classical notations: (πei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A linear form ℓ ∈ E∗ (covariant) has its components relative to (ei) (quantification) written with bottom  indices: ℓ = �n  i=1ℓiei, and its matrix representation is the row matrix [ℓ]|⃗e = ( ℓ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  82  83  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of basis formulas  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' You can also omit the sum sign � when there are repeated indices at a different position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �n  i=1xi⃗ei =noted xi⃗ei, and �n  i=1Lij⃗ei =noted Li  j⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In fact, before computers and word processors, to  print �n  i=1 was not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But with LATEX this is no more a problem, so in this manuscript the sum  sign � is not omitted (and some confusions are avoided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 Einstein’s convention is not mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Arnold doesn’t use it when he doesn’t  need it, or when it makes reading difficult, or when it induces misunderstandings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In classical mechanics,  Einstein’s convention may induce more confusion than understandings, and may be misused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' so it is  better not to use it: Golden rule: Use classical notations when in doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Do not mistake yourself  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Einstein’s convention is just meant not to confuse a linear function with a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It only deals with quantification relative to a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Classical notations are as good as duality notations, even you are told that classical notations cannot  detect obvious errors in component manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But duality notations can be misused in classical  mechanics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the paradigmatic example of the vectorial dual basis, correctly treated at § F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And thus  add confusion to the confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The convention does not admit shortcuts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with a metric: g(⃗u,⃗v) = �n  i,j=1gijuivj shows the observer  dependence on a choice of a basis thanks to the gij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And even if gij = δij you cannot write g(⃗u,⃗v) =  �n  i,j=1uivj: You have to write g(⃗u,⃗v) = �n  i,j=1gijuivj: Unmissable in physics because you need to see  the metric and bases in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Golden rule: Return to classical notations if in doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Einstein’s convention can add confusions, un-  truths, misinterpretations, absurdities, misuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Change of basis formulas  E being a finite dimension vector space, dim E = n, let (⃗eold,i) and (⃗enew,i) be two bases in E, and let  (πold,i) and (πnew,i) be the dual bases in E∗, written (ei  old) and (ei  new) with duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Change of basis endomorphism and transition matrix  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 The change of basis endomorphism P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) from (⃗eold,i) to (⃗enew,i) is the endo-  morphism (= the linear map E → E) defined by, for all j ∈ [1, n]N,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗eold,j = ⃗enew,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 The transition matrix from (⃗eold,i) to (⃗enew,i) is the matrix P := [P]|⃗eold = [Pij] of the  endomorphism P relative to the basis (⃗eold,i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' defined by, for all j,  ⃗enew,j = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗eold,j =  n  �  i=1  Pij ⃗eold,i,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗enew,j]|⃗eold = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗eold,j]|⃗eold =  �  �  �  P1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Pnj  �  �  � ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', [⃗enew,j]|⃗eold is the j-th column of P = [P]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notations: ⃗enew,j = �n  i=1P ij ⃗eold,i and  P := [P]|⃗eold = [P ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Apart from the classical and notations, you may find other “component type” notations:  ⃗enew,j =  n  �  i=1  Pij ⃗eold,i =  n  �  i=1  (Pj)i ⃗eold,i =  n  �  i=1  P i  j ⃗eold,i =  n  �  i=1  (Pj)i ⃗eold,i,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Pij = (Pj)i = P ij = (Pj)i are four notations for the i-th component of ⃗ej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗enew,j]|⃗eold =  �  �  �  P1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Pnj  �  �  � =  �  �  �  (Pj)1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Pj)n  �  �  � =  �  �  �  P 1  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  P n  j  �  �  � =  �  �  �  (Pj)1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Pj)n  �  �  �  (= the j-th column of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  83  84  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of basis formulas  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Inverse of the transition matrix  The inverse endomorphism Q := P−1 ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19), is given by, for all j ∈ [1, n]N,  ⃗eold,j = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗enew,j  (= P−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗enew,j),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Q is change of basis endomorphism from (⃗enew,i) to (⃗eold,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And Q := [Q]|⃗enew = [Qij] is the transition  matrix from (⃗enew,i) to (⃗eold,i):  ⃗eold,j =  n  �  i=1  Qij⃗enew,i,  [⃗eold,j]|⃗enew =  �  �  �  Q1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Qnj  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Q is change of basis endomorphism from (⃗enew,i) to (⃗eold,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Use other notation if you prefer: Qij = (Qj)i = Qij = (Qj)i  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19  Q = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗enew,j = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗eold,j = �n  i=1Pij⃗eold,i = �n  i=1Pij(�n  k=1Qki⃗enew,k) = �n  k=1(�n  i=1QkiPij)⃗enew,k =  �n  k=1(Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P)kj⃗enew,k for all j, thus (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P)kj = δkj for all j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 Prove  �  [P]|⃗eold = [P]|⃗enew = P,  [Q]|⃗enew = [Q]|⃗eold = Q,  �  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗enew,j = �n  i,j=1Pij⃗enew,i  (= �n  i,j=1P ij⃗enew,i = �n  i,j=1(Pj)i⃗enew,i),  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗eold,j = �n  i,j=1Qij⃗eold,i  (= �n  i,j=1Qij⃗eold,i = �n  i,j=1(Qj)i⃗eold,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Z  = [Zij] = [P]|⃗enew means P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗enew,j  = �  i Zij⃗enew,i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗enew,j  = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (�n  i=1Zij⃗enew,i) =  �n  i=1ZijQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗enew,i = �n  i=1Zij(�n  k=1Qki⃗enew,k) = �n  k=1(�n  i=1QkiZij)⃗enew,k = �n  k=1(Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Z)kj⃗enew,k for all j, thus  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Z)kj = δkj for all j, k, thus Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Z = I, thus Z = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for Q, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 P T ̸= P −1 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (⃗eold,i) = (⃗ai) is a foot-built Euclidean basis, and (⃗enew,i) =  (⃗bi) is a metre-built Euclidean basis, and ⃗bi = λ⃗ai for all i (the basis are “aligned”), so P = λI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  P T = λI and P −1 = 1  λI ̸= P T , since λ =  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3048 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus it is essential not to confuse P T and P −1 (not  to confuse covariance with contravariance), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the Mars Climate Orbiter crash (remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Change of dual basis  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 (πnew,i) and (πold,i) being the dual bases of (⃗enew,i) and (⃗eold,i), for all i ∈ [1, n]N,  πnew,i =  n  �  j=1  Qijπold,i,  and  [πnew,i]|⃗eold = ( Qi1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Qin )  (i-th row of Q),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  to compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) (matrices of linear forms are row matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notations: ei  new = �n  j=1Qijej  old and [ei  new]|⃗eold = ( Qi1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Qin ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' πnew,i(⃗eold,k) =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24) πnew,i(�  j Qjk⃗enew,j) = �  j Qjk πnew,i(⃗enew,j) = �  j Qjk δij = Qik, and  �  j Qijπold,j(⃗eold,k) = �  j Qijδjk = Qik, true for all i, k, thus πnew,i = �  j Qij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Change of coordinate system for vectors and linear forms  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23 Let ⃗x ∈ E and ℓ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  [⃗x]|⃗enew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗eold  (contravariance formula for vectors: between column matrices),  [ℓ]|⃗enew = [ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  (covariance formula for linear forms: between row matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  And the scalar value ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x is computed indifferently with one or the other basis (objective result):  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = [ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗eold = [ℓ]|⃗enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  84  85  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Bidual basis (and contravariance)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗x = �  j xj⃗eold,j = �  i yi⃗enew,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We have ⃗x = �  j xj⃗eold,j = �  j xj(�n  i=1Qij⃗enew,i) =  �  ij Qijxj⃗enew,i, thus yi = �  j Qijxj for all i, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And ℓ = �  j mjπnew,j = �  i ℓiπold,i =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27) �  ij ℓiPijπnew,j gives mj = �  i ℓiPij for all j, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Use duality notations if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Thus [ℓ]|⃗enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗enew = ([ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗eold) = [ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗eold, hence (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28) and ⃗x = �  j xj⃗eold,j = �  i yi⃗enew,i give yi = �n  j=1Qijxj, which means: yi is the  function defined by yi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn) = �n  j=1Qijxj, thus Qij = ∂yi  ∂xj (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Similarly with Pij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Which is  written  Qij = ∂yi  ∂xj  ,  and  Pij = ∂xi  ∂yj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  (Use duality notations if you prefer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Qij = ∂yi  ∂xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Bidual basis (and contravariance)  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24 The dual of E∗ is E∗∗ := (E∗)∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and is named the bidual of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E∗∗ is also called the space of contravariant vectors = the space of directional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25 Let (⃗ei) be a basis in E, let (πei) be its dual basis (basis in E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The dual basis (∂i)  of (πei) is called the bidual basis of (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Duality notations: (πei) = (ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (The notation ∂i refers to the derivation in the direction ⃗ei: ∂i(df(⃗x)) = df(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei = ∂f  ∂xi (⃗x), see § S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Thus, the linear form ∂i ∈ E∗∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) are characterized by, for all j,  ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πej = δij = πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei,  so  ℓ =  n  �  i=1  ℓiπei  iff  ℓi = ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  Indeed, ∂i(ℓ) = ∂i(�n  j=1ℓjπej) = �n  j=1ℓj∂i(πej) = �n  j=1ℓjδij = ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Duality notation: ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ej = δj  i = ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei  and ℓ = �n  i=1ℓiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark: With the natural canonical isomorphism J :  �  E → E∗∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  ⃗u → J (⃗u), where J (⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u, ∀ℓ ∈ E∗  �  see (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9), we can identify ⃗u and J (⃗u) (observer independent identification), thus ∂i = J (⃗ei) =noted ⃗ei,  and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31) reads (usual notation in differential geometry) ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πej = δij and ℓi = ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Bilinear forms  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Let E and F be vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26 • A bilinear form is a 2-multilinear form β(·, ·) :  �  E × F → R  (⃗u, ⃗w) → β(⃗u, ⃗w)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, β(⃗u1 +λ⃗u2, ⃗w) = β(⃗u1, ⃗w)+λβ(⃗u2, ⃗w) and β(⃗u, ⃗w1 +λ⃗w2) = β(⃗u, ⃗w1)+λβ(⃗u, ⃗w2) for all ⃗u, ⃗u1, ⃗u2 ∈ E,  ⃗w, ⃗w1, ⃗w2 ∈ F, λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is the set of bilinear forms E × F → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If (ℓ, m) ∈ E∗ × F ∗, then the bilinear form ℓ ⊗ m ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is defined by  (ℓ ⊗ m)(⃗u, ⃗w) = ℓ(⃗u)m(⃗w)  (= (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w))  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  for all (⃗u, ⃗w) ∈ E × F, and is called an elementary bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The transposed of a bilinear form  (Warning: Not to be confused with the subjective definition of a transposed of a linear map which requires  inner dot products to be defined, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27 If β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) then its transposed is the bilinear form βT ∈ L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) defined by,  for all (⃗w, ⃗u) ∈ F × E,  βT (⃗w, ⃗u) = β(⃗u, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  (This definition is observer independent: no basis or inner dot product is required in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  85  86  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Bilinear forms  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Symmetric and definite positive bilinear forms  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28 Here F = E (no choice), and β ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  β is semi-positive, iff for all ⃗u ∈ E,  β(⃗u, ⃗u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  β is definite positive, iff for all ⃗u ̸= ⃗0,  β(⃗u, ⃗u) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  β is symmetric iff βT = β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all ⃗u,⃗v ∈ E,  β(⃗u,⃗v) = β(⃗v, ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Inner dot product, and metric  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29 • An “inner dot product” (or “scalar inner dot product”, or “inner scalar product”, or  “inner product”) in a vector space E is a bilinear form β =noted g =noted g(·, ·) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) which is  symmetric and definite positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And then (for inner dot products)  g(·, ·) noted  =  (·, ·)g  noted  = g ·,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  g(⃗u, ⃗w) = (⃗u, ⃗w)g  noted  =  ⃗u •g ⃗w, ∀⃗u, ⃗w ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Then two vectors ⃗u, ⃗w ∈ E are (·, ·)g-orthogonal iff (⃗u, ⃗w)g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And the associated norm with (·, ·)g is the function ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g : E → R+ given by, for all ⃗u ∈ E,  ||⃗u||g =  �  (⃗u, ⃗u)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  (To prove that it is a norm, use the Cauchy–Schwarz inequality (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  An “semi-inner dot product” (·, ·)g (or “semi-scalar inner dot product”) in a vector space E is a  bilinear form β =noted g(·, ·) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) which is symmetric and semi-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the associated  semi-norm is given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30 (Cauchy–Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') (·, ·)g being an inner dot product in E,  ∀⃗u, ⃗w ∈ E,  |(⃗u, ⃗w)g| ≤ ||⃗u||g||⃗w||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  And |(⃗u, ⃗w)g| = ||⃗u||g||⃗w||g iff ⃗u and ⃗w are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let p(λ) = ||⃗u+λ⃗w||2  g = (⃗u+λ⃗w, ⃗u+λ⃗w)g, so p(λ) = aλ2 + bλ + c where a = ||⃗w||2  g, b = 2(⃗u, ⃗w)g  and c = ||⃗u||2  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With p(λ) ≥ 0 (since(·, ·)g is positive), we get b2 − 4ac ≥ 0, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39), and p(λ) = 0 iff  ⃗u+λ⃗w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31 (Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') With Rn our usual affine geometric space, n = 1, 2 or 3, and ⃗Rn = the usual  associated vector space made of bipoint vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Ω ⊂ Rn be open in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A metric in Ω is a C∞  function g :  � Ω → L(⃗Rn, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  p → g(p) noted  =  gp  �  such that gp is an inner dot product in ⃗Rn at each p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Particular Case: When the gp is independent of p (general case in continuum mechanics), a metric is  simply called a inner dot product (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a Euclidean metric is called a Euclidean dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (In a differentiable manifold Ω, a metric is a C∞ �0  2  �  tensor g s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' g(p) is an inner dot product at each  p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A Riemannian metric is a metric s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' g(p) is a Euclidean dot product at each p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Quantification: Matrice [βij] and tensorial representation  dim E = n, dim F = m, β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), (⃗ai) is a basis in E which dual basis is (πai), (⃗bi) is a basis in F  which dual basis is (πbi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (With duality notations, (πai) = (ai) and (πbi) = (bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32 The components of β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) relative to the bases (⃗ai) and (⃗bi) are the nm reals  βij := β(⃗ai,⃗bj),  and  [β]|⃗a,⃗b = [βij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  is the matrix of β relative to the bases (⃗ai) and (⃗bi), simply written [βij] if the bases are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And if F = E and (⃗bi) = (⃗ai) then [β]|⃗a,⃗b =noted [β]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  86  87  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Linear maps  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33 A bilinear form β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is known as soon as the nm scalars βij = β(⃗ai,⃗bj)  are known, and, for all (⃗u, ⃗w) ∈ E × F,  β(⃗u, ⃗w) = [⃗u]|⃗a  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [β]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗b,  written  β(⃗u, ⃗w) = [⃗u]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  so  β =  n  �  i=1  m  �  j=1  βijπai ⊗ πbj,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  and a basis in L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is made of the nm functions πai ⊗ πbj, and dim L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Duality  notations: β = �n  i=1  �m  j=1βijai ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' β being bilinear, ⃗u = �n  i=1ui⃗ai and ⃗w = �n  j=1wj⃗bj give β(⃗u, ⃗w) = �n  i,j=1uiwjβ(⃗ai,⃗bj) =  �n  i,j=1uiβijwj = ([⃗u]|⃗a)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [β]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗b, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, if the βij are known, then b is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (πai ⊗ πbj)(⃗ak,⃗bℓ) =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32) (πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ak)(πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bℓ) = δikδjℓ (all the elements of the matrix [πai ⊗ πbj]|⃗a,⃗b are  zero except the element at the intersection of row i and column j which is equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (πai ⊗  πbj)(⃗u, ⃗w) =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32) (πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)(πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = uiwj, thus β(⃗u, ⃗w) = �n  i,j=1βijuiwj = �n  i,j=1βij(πai ⊗ πbj)(⃗u, ⃗w),  thus β := �n  i,j=1βij(πai ⊗ πbj), thus the πai ⊗ πbj span L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And �  ij λij(πai ⊗ πbj) = 0 implies  0 = (�  ij λij(πai ⊗ πbj))(⃗ak,⃗bℓ) = �  ij λij(πai ⊗ πbj)(⃗ak,⃗bℓ) = λkℓ = 0 for all k, ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the πai ⊗ πbj are  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (πai ⊗ πbj) is a basis in L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and dim(L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)) = nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Duality notations:  β(⃗u, ⃗w) = �n  i,j=1βijuiwj and β := �n  i,j=1βij ai ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34 dim E = dim F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [β]|⃗a,⃗b =  �  1  2  0  3  �  means β(⃗a1,⃗b1) = β11 = 1, β(⃗a1,⃗b2) = β12 = 2,  β(⃗a2,⃗b1) = β21 = 0, β(⃗a2,⃗b2) = β22 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And β12 = [⃗a1]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[β]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗b2]|⃗b = ( 1  0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  1  2  0  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  0  1  �  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35 Let β ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), let (⃗ai) and (⃗bi) be two bases in A, and let λ ∈ R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  if, ∀i ∈ [1, n]N, ⃗bi = λ⃗ai,  then  [β]|⃗b = λ2[β]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  (A change of unit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' from foot to metre, has a “big” influence on the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗bi = λ⃗ai give β(⃗bi,⃗bj) = β(λ⃗ai, λ⃗aj) = λ2β(⃗ai,⃗aj) (bilinearity), thus [β]|⃗b = λ2[β]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36 Prove  [βT ]⃗b,⃗a = ([β]⃗a,⃗b)T ,  written  [βT ] = [β]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let[β]⃗a,⃗b = [βij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m and [βT ]⃗b,⃗a = [γij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have γij = βT (⃗bi,⃗aj) = β(⃗aj,⃗bi) = βji, qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Linear maps  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Let E and F be vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37 • A function L : E → F is linear iff L(⃗u1 + λ⃗u2) = L(⃗u1) + λL(⃗u2) for all ⃗u1, ⃗u2 ∈ E  and all λ ∈ R (distributivity type relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (distributivity notation):  L(⃗u) noted  =  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  so  L(⃗u1 + λ⃗u2) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(⃗u1 + λ⃗u2) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1 + λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  NB: This dot notation L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u is a linearity notation (distributivity type notation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is an “outer” dot product  between a (linear) function and a vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is not an “inner” dot product since L and ⃗u don’t belong to  a same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is not a matrix product since no basis has been introduced yet (no quantification has  been done yet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the set of linear maps E → F (vector space, subspace of (F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If F = E then a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is called an endomorphism in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (If F = R then a linear map E → R is called a linear form, and E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is the dual of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  87  88  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Linear maps  Vocabulary: Let Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) be the space of linear invertible linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If E is a finite dimension vector  space, dim E = n, then, in algebra, the set (Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E), ◦) of linear maps equipped with the composition  rule is named GLn(E) = “the linear group” (it is indeed a group, easy check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the “linear group” of  n ∗ n invertible matrices is GLn(Mn) := (Li(Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Mn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Li(Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Mn) with the matrix product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38 (Math exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let E = (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||E) and F = (F, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||F ) be Banach spaces, and let  Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) be the space of invertible linear continuous maps E → F, with its usual norm ||L|| =  sup||⃗x||E=1 ||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x||F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Z :  �  Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  L → L−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove dZ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M = −L−1 ◦ M ◦ L−1, for all  M ∈ Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Recall: In finite dimension, a linear map is always continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider limh→0  Z(L+hM)−Z(L)  h  = limh→0  (L+hM)−1−L−1  h  ( =noted dZ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M if the limit exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With  N = L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M we have L + hM = L(I + hN), and (I + hN) is invertible as soon as ||hN|| < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' h <  1  ||N|| =  1  ||L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M||, its inverse being I − hN + h2N − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Neumann serie);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus I + hN = I − hN + o(h), and  (L + hM)−1 = (I + hN)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1 = (I − hN + o(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1 = L−1 − hN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1 + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  (L+hM)−1−L−1  h  =  L−1−hN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1+o(h)−L−1  h  = −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1 + o(1) −→h→0 −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification: Matrices [Lij] = [Lij]  dim E = n, dim F = m, L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), (⃗ai) is a basis in E which dual basis is (πai), (⃗bi) is a basis in F  which dual basis is (πbi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (With duality notations, (πai) = (ai) and (πbi) = (bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39 The components of a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) relative to the bases (⃗ai) and (⃗bi) are  the nm reals named Lij (classical notation) = Lij (duality notation), which are the components of the  vectors L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj relative to the basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' That is:  �  �  �  �  �  �  �  �  �  �  �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  m  �  i=1  Lij⃗bi,  dual not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  m  �  i=1  Li  j⃗bi,  �  �  �  �  �  �  �  �  �  �  �  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj]|⃗b  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  �  �  �  L1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lmj  �  �  �  dual  =  �  �  �  L1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Lmj  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  And  [L]|⃗a,⃗b  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  = [Lij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  dual  = [Li  j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  is the matrix of L relative to the bases (⃗ai) and (⃗bi) (so [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj]|⃗b is the j-th column of [L]|⃗a,⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Particular case: If E = F (so L is an endomorphism) and if (⃗bi) = (⃗ai) then [L]|⃗a,⃗a =noted [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40 n = m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a,⃗b =  �  1  2  0  3  �  means L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1 = ⃗b1 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2 = 2⃗b1 + 3⃗b2 (column reading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here L11=1, L12=2, L21=0, L22=3 (duality notations: L11=1, L12=2, L21=0, L22=3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And L being linear, for all ⃗u ∈ E, ⃗u = �n  j=1uj⃗aj = �n  j=1uj⃗aj, we get, thanks to linearity,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  m  �  i=1  n  �  j=1  Lijuj⃗bi  dual  =  m  �  i=1  n  �  j=1  Li  juj⃗bi,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u]|⃗b = [L]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  Shortened notation: [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u] = [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u] when the bases are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41 A linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is known as soon as the n vectors L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj are known,  j ∈ [1, n]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the linear maps Lij ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) defined by Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ = δjℓ⃗bi (all the elements of the matrix  [Lij]|⃗a,⃗b vanish except the element at the intersection of row i and column j which is equal to 1), for  i, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', m, constitute a basis ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Duality notations: Lij =noted Lij, and  Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ = δj  ℓ⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') So, dim(L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)) = nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗u ∈ E and ⃗u = �  k uj⃗aj give L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = �  j ujL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj, since L is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus L is known iff the n vectors  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj are known for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ak = �  i Lik⃗bi together with �  ij LijLij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ak = �  ij Lijδjk⃗bi =  �  i Lik⃗bi, for all k, thus L = �  ij LijLij, thus the Lij span L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And �m  i=1  �n  j=1λijLij = 0 implies,  for all ℓ, ⃗0 = �m  i=1  �n  j=1λijLij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ = �m  i=1  �n  j=1λijδjℓ⃗bi = �m  i=1λiℓ⃗bi, thus λiℓ = 0, for all i and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  the Lij are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (Lij) i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m is a basis in L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  88  89  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Transposed matrix  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42 If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) (endomorphism), if (⃗ai) is a basis in E, prove:  if λ ∈ R∗ and ⃗bi = λ⃗ai ∀i ∈ [1, n]N,  then  [L]|⃗b = [L]|⃗a,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a change of unit has not influence on the matrix of an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Check with the change of  basis formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' NB: To compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43): Covariance and contravariance should not be confused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �n  i=1Laij⃗ai and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = �n  i=1Lbij⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then �n  i=1Lbij⃗bi = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(λ⃗aj) = λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  λ�n  i=1Laij⃗ai = λ�n  i=1Laij  ⃗bi  λ = �n  i=1Laij⃗bi, thus Lbij = Laij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of basis formula: [L]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P with P = λI here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Trace of an endomorphism  Let E be a vector space, dim E = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗u ∈ E and ℓ ∈ E∗ and call L ⃗w,ℓ ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) the endomorphism,  called an elementary endomorphism, defined by  L ⃗w,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u := ⃗w(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43 The trace of the endomorphism L ⃗w,ℓ is the real  Tr(L ⃗w,ℓ) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51)  And the trace operator is the linear map Tr :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) → R  L → Tr(L)  �  defined on elementary endomor-  phisms ℓ ⊗ ⃗w by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44 Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The real Tr(L) is objective (is intrinsic to L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is independent  of any basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (quantification) if (⃗ei) is a basis and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei for all j, then  Tr(L) =  n  �  i=1  Lii (∈ R),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Tr(L) is the trace of the matrix [L]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Duality notations L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei and Tr(L) = �n  i=1Lii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Tr(L ⃗w,ℓ) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w is a real that can be considered by any observer, and which value is the same for  all observers, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29): It is objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ai) be a basis and (πai) be its (covariant) dual basis, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �  i Lij⃗ai,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L][⃗a = [Lij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And we have (�  ik LikL⃗ai,πak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �  ik LikL⃗ai,πak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50) �  ik Lik⃗ai(πak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj) =  �  ik Lik⃗aiδkj = �  i Lij⃗ai, thus L = �  ij LijL⃗ai,πaj (sum of elementary endomorphisms), thus, Tr being  linear Tr(L) = �  ij LijTrL⃗ai,πaj = �  ij Lijδji = �  i Lii, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45 Check with the change of basis formula that Tr(L) is an invariant (the same value for  all observers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ai) and (⃗bi) be two bases, P = [Pij] be the transition matrix from (⃗ai) to (⃗bi), Q = P −1, [L][⃗a =  [(La)ij], [L][⃗b = [(Lb)ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have [L][⃗b =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='104) P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L][⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Lb)ij = �  kℓ Qik(La)kℓPℓj, thus �  i(Lb)ii =  �  ikℓ Qik(La)kℓPℓi = �  kℓ(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Q)ℓk(La)kℓ = �  kℓ δℓk(La)kℓ = �  k(La)kk, qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Alternative definition with one-one tensors: see § Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  Transposed matrix  The definition can be found in any elementary books, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Strang [18]: If M = [Mij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n is an m ∗ n  matrix then its transposed is the n ∗ m matrix M T = [(M T )ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m defined by  (M T )ij := Mji  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53)  (exchange rows and columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', M =  �  1  2  3  4  �  gives M T =  �  1  3  2  4  �  , and (M T )12=M21=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And M is symmetric iff M T = M (this requires m=n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='N = [�  k MikNkj] i  j gives (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='N)T = [�  k MjkNki] i  j = [�  k(N T )ik(M T )kj] i  j = N T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46 Prove: If M is an n ∗ n invertible matrix then M T is invertible and (M T )−1 = (M −1)T  ( =noted M −T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if moreover M is symmetric, then M −1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M −1 = I gives (M −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M T = IT = I, thus M T is invertible with (M T )−1 = (M −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Moreover if  M = M T then M −1 = (M −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  89  90  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A transposed endomorphism: depends on a chosen inner dot product  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10  A transposed endomorphism: depends on a chosen inner dot product  Not to be confused with the transposed of a matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And not to be confused with the  transposed of a bilinear form (observer independent), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, a transposed of a linear  map depends on the observer who use it (depends on the choice of an inner dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition (requires an inner dot product: Not objective)  Let E be a finite dimensional vector space equipped with an inner dot product g(·, ·) = (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47 The transpose of an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) relative to (·, ·)g is the endomorphism  LT  g ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) defined by  ∀⃗x, ⃗y ∈ E,  (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)g,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y) •g ⃗x = ⃗y •g (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  (It depends on (·, ·)g, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') If (·, ·)g is an imposed Euclidean dot product (isometric framework)  then LT  g =noted LT , thus (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y) • ⃗x = ⃗y • (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48 (Math exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The existence and uniqueness of LT  g is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' proved with a basis in E  when E is finite dimensional, see next § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' More general proof: Prove: If (E, (·, ·)g) is an infinite  dimensional Hilbert space and if L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is continuous, then LT  g exists, is unique, and is continuous  (apply the Riesz representation theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗y ∈ E, then let ℓ⃗yg : ⃗x ∈ E → ℓ⃗yg(⃗x) := (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ℓ⃗yg is linear (trivial since L is linear and (·, ·)g  is bilinear) and continuous: |ℓ⃗yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x| ≤ ||⃗y||g||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x||g ≤ ||⃗y||g||L|| ||⃗x||g gives ||ℓ⃗yg||E∗ ≤ ||L|| ||⃗y||g < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗ℓ⃗yg ∈ E  be the (·, ·)g-Riesz representation of ℓ⃗yg ∈ E∗: ℓ⃗yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗ℓ⃗yg, ⃗x)g for all ⃗x, with ||⃗ℓ⃗yg||g = ||ℓ⃗yg||E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have thus  defined LT  g : ⃗y ∈ E → LT  g (⃗y) := ⃗ℓ⃗yg ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with (LT  g (⃗y), ⃗x)g = (⃗ℓ⃗yg, ⃗x)g = ℓ⃗yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)g, thus LT  g is linear (since  (·, ·)g is bilinear) and continuous: ||LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y||g = ||⃗ℓ⃗yg||g = ||ℓ⃗yg||E∗ ≤ ||L|| ||⃗y||g gives ||LT  g || ≤ ||L||L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='E) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49 Recall: The transposed βT of a bilinear form β is objective, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33): We don’t need  any tool like an inner dot product to define βT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Not to be confused with: The transposed LT  g =noted LT  of a linear map L is subjective: It depends on a choice of an inner dot products (·, ·)g by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In  particular it is dangerous to represent a linear map in a basis with its “bilinear tensorial representation”  when dealing with the transposed: L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is naturally canonically represented by the bilinear form  βL ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and thus (βL)T ∈ L(E, F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  Li  j⃗bi gives βL =  n  �  i,j=1  Li  j⃗bi ⊗ aj, thus (βL)T (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  =  n  �  i,j=1  Lj  iai ⊗⃗bj,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55)  while LT ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is naturally canonically represented by the bilinear form β(LT ) ∈ L(E∗, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and  LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj =  n  �  i=1  (LT )i  j⃗ai gives b(LT ) =  n  �  i,j=1  (LT )i  j⃗ai ⊗ bj, thus  β(LT ) ̸= (βL)T  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56)  for two reasons: 1- ⃗ai ⊗ bj ̸= ai ⊗ ⃗bj, and 2- LT := LT  gh depends on chosen inner dot products (·, ·)g  and (·, ·)h by observers in E and F, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='73): (LT )ij =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='73) �n  k,ℓ=1([g]−1)ikLℓk hℓj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While (βL)T is  independent of any inner dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (In fact (βL)T ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is the tensorial representation of the  adjoint L∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) of L: With L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='bj = �n  i=1(L∗)ijai we get �  (L∗) = �n  i,j=1(L∗)ijai ⊗⃗bj = (βL)T ,  see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  So in continuum mechanics it is strongly advised not to use the tensorial notation for linear maps  when dealing with transposed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' when using F T the transposed of the deformation gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification with bases  Let (⃗ei) be a basis in E, let gij := g(⃗ei,⃗ej), so [g]|⃗e := [gij] =noted [g], and let (classical notation)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  i=1  Lij⃗ei,  LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  i=1  (LT  g )ij,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [L]|⃗e = [Lij] noted  =  [L],  [LT  g ]|⃗e = [(LT  g )ij] noted  =  [LT  g ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='57)  90  91  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A transposed endomorphism: depends on a chosen inner dot product  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54) gives [⃗x]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y] = [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y] for all ⃗x, ⃗y, thus  [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  g ] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  n  �  k=1  gik(LT  g )kj =  n  �  k=1  Lki gkj  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  [LT  g ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g] ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (LT  g )ij =  n  �  k,ℓ=1  ([g]−1)ikLℓkgℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)  To compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If and only if (⃗ei) is (·, ·)g-orthonormal then [g] = [δij] and (LT  g )ij = Lji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With  duality notations, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei, LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1(LT  g )ij, [L]|⃗e = [Lij], [LT  g ]|⃗e = [(LT  g )ij], and  n  �  k=1  gik(LT  g )k  j =  n  �  k=1  Lk  i gkj,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (LT  g )i  j =  n  �  k,ℓ=1  ([g]−1)ikLℓ  k gℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60)  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50 The last equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60)2 is also written  (LT  g )i  j =  n  �  k,ℓ=1  gikLℓ  k gℓj  when  ([g]⃗e)−1 = [gij]−1 noted  =  [gij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61)  Don’t be fooled by the notation gij, defined by [gij] := [gij]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (It is also the short notation for (g♯)ij,  see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Use classical notations to avoid misuses and misinterpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51 A bilinear form β ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) satisfies [βT ] = [β]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A linear endomorphism L ∈  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) satisfies [LT  g ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g] ̸= [L]T in general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' take [L] =  �  0  1  1  0  �  and [g] =  �  1  0  0  2  �  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So do not confuse a bilinear on E (objective) with a linear endomorphism on E (subjective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52 In ⃗R2, let (⃗e1,⃗e2) be a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let L ∈ L( ⃗R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗R2) be defined by [L]|⃗e =  �  0  1  1  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Find  two inner dot products (·, ·)g and (·, ·)h in ⃗R2 such that LT  g ̸= LT  h (a transposed endomorphism is not  unique, is not intrinsic to L, since it depends on a choice of an inner dot product by an observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Calculations with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58):  Choose (·, ·)g given by [g]|⃗e =  � 1  0  0  1  �  = [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus [LT  g ]|⃗e = [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [I] =  � 0  1  1  0  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So LT  g = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Choose (·, ·)h given by [h]|⃗e =  � 1  0  0  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus [LT  h ]|⃗e = [h]−1  |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h]|⃗e =  � 0  2  1  2  0  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So LT  h ̸= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus LT  h ̸= LT  g , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗e2 = LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1 ̸= LT  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1 = 1  2⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53 Prove: If L is invertible then LT  g is invertible, and (LT  g )−1 = (L−1)T  g (written L−T  g  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Suppose: ∃⃗y ∈ E, ⃗y ̸= ⃗0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L being invertible, ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ⃗y, with ⃗x ̸= ⃗0 since ⃗y ̸= ⃗0  and L is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y = 0 gives LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = 0, thus (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗x)g = 0, thus ||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x||2  g = 0, thus L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = 0, thus  ⃗x = 0 since L is linear bijective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus Ker(LT  g ) = {⃗0}, thus LT  g is invertible since it is an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L−1)T  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  = ((L−1)T  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)g  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  = (⃗x, (L−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)g = (⃗x, ⃗y)g = (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  g )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g, true ∀⃗x, ⃗y, thus  LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L−1)T  g = LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  g )−1, thus (L−1)T  g = (LT  g )−1 since LT  g is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54 Special case of proportional inner dot products (·, ·)a and (·, ·)b: ∃λ > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (·, ·)a =  λ2(·, ·)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove: LT  a = LT  b : Two proportional inner dot products give the same transposed endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)b = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)b = λ2(⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)a = λ2(LT  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)a = (LT  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)b, for all ⃗x, ⃗y, so LT  b = LT  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Symmetric endomorphism  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55 An endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is (·, ·)g-symmetric iff LT  g = L:  L (·, ·)g-symmetric  ⇐⇒  LT  g = L  ⇐⇒  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g = (⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)g,  ∀⃗x, ⃗y ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='62)  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56 The symmetric character of an endomorphism L is not intrinsic to the endomorphism:  It depends on (·, ·)g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See exercise A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52 where L is (·, ·)g-symmetric while it is not (·, ·)h-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  91  92  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A transposed of a linear map: depends on chosen inner dot products  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  The general flat ♭ notation for an endomorphism: Relative to a (·, ·)g  Let (·, ·)g be an inner dot product in a vector space E, and let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) (a C0 endomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='57 The bilinear form L♭  g ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) which is (·, ·)g-associated to the endomorphism  L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is defined by, for all ⃗u, ⃗w ∈ E,  L♭  g(⃗u, ⃗w) := (⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='63)  (L♭  g depends on a choice of a (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') We have thus defined the (·, ·)g-dependent operator:  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' )♭  g = Jg(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) → L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  L → Jg(L) := L♭  g,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='64)  If (·, ·)g is imposed, then L♭  g =noted L♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (The bilinearity of L♭  g is trivial since L is linear and (·, ·)g is bilinear, and the bilinear form L♭  g  continuous as soon as L and (·, ·)g are since |L♭  g(⃗u,⃗v)| ≤ ||g|| ||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u|| ||⃗v|| ≤ (||g|| ||L||) ||⃗u|| ||⃗v||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58 With the natural canonical isomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) ≃ TL ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) given by  TL(ℓ, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, and with TL =noted L, the function (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' )♭  g is the change of contravariance to covariance  mapping given by  L♭  g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Recall: The contraction of an elementary  �0  2  �  tensor ℓ1 ⊗ ℓ2 with an elementary  �1  1  �  tensor ⃗v ⊗ ℓ3  is the  �0  2  �  tensor (ℓ1 ⊗ℓ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗v ⊗ℓ3) := (ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)ℓ1 ⊗ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the contraction on any tensors is the bilinear map  defined on elementaty tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, with a basis (⃗ei) in E and its dual basis (ei) in E∗, if g = �  ij gijei⊗ej  and L = �  ij Lij⃗ei ⊗ ej then g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L = �  ijk gikLkj⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L)(⃗u, ⃗w) = �  ij uiwj(g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L)(⃗ei,⃗ej) =  �  ijk uiwjgikLkj = �  ik uigik(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)k = �  ik uigik(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)k = g(⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = L♭  g(⃗u, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: Let (⃗ei) be a basis in E, and, with duality notations motivated by the flat notation  “i top changed into i bottom” in the components Lij of L, let gij := g(⃗ei,⃗ej), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei and  L♭  g,ij = L♭  g(⃗ei,⃗ej), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with tensorial notations for calculations  g =  �  ij  gijei ⊗ ej,  L =  �  ij  Li  j⃗ei ⊗ ej,  L♭  g =  �  ij  L♭  g,ijei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='66)  So [g]|⃗e = [gij], [L]|⃗e = [Lij] and [L♭  g]|⃗e = [L♭  g,ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65) gives (or see next exercise)  [L♭  g] = [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67)  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59 Prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='63) we get L♭  g,ij = L♭  g(⃗ei,⃗ej) = (⃗ei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)g = (⃗ei,  �  k  Lk  j⃗ek)g =  �  k  Lk  jgik = ([g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L])ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60 A change of variance, here from the  �1  1  �  type tensor L to the  �0  2  �  tensor L♭  g, is necessarily  observer dependent: There is no natural canonical isomorphism between a vector space E and its dual E∗,  see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Details: Here fix ⃗w and write ℓg,⃗w(⃗u) = (⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)g (= L♭  g(⃗u, ⃗w));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ℓg,⃗w ∈ E∗ is the (·, ·)g-  representation function (linear form) of the vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ℓg,⃗w = ⃗Rg(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) where ⃗Rg is the (·, ·)g-Riesz-  representation operator (the change of variance operator, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11  A transposed of a linear map: depends on chosen inner dot products  This paragraph is needed to define the transposed of the deformation gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Not to be confused with the transposed of a matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And not to be confused with the  objective transposed of a bilinear form, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a transposed of a linear map is not objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  92  93  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A transposed of a linear map: depends on chosen inner dot products  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition (subjective)  (E, (·, ·)g) and (F, (·, ·)h) are Hilbert spaces, and L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) (which is supposed to be continuous if E  and F are infinite dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', E = ⃗Rn  t0, F = ⃗Rn  t , L = dΦt0  t (P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) = the deformation  gradient, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), (·, ·)g is the foot built Euclidean dot product chosen by the observer who made the  measurements at t0, (·, ·)h is the metre built Euclidean dot product chosen by the observer who makes  the measurements at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61 The transposed of L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) relative to (·, ·)g and (·, ·)h is the linear map LT  gh ∈  L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) defined by, for all (⃗x, ⃗y) ∈ E × F,  (LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)h,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68)  where we used the dot notation LT  gh(⃗y) =noted LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y since LT  gh is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines the map  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' )T  gh :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E)  L → (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' )T  gh(L) := LT  gh  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='69)  NB: So a linear map has an infinite number of transposed (it depends on inner dot products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation: If (·, ·)g and (·, ·)h are imposed then LT  gh =noted LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And if F = E and (·, ·)h = (·, ·)g then LT  gh = LT  g , see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification with bases  Let (⃗ai) and (⃗bi) be bases in E and F, let gij := g(⃗ai,⃗aj), hij := h(⃗bi,⃗bj), [g]|⃗a = [gij], [h]|⃗b = [hij], and  let (classical notation)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  m  �  i=1  Lij⃗bi,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [L]|⃗a,⃗b = [Lij] noted  =  [L],  LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj =  n  �  i=1  (LT  gh)ij⃗ai,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [LT  gh]|⃗b,⃗a = [(LT  gh)ij] noted  =  [LT  gh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='70)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68) gives [⃗x]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y]|⃗y = ([L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x]|⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y]|⃗b for all ⃗x, ⃗y, thus, [g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  gh]|⃗b,⃗a = ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h]|⃗b and  [LT  gh]|⃗b,⃗a = [g]−1  |⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Shortened notation:  [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT ] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  n  �  k=1  gik(LT  gh)kj =  m  �  k=1  Lki hkj,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='71)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [LT ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h] ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (LT  gh)ij =  n  �  k=1  m  �  ℓ=1  ([g]−1)ikLℓkhℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='72)  With duality notations, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei, [L]|⃗e = [Lij], LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1(LT  gh)ij, [LT  gh]|⃗e = [(LT  gh)ij], and  n  �  k=1  gik(LT  gh)k  j =  n  �  k=1  Lk  i hkj,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (LT  gh)i  j =  n  �  k,ℓ=1  ([g]−1)ikLℓ  k hℓj  ( noted  =  n  �  k,ℓ=1  (gikLℓ  k hℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='73)  (Be careful with the notation ([g]−1)ik =noted gij, see remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='62 Prove: If L is invertible then (LT  gh)−1 = (L−1)T  hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L−1)T  hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g = ((L−1)T  hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)h = (⃗x, L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)g = (⃗x, ⃗y)g = (LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  gh)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g, true ∀⃗x, ⃗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Deformation gradient symmetric: Absurd  The symmetry of a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is a nonsense if E ̸= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : The gradient of deformation F t0  t (pt0) = dΦt0  t (pt0) =noted F ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) cannot be symmetric  since F T ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for the first Piola–Kirchhoff tensor PKt0  t , which motivates the introduction  of the symmetric second Piola–Kirchhoff tensor SKt0  t , see Marsden–Hughes [12] or § M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  93  94  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The adjoint of a linear map (objective)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Isometry  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='63 A linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is an isometry relative to (·, ·)g and (·, ·)h iff  ∀⃗x, ⃗y ∈ E,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)h = (⃗x, ⃗y)g,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  LT  gh ◦ L = IE (identity in E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='74)  In particular, an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is a (·, ·)g-isometry iff  ∀⃗x, ⃗y ∈ E,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y)g = (⃗x, ⃗y)g,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  LT  g ◦ L = IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='75)  Thus, if L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is an isometry and (⃗ei) is a (·, ·)g-orthonormal basis, then (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei) is a (·, ·)h-  orthonormal basis, since (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)h = (⃗ei,⃗ej)g = δij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='64 Let ⃗f : E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  if  ⃗f is an isometry then ⃗f is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='76)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ei) be a (·, ·)g-orthonormal basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (⃗f(⃗ei)) is a (·, ·)h-orthonormal basis (since ⃗f is an isometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, if ⃗x = �n  i=1xi⃗ei then ⃗f(⃗x)  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  (⃗f(⃗x), ⃗f(⃗ei))h ⃗f(⃗ei)  hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  (⃗x,⃗ei)g ⃗f(⃗ei)  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  i=1  xi ⃗f(⃗ei), thus ⃗f(⃗x+λ⃗y) =  n  �  i=1  (xi + λyi)⃗f(⃗ei) =  n  �  i=1  xi ⃗f(⃗ei) + λ  n  �  i=1  yi ⃗f(⃗ei) = ⃗f(⃗x) + λ⃗f(⃗y), thus ⃗f is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65 Rn is an affine space, ⃗Rn is the usual associated vector space, and (·, ·)g is an inner dot  product in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Definition: A distance-preserving function f : p ∈ Rn → f(p) ∈ Rn is a function s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ||−−−−−→  f(p)f(q)||g = ||−→  pq||g,  ∀p, q ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='77)  Prove: If f is a distance-preserving function, then f is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let O ∈ Rn (an origin) and ⃗f : ⃗x = −→  Op ∈ ⃗Rn → ⃗f(⃗x) := −−−−−−→  f(O)f(p) (vectorial associated function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  ⃗x = −→  Op and ⃗y = −→  Oq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the remarkable identity 2(⃗f(⃗x), ⃗f(⃗y))g = ||⃗f(⃗x)||2  g + ||⃗f(⃗y)||2  g − ||⃗f(⃗x)−⃗f(⃗y)||2  g gives  2(⃗f(⃗x), ⃗f(⃗y))g = ||⃗f(⃗x)||2  g+||⃗f(⃗y)||2  g−||−−−−−→  f(q)f(p)||2  g = ||⃗f(⃗x)||2  g+||⃗f(⃗y)||2  g−||−→  qp||2  g = ||⃗x||2  g+||⃗y||2  g−||⃗x−⃗y||2  g = 2(⃗x, ⃗y)g,  thus ⃗f is an isometry, thus ⃗f is linear cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='76), thus f is affine since f(p) = f(O) + ⃗f(−→  Op).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12  The adjoint of a linear map (objective)  (For mathematicians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' May produce misunderstandings, misuses, problematic mechanical interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  No inner dot product is required here: A linear map L has only one adjoint L∗ (intrinsic to L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While  L has many transposed LT = LT  gh which depend on inner dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  E and F are vector spaces, and E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and F ∗ = L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) are the dual spaces (made of linear  continuous forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (If E and F are finite dimensional, the continuity is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='66 Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) (linear and continuous);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its adjoint is the linear map L∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗)  canonically defined by  L∗ :  �  F ∗ → E∗  m → L∗(m) := m ◦ L,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='78)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all (⃗x, m) ∈ E × F ∗,  (L∗(m))(⃗x) := m(L(⃗x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='79)  (The adjoint L∗ cannot be confused with a transposed LT which requires inner dot products, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  The linearity of L∗ is trivial, thus, together with the linearity of m and L, we can use the dot notation:  L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L,  and  (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='80)  And ||L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m||E∗ = ||m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L||E∗ ≤ ||m||F ∗||L||L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ) gives ||L∗||L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='E∗) ≤ ||L||L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ) < ∞, thus L∗ is  continuous (when L is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  94  95  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Tensorial representation of a linear map  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification  E and F are finite dimensional, dim E = n, dim F = m, and (⃗ai) and (⃗bi) are bases in E and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  [L]|⃗a,⃗b =noted [L], [L∗]|b,a =noted [L∗], [m]|b =noted [m] and [⃗x]|⃗a =noted [⃗x] be the matrices relative to the  chosen bases: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='80) gives ([L∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x] = [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x] for all ⃗x ∈ E and m ∈ F ∗, thus, for all m ∈ F ∗  (recall that [m] is a line matrix), thus [L∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [m]T = ([L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [m]T , thus  [L∗] = [L]T  (transposed matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='81)  (Full notation: [L∗]|b,a = ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Details: With the dual bases (πai) and (πbi), with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �m  i=1Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a,⃗b = [Lij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n , and  with L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbj = �n  i=1(L∗)ijπai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L∗]|b,a = [(L∗)ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='80) gives, for all (i, j) ∈ [1, n]N × [1, m]N,  (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai = πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai),  thus  (L∗)ij = Lji  and  [L∗] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='82)  Duality notations (warning: can be misused): L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �m  i=1Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a,⃗b = [Lij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n , and L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='bj =  �n  i=1(L∗)i jai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L∗]|b,a = [((L∗)i j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m , thus, for all (i, j) ∈ [1, n]N × [1, m]N,  (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai = bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai),  thus  (L∗)i  j = Lj  i  and  [L∗] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='83)  (Recall: Use classical notations if in doubt, or, preferably, don��t use duality notations here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67 Reminder: The transposed bT of a bilinear b form is intrinsic to b, and the adjoint L∗ of  a linear map L is intrinsic to L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But a transposed LT of a linear form L is not intrinsic to the linear form  (it depends on chosen inner dot products): Watch out for the (unfortunate) vocabulary “transpose”!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Relation with the transposed when inner dot products are introduced  let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We need inner dot products (·, ·)g and (·, ·)h in E and F to define LT = LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To  have a functional relation between L∗ and LT  gh, we use the (·, ·)g-Riesz representation mapping ⃗Rg :  �  E∗ → E  ℓ → ⃗Rg(ℓ) = ⃗ℓg  �  , where ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗ℓg, ⃗x)g for all ⃗x ∈ E, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' idem with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) (continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' For all ⃗x ∈ E and all m ∈ F ∗ we have  (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='79)  =  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x),  thus  (⃗Rg(L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m), ⃗x)g = (⃗Rh(m), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)h,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='84)  thus ((⃗Rg ◦ L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m), ⃗x)g = ((LT  gh ◦ ⃗Rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗Rg ◦ L∗ = LT  gh ◦ ⃗Rh, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  LT  gh = ⃗Rg ◦ L∗ ◦ (⃗Rh)−1  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E  LT  gh  ←−  F  ⃗Rg ↑  ↑ ⃗Rh  E∗ ←−  L∗ F ∗  is a commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='85)  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68 From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='85), recover (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='71), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  gh] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  gh] =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='85) [⃗Rg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗Rh]−1 =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13  Tensorial representation of a linear map  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  A tensorial representation  Consider the natural canonical isomorphism (between linear maps E → F and bilinear forms F ∗×E → R)  �  J :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  L → βL = �  J (L)  �  where  βL(m, ⃗u) := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u),  ∀(m, ⃗u) ∈ F ∗ × E,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='86)  see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And βL is also named L for calculations purposes, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (NB: It can be dangerous to substitute L with βL, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  95  96  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of basis formulas for bilinear forms and linear maps  Quantification: Let (⃗ai)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n be a basis in E, (⃗bi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m be a basis in F which dual basis is (πbi),  L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  βL(πbi,⃗ai) = πbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='87)  Thus, if  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  m  �  i=1  Lij⃗bi  then  βL =  m  �  i=1  n  �  j=1  Lij⃗bi ⊗ πaj  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='88)  Indeed,  (�  ij Lij⃗bi ⊗ πaj)(πbk,⃗aℓ)  =  �  ij Lij(⃗bi ⊗ πaj)(πbk,⃗aℓ)  =  �  ij Lij(⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbk)(πaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ)  =  �  ij Lij(⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbk)(πaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ) = �  ij Lijδkiδjℓ = Lkℓ = πbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ, so (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='87) gives (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notations: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �m  i=1Lij⃗bi and βL = �m  i=1  �n  j=1Lij⃗bi ⊗ aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Contraction rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If you write L = �m  i=1  �n  j=1Lij⃗bi ⊗πaj (≃ βL), then the vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u ∈ F is computed  thanks to the “contraction rule”:  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (  m  �  i=1  n  �  j=1  Lij⃗bi ⊗ πaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u  � �� �  contraction  :=  m  �  i=1  n  �  j=1  Lij⃗bi(πaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) =  m  �  i=1  n  �  j=1  Lijuj⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='89)  (With duality notations: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (  m  �  i=1  n  �  j=1  Li  j⃗bi ⊗ aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u  � �� �  contraction  =  m  �  i=1  n  �  j=1  Li  j⃗bi(aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) =  m  �  i=1  n  �  j=1  Li  juj⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='69 Warning: The bilinear form βL should not be confused with the linear map L: The  domain of definition of βL is F ∗ × E, and βL acts on the two objects ℓ (linear form) and ⃗u (vector) to  get a scalar result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While the domain of definition of L is E, and L acts one object ⃗u to get a vector  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' However, you can use the tensorial notation for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' only to calculate L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Warning: Confusion between transposed and adjoint  The transposed LT ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) of a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) needs inner dot products to be defined,  cf (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68): It is not intrinsic to L, not objective ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' While the transposed bT ∈ L(B, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) of a bilinear  form b ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is intrinsic to L (it does not need inner dot products to be defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So if you represent a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) by its tensorial representation βL ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R),  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='88), then  1- you know the transposed (βL)T (given by (βL)T (⃗w, ⃗u) = βL(⃗u, ⃗w)),  2- but you cannot deduce the transposed LT ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) from (βL)T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', to start with (βL)T is  misleading): You need to choose inner dot products, and then use the formula (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x)h  where LT := LT  gh to get [LT ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [h] (and [LT ] ̸= [L]T in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- In particular: If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is symmetric (relative to the chosen inner dot products), then  βL ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is never symmetric because E∗ ̸= E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Recall: there is no natural canonical isomorphism  between E and E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14  Change of basis formulas for bilinear forms and linear maps  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Notations for transitions matrices for bilinear forms and linear maps  Let A and B be finite dimension vector spaces, dim A = n, dim B = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' application to the change  of basis formula for the deformation gradient A=⃗Rn  t0 → B=⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let (⃗aold,i) and (⃗anew,i) be two bases in A, and (⃗bold,i) and (⃗bnew,i) be two bases in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let PA  and PB be the change of basis endomorphisms from old to new bases, and PA := [PA]|⃗aold = [PAij] and  PB := [PB]|⃗bold = [PBij] be the associated transition matrices, and QA = PA  −1 and QB = PB  −1:  ⃗anew,j = PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aold,i =  n  �  i,j=1  PAij⃗aold,i,  πanew,j =  n  �  i=1  QAijπaold,i,  ⃗bnew,j = PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bold,i =  m  �  i,j=1  PBij⃗bold,i,  πbnew,j =  n  �  i,j=1  QBijπbold,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='90)  Duality notations: ⃗anew,j = �n  i=1PA  i  j⃗aold,i and ai  new = �n  j=1QA  i  jaj  old and ⃗bnew,j = �n  i=1PB  i  j⃗bold,i and  bi  new = �n  j=1QB  i  jbj  old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  96  97  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of basis formulas for bilinear forms and linear maps  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Change of coordinate system for bilinear forms ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  Let g ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and, for all (i, j) ∈ [1, n]N × [1, m]N,  g(⃗aold,i,⃗bold,j) = Mij,  g(⃗anew,i,⃗bnew,j) = Nij,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  [g]|olds = M = [Mij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m ,  [g]|news = N = [Nij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='91)  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='70 Change of basis formula:  [g]|news = PA  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = PA  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='92)  In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and  [g]|new = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='93)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Nij = g(⃗anew,i,⃗bnew,j) = �  kℓ PA  k  iPB  ℓ  jg(⃗aold,k,⃗bold,ℓ) = �  kℓ PA  k  iMkℓPB  ℓ  j = �  kℓ(PA  T )ikMkℓPB  ℓ  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='71 Prove (objective result):  g(⃗u, ⃗w) = [⃗u]T  |⃗anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗bnew = [⃗u]T  |⃗aold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='94)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]T  |⃗anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗bnew = (PA  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗aold)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (PA  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (PB  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Change of coordinate system for bilinear forms ∈ L(A∗, B∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  Let z ∈ L(A∗, B∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and, for all (i, j) ∈ [1, n]N × [1, m]N,  z(ai  old, bj  old) = M ij,  z(ai  new, bj  new) = N ij,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  [z]|olds = M = [M ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m ,  [z]|news = N = [N ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='95)  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='72 Change of basis formula:  [z]|news = PA  −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [z]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB  −1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = PA  −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='96)  In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and  [z]|new = P −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [z]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1 ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = P −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='97)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Nij = z(ai  new, bj  new) = �  kℓ QA  k  iQB  ℓ  jz(ak  old, bℓ  old) = �  kℓ QA  k  iM kℓQB  ℓ  j = �  kℓ(QA  T )ikM kℓQB  ℓ  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Change of coordinate system for bilinear forms ∈ L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (Toward linear maps L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) ≃ L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) thanks to the natural canonical isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let T ∈ L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and, for all (i, j) ∈ [1, n]N × [1, m]N,  T(bi  old,⃗aold,j) = M i  j,  T(bi  new,⃗anew,j) = N i  j,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  [T]|olds = M = [M i  j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m ,  [T]|news = N = [N i  j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='98)  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='73 Change of basis formula:  [T]|news = PB  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [T]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PA,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='99)  In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and  [T]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [T]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N i  j =  n  �  k,ℓ=1  Qi  kM k  ℓP ℓ  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='100)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' N ij = T(bi  new,⃗anew,j) = �  kℓ QB  i  kPA  ℓ  jT(bi  old,⃗aold,j) = �  kℓ QB  i  kM ijPA  ℓ  j  97  98  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of basis formulas for bilinear forms and linear maps  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Change of coordinate system for tri-linear forms ∈ L(A∗, A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (Toward d2⃗u:  For a vector field ⃗u ∈ Γ(U) ≃ T 1  0 (U), ⃗u(p) ∈ ⃗Rn, its differential satisfies d⃗u(p) ∈  L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn) ≃ L(Rn∗, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and d2⃗u(p) ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn)) ≃ L(Rn∗, ⃗Rn, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), see § S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Consider a tri-linear form T ∈ L(A∗, A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), and  M i  jk = T(ai  old,⃗aold,j,⃗aold,k),  N i  jk = T(ai  new,⃗anew,j,⃗anew,k),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [T]|⃗aold = [M i  jk],  [T]|⃗anew = [N i  jk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='101)  Then  N i  jk =  n  �  λ,µ,ν=1  Qi  λP µ  j P ν  k M λ  µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='102)  Indeed �  λµν M λ  µν⃗aold,λ ⊗ aµ  old ⊗ aν  old = �  λµνijk M λ  µνQi  λP µ  j P ν  k⃗anew,i ⊗ aj  new ⊗ ak  new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Change of coordinate system for linear maps ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B)  Notation of § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) be a linear map, and let, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  �  �  �  �  �  �  �  �  �  �  �  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aold,j =  m  �  i=1  Mij⃗bold,i =  m  �  i=1  M i  j⃗bold,i  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [L]|olds = M = [Mij] = [M i  j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n ,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗anew,j =  m  �  i=1  Nij⃗bnew,i =  m  �  i=1  N i  j⃗bnew,i  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [L]|news = N = [Nij] = [N i  j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='103)  with classical and duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='74 Change of bases formula:  [L]|news = PB  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PA,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = PB  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='104)  In particular, if A = B, if (⃗aold,i) = (⃗bold,i), (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P and  [L]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  N = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Nij =  n  �  k,ℓ=1  QikMkℓPℓj,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='105)  with Q = P −1, and with duality notations N ij = �  kℓ QikM kℓP ℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗anew,j  =  �  i N ij⃗bnew,i  =  �  ik N ijPB  k  i⃗bold,k  =  �  k(PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='N)kj⃗bold,k  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗anew,j  =  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(�  i PA  i  j⃗aold,i) = �  i PA  i  j  �  k M ki⃗bold,k = �  k(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PA)kj⃗bold,k, for all j, thus PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='N = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='75 Prove:  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = [ℓ]|⃗bnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗anew = [ℓ]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗aold  (objective result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='106)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ℓ]|⃗bnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗anew = ([ℓ]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (PB  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[L]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (PA  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗aold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='76 Bilinear forms in L(A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and endomorphisms in L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A) behave differently: The  formulas (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='93) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='105) should not be confused since P −1 ̸= P T in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if an English  observer uses a Euclidean (old) basis (⃗ai) = (⃗aold,i) in foot, if a French observer uses a Euclidean (new)  basis (⃗bi) = (⃗anew,i) in metre, and if (simple case) ⃗bi = λ⃗ai for all i (change of unit), then  [L]|new = [L]|old,  while  [g]|new = λ2  ����  >10  [g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='107)  Quite different results!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P ̸= P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P for a general change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See the Mars  Climate Orbiter crash, remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14, where someone forgot that 1 foot ̸= 1 metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B  Euclidean Frameworks  Time and space are decoupled (classical mechanics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Rn is the geometric affine space, n = 1, 2, 3, and ⃗Rn  is the associated vector space made of “bi-point vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  98  99  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Euclidean basis  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Euclidean basis  Manufacturing of a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  An observer chooses a unit of measure (foot, metre, a unit of length used by Euclid, the diameter a  of pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') and makes a “unit rod” of length 1 in this unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Postulate: The length of the rod does not depend on its direction in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Space dimension n = 1: This rod models a vector ⃗e1 which makes a basis (⃗e1) called the Euclidean  basis relative to the chosen unit of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Space dimension n ≥ 2:  The observers makes three rods of length 3, 4 and 5, and makes a triangle (A, B, C) with A, B and  C are the vertices and A not on the side on length 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Pythagoras: 32 + 42 = 52 gives: The triangle (A, B, C) is said to have a right angle at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Two vectors ⃗u and ⃗w in ⃗Rn are orthogonal iff the triangle (A, B, C) can be positioned such that ⃗  AB  and ⃗  AC are parallel to ⃗u and ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A basis (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n is Euclidean relative to the chosen unit of measurement iff the ⃗ei are two to two  orthogonal and their length is 1 (relative to the chosen unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 An English observer defines a Euclidean basis (⃗ai) using the foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A French observer  defines a Euclidean basis (⃗bi) using the metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have  1 foot = µ metre,  µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3048,  and  1 metre = λ foot,  λ = 1  µ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  (µ = 0, 3048 is the official length in metre for the English foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', the bases are “aligned” iff, for all i,  ⃗bi = λ⃗ai  (change of measurement unit),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  thus the transition matrix from (⃗ai) to (⃗bi) is P = λI, thus P T = P, P −1 = 1  λI and P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = λ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The bases used in practice are not all Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13, especially if you fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Euclidean dot product  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 An observer who has built) his Euclidean basis (⃗ei), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The associated Euclidean  dot product is the bilinear form g(·, ·) = (·, ·)g ∈ L(⃗Rn, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) defined by  (gij =)  g(⃗ei,⃗ej) = δij,  ∀i, j,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [g]|⃗e = [δij] = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  In other words,  (·, ·)g :=  n  �  i=1  πei ⊗ πei =  n  �  i=1  ei ⊗ ei,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  with classical and duality notations, (πei) = (ei) being the dual basis of (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if you want to use the  Einstein convention you have to write (·, ·)g := �n  i,j=1δijei ⊗ ej: You cannot avoid writing δij = gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, for all ⃗x, ⃗y ∈ ⃗Rn, with ⃗x = �n  i=1xi⃗ei and ⃗y = �n  i=1yi⃗ei (classical notations),  (⃗x, ⃗y)g =  n  �  i=1  xiyi = [⃗x]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  With duality notations, ⃗x = �n  i=1xi⃗ei, ⃗y = �n  i=1yi⃗ei and (⃗x, ⃗y)g = �n  i=1xiyi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if you want to use  the Einstein convention then write (⃗x, ⃗y)g := �n  i,j=1δijxiyj: You cannot avoid writing δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 The associated norm is ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g :=  �  (·, ·)g, and the length of a vector ⃗x relative to the  chosen Euclidean unit of measurement is ||⃗x||g :=  �  (⃗x, ⃗x)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus with the Euclidean basis (⃗ei) (used to build (·, ·)g), if ⃗x = �n  i=1xi⃗ei, then ||⃗x||g =  ��n  i=1x2  i is  the length of ⃗x relative to the chosen Euclidean unit of measure (Pythagoras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (With duality notations  ||⃗x||g =  ��n  i=1(xi)2, and if you want to use the Einstein convention: ||⃗x||g =  ��n  i,j=1δijxixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The angle θ(⃗x, ⃗y) between two vectors ⃗x, ⃗y ∈ ⃗Rn − {⃗0} is defined by  cos(θ(⃗x, ⃗y)) = (  ⃗x  ||⃗x||g  ,  ⃗y  ||⃗y||g  )g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  (With a calculator, this formula gives θ(⃗x, ⃗y) = arccos((  ⃗x  ||⃗x||g ,  ⃗y  ||⃗y||g )g) a value in [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  99  100  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of Euclidean basis  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Change of Euclidean basis  Let (⃗ai) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' English observer basis built with the foot) and (⃗bi) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' French observer basis built with  the metre) be Euclidean bases in ⃗Rn, and let (·, ·)g and (·, ·)h be the associated Euclidean dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Two Euclidean dot products are proportional  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 If λ = ||⃗b1||g, then ||⃗bi||g = λ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n (change of unit) and  (·, ·)g = λ2(·, ·)h,  and  ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' By definition of a Euclidean basis, the length of the rod that enabled to define (⃗bi) is independent  of i, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1, thus ||⃗bi||g = ||⃗b1||g for all i, and here ||⃗bi||g =noted λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ||⃗bi||2  g = λ2 = λ2||⃗bi||2  h for all i,  since ||⃗bi||2  h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if i ̸= j then (⃗bi,⃗bj)g = 0 = (⃗bi,⃗bj)h since ⃗bi and ⃗bj form a right angle (Pythagoras),  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence (⃗bi,⃗bj)g = λ2(⃗bi,⃗bj)h for all i, j, thus (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 Continuation of example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1: (·, ·)a = �n  i=1ai ⊗ ai is the English Euclidean dot product  (foot), and (·, ·)b = �n  i=1bi ⊗ bi is the French Euclidean dot product (metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) give:  (·, ·)a = λ2(·, ·)b  and  ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||a = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||b,  with  λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28  and  λ2 ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  In particular, if ⃗w is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ||⃗w||b = 1 (its length is 1 metre), then ||⃗w||a = λ (its length is λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28 foot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Counterexample : non existence of a Euclidean dot product  1- Thermodynamic: Let T be the temperature and P the pressure, and consider the Cartesian vector  space {(T, P)} = {(temperature,pressure)} = R × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' There is no associated Euclidean dot product: An  associated norm would give ||(T, P)|| =  √  T 2 + P 2 ∈ R which is meaningless (incompatible dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  See § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Polar coordinate system ⃗q = (r, θ) ∈ R × R: There is no Euclidean norm  √  r2 + θ2 for ⃗q that is  physically meaningful (incompatible dimensions), see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Euclidean transposed of the deformation gradient  Let n ∈ {1, 2, 3} and consider a linear map L ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L = F t0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (·, ·)G be a Euclidean dot product in ⃗Rn  t0 (used in the past by someone), and let (·, ·)g and (·, ·)h  be Euclidean dot products in ⃗Rn  t (the actual space where the results are obtained by two observers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  (·, ·)g built with a foot and (·, ·)h built with a metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let LT  Gg and LT  Gh be the transposed of L relative  to the dot products, that is, LT  Gg and LT  Gh in L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) are characterized by, for all ( ⃗X, ⃗y) ∈ ⃗Rn  t0 × ⃗Rn  t ,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68),  (LT  Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗X)G = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗X, ⃗y)g  and  (LT  Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗X)G = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗X, ⃗y)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  if  (·, ·)g = λ2(·, ·)h  then  LT  Gg = λ2LT  Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  NB: Do not forget λ2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 (Mars Climate Orbiter crash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗X)G  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗X, ⃗y)g  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)1  =  λ2(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗X, ⃗y)h  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  = λ2(LT  Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗X)G for all ⃗X ∈ ⃗Rn  t0 and all ⃗y ∈ ⃗Rn  t ,  thus LT  Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y = λ2LT  Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y for all ⃗y ∈ ⃗Rn  t , thus (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  The Euclidean transposed for endomorphisms  Let n ∈ {1, 2, 3} and consider an endomorphism L ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L = d⃗vt(p) ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) the  differential of the Eulerian velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (·, ·)g and (·, ·)h be dot products in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let LT  g and LT  h be the  transposed of L relative to (·, ·)g and (·, ·)h, that is, LT  g and LT  h in L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) are the endomorphisms  defined by, for all ⃗x, ⃗y ∈ ⃗Rn  t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54),  (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g,  and  (LT  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)h = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  100  101  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Unit normal vector, unit normal form  Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  if  (·, ·)g = λ2(·, ·)h  then  LT  g = LT  h  noted  =  LT  ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  (an endomorphism type relation): Thus we can speak of “the Euclidean transposed of an endomorphism”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)g  hyp  = λ2(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x, ⃗y)h  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  =  λ2(LT  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)h  hyp  = (LT  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗x)g for all ⃗x, ⃗y ∈ ⃗Rn, thus  LT  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y = LT  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y for all ⃗y ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Unit normal vector, unit normal form  The results in this § are not objective: We need a Euclidean dot product (need a unit of length: Foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Meter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') to get a unit (Euclidean) normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Framework  (·, ·)g is a Euclidean dot product (needed to define Euclidean orthonormality) and, for all ⃗u, ⃗w ∈ ⃗Rn,  (⃗u, ⃗w)g  noted  =  ⃗u •g ⃗w  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  (or =noted ⃗u • ⃗w when one chosen Euclidean dot product is imposed to all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Ω is a regular open bounded set in Rn, n = 2 or 3, and Γ := ∂Ω is its regular surface (dimension n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If p ∈ Γ then TpΓ is the tangent plane at p to Γ, and a basis (⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗βn−1(p)) in TpΓ is known (usually  obtained thanks to a coordinate system describing Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And, to lighten the writings, (⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗βn−1(p))  is written (⃗β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗βn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Unit normal vector  Call ⃗ng(p) the unit outward normal vector at p ∈ Γ at TpΓ relative to (·, ·)g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So ⃗ng(p) •g ⃗βi(p) = 0 for all  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n−1, and ||⃗ng(p)||g = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗ng is defined on Γ by (up to its sign)  ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n−1, ⃗βi •g ⃗ng = 0,  and  ⃗ng •g ⃗ng = 1  (= ||⃗ng||2  g),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', at any p ∈ Γ, ⃗ng(p) is orthogonal to the hyperplane Vect{⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗βn−1(p)} and ⃗ng(p) is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  (⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗βn−1(p),⃗ng(p)) is a basis at p in ⃗Rn, written in short (⃗β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗βn−1,⃗ng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, for all ⃗w ∈ ⃗Rn, if ⃗w = �n−1  i=1 wi⃗βi + wn⃗ng (classical notations) then  wn = ⃗w •g ⃗ng = the normal component of ⃗w at p at Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  (wn depends on (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') (Duality notations: ⃗w = �n−1  i=1 wi⃗βi + wn⃗ng and wn = ⃗w •g ⃗ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Let (⃗ai) be a basis in ⃗Rn, ⃗βj = �n  i=1Bij⃗ai for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n−1, and ⃗ng = �n  i=1ni⃗ai, and  gij = g(⃗ai,⃗aj) for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' What equations satisfy the nj?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And particular case (⃗ai) is (·, ·)g-orthonormal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14) gives [⃗βi]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ng]|⃗a = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n−1 (so n−1 equations), with [⃗ng]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ng]|⃗a = 1 (so 1  equation), and ⃗ng is obtained up to its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If (⃗ai) is (·, ·)g-orthonormal, then �n  j=1Bijnj = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n−1, with �n  i=1n2  i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Let (⃗ai) be a Euclidean basis in foot, (⃗bi) a Euclidean basis in metre, (·, ·)a and (·, ·)b  the associated Euclidean dot products, so (·, ·)a = λ2(·, ·)b with λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗na(p) and ⃗nb(p)  be the corresponding unit outward normal vectors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- Prove (up to the sign):  ⃗nb = λ⃗na,  and  (⃗w,⃗na)a = λ(⃗w,⃗nb)b  ∀⃗w ∈ ⃗Rn  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  2- Then let ⃗na = �m  i=1nai⃗ai and ⃗nb = �m  i=1nbi⃗bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  If, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, ⃗bi = λ⃗ai  then  ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, nai = nbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  So the vectors ⃗na and ⃗nb are different (λ > 1), and their respective components are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' relative to  different bases!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And of course 1 = ||⃗na||2  a = �n  i=1(nai)2 = �n  i=1(nbi)2 = ||⃗nb||2  b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗na(p) ∥ ⃗nb(p), since the vectors are Euclidean and orthogonal to TpΓ cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||a = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||b  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8), thus ||⃗nb||b = 1 = ||⃗na||a = λ||⃗na||b = ||λ⃗na||b, so ⃗nb = ±λ⃗na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And they both are outward vectors, so  ⃗nb = +λ⃗na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (⃗w,⃗na)a = λ2(⃗w,⃗na)b = λ2(⃗w, ⃗nb  λ )b = λ(⃗w,⃗nb)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And if ⃗bi = λ⃗ai (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) gives �n  i=1ni  b⃗bi = λ�n  i=1ni  a⃗ai = �n  i=1ni  a(λ⃗ai) = �n  i=1ni  a⃗bi, then ni  a = ni  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  101  102  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Integration by parts (Green–Gauss–Ostrogradsky)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Unit normal form n♭ associated to ⃗n  (For mathematicians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' May produce misunderstandings and lack of mechanical interpretations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Don’t  forget: n♭ is obtained after ⃗n has been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  At p ∈ Γ, once you have computed ⃗ng(p), you can define the associated unit normal form n♭  g(p) ∈ Rn∗:  It is the linear form defined by n♭  g(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w := ⃗ng(p) •g ⃗w for all ⃗w ∈ ⃗Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' on Γ, for all ⃗w ∈ ⃗Rn,  n♭  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w := ⃗ng •g ⃗w  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  ( =noted ⃗n • ⃗w if one chosen Euclidean dot product is imposed to all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus [n♭  g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] = [⃗ng]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: Let (⃗ei) be a basis in ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) gives [n♭  g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗e = [⃗ng]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗e simply  written [n♭  g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] = [⃗ng]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] if the basis (⃗ei) is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, with duality notations to justify the ♭ notation, with (ei) the dual basis of (⃗ei), let  ⃗ng =  n  �  i=1  ni  g⃗ei  and  n♭  g =  n  �  i=1  ngiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ni  g and ngi are the components of ⃗ng and n♭  g relative to the basis (⃗ei) and (ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) gives  n♭  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei := ⃗ng •g ⃗ei for all i, we get, for all i,  nig =  n  �  j=1  gijnj  g  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Particular case (⃗ei) is a (·, ·)g-Euclidean basis, then nig = ni  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Classical notations: ⃗ng = �n  i=1(⃗ng)i⃗ei, dual basis (πei), n♭  g = �n  i=1(n♭  g)iπei, (n♭  g)i = �n  j=1gij(⃗ng)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: In physics don’t forget to write the gij in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) even if gij = δij, since you need to see the chosen  metric and basis (and verify the Einstein convention), although (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20) is simply written ni = �n  j=1gijnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Integration by parts (Green–Gauss–Ostrogradsky)  Let Ω be a regular bounded open set in Rn and Γ = ∂Ω its frontier, let ϕ ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), let (⃗ei) be a  Euclidean basis and (·, ·)g ites associated Euclidean dot product, let  ∂ϕ  ∂xi (p) := dϕ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei (usual notation),  let ⃗ng(p) = ⃗n(p) = �n  i=1ni(p)⃗ei (classical notations) be the unit outward normal at p ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  �  p∈Ω  ∂ϕ  ∂xi  (p) dΩ =  �  p∈Γ  ϕ(p)ni(p) dΓ,  in short  �  Ω  ∂ϕ  ∂xi  dΩ =  �  Γ  ϕni dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Thus, for any v ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), with ϕv instead of ϕ in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21), we get the integration by parts formula  (Green formula):  �  Ω  ∂ϕ  ∂xi  v dΩ = −  �  Ω  ϕ ∂v  ∂xi  dΩ +  �  Γ  ϕvni dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Thus, for any ⃗v ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn) (vector field), with ⃗v(p) = �n  i=1vi(p)⃗ei) we get  �  Ω  ∂ϕ  ∂xi  vi dΩ = −  �  Ω  ϕ ∂vi  ∂xi  dΩ +  �  Γ  ϕvini dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  Thus, with the gradient vector  ⃗  gradϕ(p) = �n  i=1  ∂ϕ  ∂xi⃗ei and with div⃗v = �n  i=1  ∂vi  ∂xi , we get the Gauss–  Ostrogradsky formula:  �  Ω  ⃗  gradϕ • ⃗v dΩ = −  �  Ω  ϕ div⃗v dΩ +  �  Γ  ϕ⃗v • ⃗n dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  (And  �  Γ ϕ⃗v • ⃗n dΓ gives the flux through Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Use the differential dϕ instead of the gradient  ⃗  gradϕ (which is the (·, ·)g-Riesz represen-  tation vector of dϕ) to express (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Is the use of n♭ useful in that case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  Ω dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v dΩ = −  �  Ω ϕ div⃗v dΩ +  �  Γ ϕ⃗v • ⃗n dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since n♭ depends on ⃗n (definition), there is no reason that  justifies the use of n♭ (unless you want to introduce useless notations here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  102  103  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The symmetric and antisymmetric parts of d⃗v  C  Rate of deformation tensor and spin tensor  Let �Φ : [t1, t2] × Obj → Rn be a regular motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5), and let ⃗v : C → ⃗Rn be the Eulerian velocity  field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4), that is, ⃗v(t, p) = ∂Φ  ∂t (t, PObj) when p = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its differential d⃗v is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  At t, an observer chooses a unit of measurement (foot, metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') and builds the associated Euclidean  dot product (·, ·)g in ⃗Rn  t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (We loose the objective point of view here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the same (·, ·)g is  used at all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  The symmetric and antisymmetric parts of d⃗v  With the imposed chosen Euclidean dot product (·, ·)g in ⃗Rn  t , we can consider the transposed endomor-  phism d⃗vt(p)T  g =noted d⃗vt(p)T ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ), which is defined by, for all ⃗w1, ⃗w2 ∈ ⃗Rn  t vectors at p,  (d⃗vt(p)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g = (⃗w1, d⃗vt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)g  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have thus defined  d⃗vT  t :  �  Ωt → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  p → d⃗vT  t (p) := d⃗vt(p)T  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Other usual notations (definitions): d⃗vt(p)T =noted d⃗v(t, p)T =noted d⃗vT (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The (Eulerian) rate of deformation tensor, or stretching tensor, is the (·, ·)g-symmetric  part of d⃗v:  D = d⃗v + d⃗vT  2  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  ∀(t, p) ∈  �  t∈R  ({t} × Ωt),  D(t, p) = d⃗v(t, p) + d⃗v(t, p)T  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  The (Eulerian) spin tensor is the (·, ·)g-antisymmetric part of d⃗v:  Ω = d⃗v − d⃗vT  2  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  ∀(t, p) ∈  �  t∈R  ({t} × Ωt),  Ω(t, p) = d⃗v(t, p) − d⃗v(t, p)T  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  (So d⃗v = D + Ω with D the rate of deformation tensor and Ω = ⃗ω∧ a rotation times a dilation, see the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  NB: The same notation is used for the set of points Ωt = Φt0  t (Ωt0) ⊂ Rn and for the spin tensor  Ωt = d⃗vt−d⃗vT  t  2  : The context removes ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification with a basis  With a basis (⃗ei) in ⃗Rn  t , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) gives  [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗vT ]|⃗e = [d⃗v]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e,  and  [d⃗vT ]|⃗e = [g]−1  |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗v]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  In particular, if (⃗ei) is a (·, ·)g-orthonormal basis, then [d⃗vT ]|⃗e = [d⃗v]T  |⃗e (orthonormal basis case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus for  the endomorphisms D and Ω, and with the above Euclidean framework and its Euclidean orthonormal  basis, we have D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Dij⃗ei and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Ωij⃗ei with Dij = 1  2( ∂vi  ∂xj + ∂vj  ∂xi ) and Ωij = 1  2( ∂vi  ∂xj − ∂vj  ∂xi ),  that is,  [D]|⃗e =  [d⃗v]|⃗e + [d⃗v]T  |⃗e  2  and  [Ω]|⃗e =  [d⃗v]|⃗e − [d⃗v]T  |⃗e  2  (Euclidean framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Duality notations: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Di  j⃗ei, Di  j = 1  2( ∂vi  ∂xj + ∂vj  ∂xi ) and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Ωij⃗ei, Ωij = 1  2( ∂vi  ∂xj − ∂vj  ∂xi ), so  with Di  j = Dj  i and Ωij = −Ωji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  103  104  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Affine motions and rigid body motions  D  Interpretation of the rate of deformation tensor  We are interested in the evolution of the deformation gradient F(t) := F t0  pt0 (t) along the trajectory of a  particle PObj which was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So:  Let ⃗A = ⃗a(t0, pt0) and ⃗B = ⃗b(t0, pt0) be vectors at t0 at pt0 in Ωt0, and consider their push-forwards  by the flow Φt0  t (the transported vectors), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the vectors at t at p(t) = Φt0  pt0 (t) given by  ⃗a(t, p(t)) := F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A  and  ⃗b(t, p(t)) := F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) and figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' They define the function  (⃗a,⃗b)g :  �  C → R  (t, pt) → (⃗a,⃗b)g(t, pt) := (⃗a(t, pt),⃗b(t, pt))g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The rate of deformation tensor D = d⃗v+d⃗vT  2  gives (half) the evolution rate between  two vectors deformed by the flow, that is, along trajectories,  D(⃗a,⃗b)g  Dt  = 2(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a,⃗b)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' f(t) := (⃗a(t, p(t)),⃗b(t, p(t)))g = (F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)g gives  f ′(t) = (F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)g + (F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  And F ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, with ⃗a(t, p(t)) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A and ⃗b(t, p(t)) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B,  f ′(t) = (d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)g + (F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)g  = (d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a(t, p(t)),⃗b(t, p(t)))g + (⃗a(t, p(t)), d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b(t, p(t)))g  = ((d⃗v(t, p(t)) + d⃗v(t, p(t))T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a(t, p(t)),⃗b(t, p(t)))g,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3), since f(t) = (⃗a,⃗b)g(t, p(t)) gives f ′(t) = D(⃗a,⃗b)g  Dt  (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E  Rigid body motions and the spin tensor  Choose a Euclidean dot product (·, ·)g (required to characterize a rigid body motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Result: A rigid body motion is a motion whose Eulerian velocity satisfies d⃗v + d⃗vT = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', D = 0  (Eulerian approach independent of any initial time t0 chosen by some observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  But the usual classical introduction to rigid body motion relies on some initial time t0 (Lagrangian  approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, to begin with, let us do it with the Lagrangian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Recall: T the first order Taylor  expansion of Φt0  t in the vicinity of a pt0 ∈ Ωt0 is  Φt0  t (qt0) = Φt0  t (pt0) + F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0 + o(−−−→  pt0qt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Affine motions and rigid body motions  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Affine motions  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 Φt0 is an affine motion (understood “affine motion in space”) iff Φt0  t is an “affine motion”,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' iff Φt0  t is a C1 diffeomorphism (in space), and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) reads, for all pt0, qt0 ∈ Ωt0 and all t ∈ [t1, t2],  Φt0  t (qt0) = Φt0  t (pt0) + F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Marsden–Hughes notations: Φ(Q) = Φ(P) + F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−→  PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 and definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If Φt0 is an affine motion, then F t0  t (pt0) is independent of pt0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  for all t ∈]t1, t2[ and all pt0 ∈ Ωt0 and all qt0 ∈ Ωt0,  F t0  t (pt0) = F t0  t (qt0) noted  =  F t0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  And then dF t0  t (pt0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d2Φt0  t (pt0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And for all t ∈]t1, t2[, Φt is an affine motion: For all  τ ∈]t1, t2[ and all pt, qt ∈ Ωt,  Φt  τ(qt) = Φt  τ(pt) + F t  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−→  ptqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  And �Φ is said to be an affine motion (understood “affine motion in space”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  104  105  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Affine motions and rigid body motions  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' qt0 = pt0 + −−−→  pt0qt0 gives Φt0  t (qt0) = Φt0  t (pt0 + −−−→  pt0qt0) = Φt0  t (pt0) + dΦt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0, and, similarly,  Φt0  t (pt0) = Φt0  t (qt0 +−−−→  qt0pt0) = Φt0  t (qt0)+dΦt0  t (qt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  qt0pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (addition) Φt0  t (qt0)+Φt0  t (pt0) = Φt0  t (pt0)+  Φt0  t (qt0) + (dΦt0  t (pt0) − dΦt0  t (qt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0, thus (dΦt0  t (pt0) − dΦt0  t (qt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0 = 0, true for all pt0, qt0, thus  dΦt0  t (pt0) − dΦt0  t (qt0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus d2Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ut0 = limh→0  dΦt0  t (pt0+h⃗ut0)−dΦt0  t (pt0)  h  = limh→0  dΦt0  t −dΦt0  t  h  = 0 for all pt0 and all ⃗ut0,  thus d2Φt0  t (pt0) = 0 for all pt0, thus d2Φt0  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17) gives (Φt  τ ◦ Φt0  t )(pt0) = Φt0  τ (pt0), thus, with pt = Φt0  t (pt0), we get dΦt  τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0  t (pt0) =  dΦt0  τ (pt0), thus dΦt  τ(pt) = dΦt0  τ (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0  t (pt0)−1, and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives  dΦt  τ(pt) = dΦt0  τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦt0  t  −1 noted  =  dΦt  τ  (independent of pt),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 If �Φ is affine then, ⃗vt is affine for all t, and ⃗V t0  t  is affine for all t0, t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all pt ∈ Ωt we  have d⃗vt(pt) = d⃗vt (independent of pt), and for all pt0 ∈ Ωt0 we have d⃗V t0  t (pt0) =noted d⃗V t0  t  (independent  of pt0): For all qt ∈ Ωt and all qt0 ∈ Ωt0,  �  ⃗vt(qt) = ⃗vt(pt) + d⃗vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−→  ptqt,  ⃗V t0  t (qt0) = ⃗V t0  t (pt0) + d⃗V t0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives Φt0(t, qt0) = Φt0(t, pt0) + F t0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−−→  pt0qt0, and the derivation in time gives (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)2,  then (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)1 thanks to pt = Φt0  t (pt0), qt = Φt0  t (qt0) and −−−→  pt0qt0 = (F t0  t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−−→  ptqt, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 In R2, with a basis ( ⃗E1, ⃗E2) in ⃗Rn  t0 and a basis (⃗e1,⃗e2) ∈ ⃗Rn  t , then F t0  t  given by [F t0  t ]| ⃗E,⃗e =  �  1 + t  2t2  3t3  et  �  derives from the affine motion [−−−−−−−−−−−→  Φt0  t (pt0)Φt0  t (qt0)]|⃗e =  �  1 + t  2t2  3t3  et  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [−−−→  pt0qt0]| ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Rigid body motion  A Euclidean dot product (·, ·)g in ⃗Rn  t is chosen, the same at all time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Φ := Φt0  t  and F := F t0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Recall: If P ∈ Ωt0 and p = Φ(P) (∈ Ωt) then the transposed of the linear map F(P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) relative  to (·, ·)g is the linear map F T (p) := F(P)T ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) defined by  F T (p) := F(P)T :  � ⃗Rn  t  → ⃗Rn  t0  ⃗wp → F T (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F T (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp, ⃗UP )g = (⃗wp, F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗UP )g, ∀⃗UP ∈ ⃗Rn  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  We have thus defined the function F T : Ωt → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Particular case: For an affine motion, since F is independent of P, we get F T is independent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 A rigid body motion is an affine motion �Φ such that, for all t0, t ∈ R, P ∈ Ωt0, ⃗UP , ⃗WP ∈  ⃗Rn  t0, and with p = Φt0  t (P),  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗UP , F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗WP )g = (⃗UP , ⃗WP )g,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗UP , ⃗WP )g = (⃗UP , ⃗WP )g,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  (Angles and lengths are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  In other words, with the Cauchy strain tensor C ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) defined by C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F, the motion is  rigid iff it is affine and  C = I ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F −1 = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 If Φt0 is a rigid body motion, if ( ⃗Ai) is a (·, ·)g-Euclidean basis in ⃗Rn  t0, if P ∈ Ωt0, if  t ∈ [t0, T] and p = Φt0  t (P), and if ⃗ai(t, p) = F t0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ai for all i, then ⃗ai(t, p) =noted ⃗ai,t is independent  of p, and (⃗ai,t) is a (·, ·)g-Euclidean basis with the same orientation than ( ⃗Ai) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  105  106  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Representation of the spin tensor Ω: vectors, and pseudo-vectors  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Φt0  t  is affine, thus, for all t, P, F t0  t (P) = F t0  t  (independent of P), thus ⃗ai,t(p) = F t0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ai ∈ ⃗Rn  t  is independent of p, this at all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let t be fixed and ⃗ai,t =noted ⃗ai (= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We get (⃗ai,⃗aj)g =  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ai, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Aj)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ai, ⃗Aj)g = (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ai, ⃗Aj)g = ( ⃗Ai, ⃗Aj)g = δij for all i, j, thus (⃗ai) is (·, ·)g-orthonormal  basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And det(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗an) = det(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗An) = det(F) det( ⃗A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗An) = det(F) since ( ⃗Ai) is a (·, ·)g-  orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And, Φt0  t  being a diffeomorphism, t → det(F t0  t ) is continuous, does not vanish,  moreover with det(F t0  t0 ) = det(I) = 1 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus det(F t0  t ) > 0 for all t, hence det(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗an) > 0: The  bases have the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 In R2, a rigid body motion is given by F t0  t  =  �  cos(θ(t))  − sin(θ(t))  sin(θ(t))  cos(θ(t))  �  with θ a regular  function s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' θ(t0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Let �Φ be a rigid body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove  (F T )′(t) = (F ′(t))T ,  and  F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ′ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let t ∈ R, p(t) = Φt0  t (P), ⃗U, ⃗W ∈ ⃗Rn  t0 and ⃗w(t, p(t)) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And recall that the function  F T : t → F T (t) is defined (as usual) by F T (t) := (F(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have (F(t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)), ⃗U)g = (⃗w(t, p(t)), F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus ((F T )′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t))+F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w  Dt (t, p(t)), ⃗U)g = ( D ⃗w  Dt (t, p(t)), F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U)g+(⃗w(t, p(t)), F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U)g, which simplifies  into ((F T )′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)), ⃗U)g = (⃗w(t, p(t)), F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U)g = ((F ′(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(t, p(t)), ⃗U)g, thus (F T )′(t) = (F ′(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) reads F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) = It0, thus (F T )′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)+F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ′(t) = 0, thus (F ′)T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)+F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ′(t) = 0,  thus F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ′ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Alternative definition of a rigid body motion: d⃗v + d⃗vT = 0  The stretching tensor Dt = d⃗vt+d⃗vT  t  2  and the spin tensor Ωt = d⃗vt−d⃗vT  t  2  have been defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)-(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 If �Φ is a rigid body motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8), then the endomorphism d⃗vt ∈ L( ⃗  Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t ) is  antisymmetric at all t:  d⃗vt = Ωt,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Conversely, if d⃗vt + d⃗vT  t = 0 at all t, then �Φ is a rigid body motion (here no initial time is required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So the relation « d⃗vt + d⃗vT  t = 0 for all t » gives an equivalent definition to the definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let F(t)  :=  F t0  pt0 (t) and F T (t)  :=  F(t)T  and V (t)  :=  ⃗V t0  pt0 (t)  =  (Φt0  pt0 )′(t)  =  ⃗v(t, pt) (the Lagrangian and Eulerian velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) gives (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T )′(t) = 0 = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)T +  F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F T )′(t)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  =  F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)T + (F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)T )T  =  dV (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)−1 + (dV (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)−1)T (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  =  d⃗v(t, pt) +  d⃗v(t, pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Conversely, suppose d⃗v + d⃗vT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) gives D(⃗a,⃗b)g  Dt  = 0, thus (⃗a,⃗b)g(t, pt) = (⃗a,⃗b)g(t0, pt0)  for all t, t0 and all pt0 = Φt0  t (pt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)g = ( ⃗A, ⃗B)g for all t, t0, all pt0 and all  ⃗A, ⃗B ∈ ⃗Rn  t0: Thus �Φ is a rigid body motion, cf (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Representation of the spin tensor Ω: vectors, and pseudo-vectors  We are dealing here with concepts that are sometimes misunderstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Framework: Rn = R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Reminder  The determinant det|⃗e associated with a basis (⃗ei) in R3 is the alternating multilinear form defined  by det|⃗e(⃗e1,⃗e2,⃗e3) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The algebraic volume (or signed volume) limited by three vectors ⃗u1, ⃗u2, ⃗u3 is  det|⃗e(⃗u1, ⃗u2, ⃗u3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the (positive) volume is | det|⃗e(⃗u1, ⃗u2, ⃗u3)|, see § K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let A and B be two observers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A=English and B=French), let (⃗ai) be a Euclidean basis chosen  by A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' based on the foot), let (⃗bi) be a Euclidean basis chosen by B (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' based on the metre), see § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  106  107  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Representation of the spin tensor Ω: vectors, and pseudo-vectors  Let λ = ||⃗b1||a > 0 (change of unit of length coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The relation between the determinants is:  det  |⃗a = ±λ3 det  |⃗b  with  �  �  �  �  �  + if det  |⃗a (⃗b1,⃗b2,⃗b3) > 0  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if the bases have the same orientation),  − if det  |⃗a (⃗b1,⃗b2,⃗b3) < 0  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if the bases have opposite orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  In particular, if A and B use the same unit of length (or if A uses two (·, ·)g-Euclidean basis (⃗ai) and (⃗bi)),  then λ = 1 and det|⃗a = ± det|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With an imposed Euclidean dot product (·, ·)g: An endomorphism L is (·, ·)g-antisymmetric iff  ∀���u,⃗v, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,⃗v)g + (⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)g = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  LT = −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Definition of the vector product (cross product)  Let (⃗ei) be a (·, ·)g-orthonormal basis, let ⃗u,⃗v ∈ ⃗R3, and let ℓ⃗e,⃗u,⃗v ∈ L( ⃗R3, R) be the linear form defined  by  ℓ⃗e,⃗u,⃗v :  �  �  �  ⃗R3 → R  ⃗z → ℓ⃗e,⃗u,⃗v(⃗z) := det  |⃗e (⃗u,⃗v, ⃗z)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  (the algebraic volume of the parallelepiped limited by ⃗u,⃗v, ⃗z in the Euclidean chosen unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 The vector product, or cross product, ⃗u ∧e ⃗v of two vectors ⃗u and ⃗v is the (·, ·)g-Riesz  representation vector of ℓ⃗e,⃗u,⃗v, that is, ⃗u ∧e ⃗v ∈ ⃗R3 is characterized by ℓ⃗e,⃗u,⃗v(⃗z) = (⃗u ∧e ⃗v, ⃗z)g for all  ⃗z ∈ ⃗R3, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∀⃗z ∈ ⃗R3,  (⃗u ∧e ⃗v, ⃗z)g = det  |⃗e (⃗u,⃗v, ⃗z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  NB: ⃗u ∧e ⃗v depends on (·, ·)g since we need a (·, ·)g-Euclidean basis (⃗ei) (and depends on the orientation  of (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We have thus defined the bilinear cross product operator  ∧e :  � ⃗R3 × ⃗R3 → ⃗R3  (⃗u,⃗v) → ∧e(⃗u,⃗v) := ⃗u ∧e ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  (The bilinearity is trivial thanks to the multilinearity of the determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') If one Euclidean basis is  imposed by one observer to all the other observers, then ⃗u ∧e ⃗v is written ⃗u ∧ ⃗v (non objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Calculation of the vector product  ⃗u = �3  i=1 ui⃗ei, ⃗v = �3  i=1 vi⃗ei and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15) give  (⃗u ∧e ⃗v,⃗e1)g = det  |⃗e (⃗u,⃗v,⃗e1) = det  �  �  u1  v1  1  u2  v2  0  u3  v3  0  �  � = det  �  u2  v2  u3  v3  �  = u2v3 − u3v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  Similar calculation for (⃗u ∧e ⃗v,⃗e2)e and (⃗u ∧e ⃗v,⃗e3)e, thus  ⃗u ∧e ⃗v =  3  �  i=1  (ui+1vi+2 − ui+2vi+1)⃗ei,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗u ∧e ⃗v]|⃗e =  �  �  u2v3 − u3v2  u3v1 − u1v3  u1v2 − u2v1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  with the generic notation w4 := w1 and w5 = w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (In particular ⃗ei ∧e ⃗ei+1 = ⃗ei+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 1- ⃗u ∧e ⃗v = −⃗v ∧e ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- ⃗u ∥ ⃗v iff ⃗u ∧e ⃗v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- If ⃗u and ⃗v are independent then ⃗u ∧e ⃗v is orthogonal to the linear space Vect{⃗u,⃗v} generated by ⃗u  and ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- ⃗u ∧e ⃗v depends on the unit of measurement and on the orientation of (⃗ei): If (·, ·)a and (·, ·)b are  two Euclidean dot products, let λ > 0 such that (·, ·)a = λ2(·, ·)b, and then  ⃗u ∧a ⃗v = ±λ⃗u ∧b ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  107  108  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Representation of the spin tensor Ω: vectors, and pseudo-vectors  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- det|⃗e(⃗u,⃗v, ⃗z) = − det|⃗e(⃗v, ⃗u, ⃗z) (since det|⃗e is alternated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- If ⃗u ∥ ⃗v then det|⃗e(⃗u,⃗v, ⃗z) = 0 = (⃗u ∧e ⃗v, ⃗z)e, so ⃗u ∧e ⃗v ⊥g ⃗z, for all ⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if ⃗u ∧e ⃗v = 0 then (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  gives ⃗u ∥ ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- If ⃗z ∈ Vect{⃗u,⃗v} then det|⃗e(⃗u,⃗v, ⃗z) = 0 = (⃗u ∧e ⃗v,⃗z)g thus ⃗u ∧e ⃗v ⊥g ⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- (⃗u ∧a ⃗v, ⃗z)a  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  =  det  |⃗a (⃗u,⃗v, ⃗z)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  =  ±λ3 det  |⃗b  (⃗u,⃗v, ⃗z)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  =  ±λ3(⃗u ∧b ⃗v, ⃗z)b = ±λ3 1  λ2 (⃗u ∧b ⃗v, ⃗z)a, true  for all ⃗z, thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Prove that ⃗u ∧e ⃗v is a contravariant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is a vector (Riesz representation vector) in ⃗R3, so it is contravariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Or calculation: It satisfies the  contravariance change of basis formula, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Antisymmetric endomorphism represented by a vector  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 Let (⃗ei) be a chosen (·, ·)g-Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If an endomorphism Ω ∈ L( ⃗R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗R3) is  (·, ·)g-antisymmetric then there exists a unique vector ⃗ωe ∈ ⃗R3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all ⃗y, ⃗z ∈ ⃗R3,  (Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y, ⃗z)g = det  |⃗e (⃗ωe, ⃗y,⃗z),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', there exists a unique vector ⃗ωe ∈ ⃗R3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all ⃗y, ⃗z ∈ ⃗R3,  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y = ⃗ωe ∧e ⃗y ,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  And  [Ω]|⃗e =  �  �  0  −c  b  c  0  −a  −b  a  0  �  �  iff  [⃗ωe]|⃗e =  �  �  a  b  c  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  In particular Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ωe = ⃗0 (= ⃗ωe ∧e ⃗ωe), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗ωe is an eigenvector associated with the eigenvalue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ω is antisymmetric, thus [Ω]|⃗e is given as in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular [Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1]|⃗e = [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗e1]|⃗e =  �  �  0  c  −b  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Calculation of the components of ⃗ωe if it exists: Let ⃗ω = ω1⃗e1 + ω2⃗e2 + ω3⃗e3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' thus [⃗ω ∧ ⃗e1]|⃗e =  �  �  0  ω3  −ω2  �  �,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18), thus ω3 = c and ω2 = b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem with ⃗e2 so that ω1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus if it exists ⃗ω is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗ωe  given in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22) satisfies (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21): It exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 Let (·, ·)a and (·, ·)b be two Euclidean dot products (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' in foot and metre), let (⃗ai)  and (⃗bi) be Euclidean associated bases, let ||⃗b1||a = λ (change of unit coefficient), so (·, ·)a = λ2(·, ·)b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Suppose [Ω]|⃗a =  �  �  0  −c  b  c  0  −a  −b  a  0  �  �, thus [⃗ωa]|⃗a =  �  �  a  b  c  �  �, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then (change of representation  vector for Ω):  If (⃗bi) and (⃗ai) have the same orientation, then  ⃗ωb = λ⃗ωa,  If (⃗bi) and (⃗ai) have opposite orientation, then  ⃗ωb = −λ⃗ωa,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if ⃗bi = λ⃗ai for all i (change of unit, same orientation) then ⃗ωb = λ⃗ωa, and if ⃗b1 = −λ⃗a1, ⃗b2 = λ⃗a2,  ⃗b3 = λ⃗a3 (change of unit, opposite orientation) then ⃗ωb = −λ⃗ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: The formula ⃗ωb = ±λ⃗ωa is a change of vector formula, not a change of basis formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Apply (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation of ⃗ωe: Suppose [Ω]|⃗e = α  �  �  0  −1  0  1  0  0  0  0  0  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So Ω is the rotation with angle  π  2 in the  horizontal plane composed with the dilation with ratio α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And [⃗ωe]|⃗e = α  �  �  0  0  1  �  � = α⃗e3 is orthogonal to  the horizontal plane and gives the rotation axis and the dilation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  108  109  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Pseudo-cross product, and pseudo-vector  Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 Let Ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ω]|⃗e =  �  �  0  −c  b  c  0  −a  −b  a  0  �  � (see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Find a direct orthonormal basis (⃗bi)  (relative to (⃗ei)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ω]|⃗b =  √  a2+b2+c2  �  �  0  −1  0  1  0  0  0  0  0  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗b3 =  ⃗ωe  ||⃗ωe||e , that is, [⃗b3]|⃗e =  1  √  a2+b2+c2  �  �  a  b  c  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then choose ⃗b1 ⊥ ⃗b3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗b1]|⃗e =  1  √  a2+b2  �  �  −b  a  0  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then choose ⃗b2 = ⃗b3 ∧e ⃗b1, that is, [⃗b2]|⃗e =  1  √  a2+b2  1  √  a2+b2+c2  �  �  −ac  −bc  a2 + b2  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (⃗bi) is a direct orthonormal  basis, and the transition matrix from (⃗ei) to (⃗bi) is P =  �  [⃗b1]|⃗e  [⃗b2]|⃗e  [⃗b3]|⃗e  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And [Ω]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P (change  of basis formula), with P −1 = P T (change of orthonormal basis), thus [Ω]|⃗b = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  With [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗b1]|⃗e =  1  √  b2+c2  �  �  0  −c  b  c  0  −a  −b  a  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  −b  a  0  �  � =  1  √  b2+c2  �  �  −ac  −bc  a2 + b2  �  � =  √  a2+b2+c2[⃗b2]|⃗e (expected),  [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗b2]|⃗e =  1  √  b2+c2  1  √  a2+b2+c2  �  �  0  −c  b  c  0  −a  −b  a  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  −ac  −bc  a2 + b2  �  � =  1  √  b2+c2  1  √  a2+b2+c2  �  �  bc2 + b(a2 + b2)  −ac2 − a(a2 + b2)  abc − abc  �  � =  −  √  a2+b2+c2[⃗b1]|⃗e  (expected),  and  [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗b3]|⃗e  =  [⃗0]  (expected  since ⃗b3  ∥  ⃗ωe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  =  √  a2+b2+c2 �  [⃗b2]|⃗e  −[⃗b1]|⃗e  [⃗0]|⃗e  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P)ij = [⃗bi]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗bj]|⃗e gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Curl  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 If ⃗v is a C1 vector field, if (⃗ei) is a Euclidean basis in ⃗R3, and if ⃗v = �3  i=1 vi⃗ei, then  the curl (or rotational) of ⃗v relative to (⃗ei) is the vector field  ⃗  curle⃗v = ⃗  rote⃗v given by  ⃗  curle⃗v =  3  �  i=1  ( ∂vi+2  ∂xi+1  − ∂vi+1  ∂xi+2  )⃗ei,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [ ⃗  curle⃗v]|⃗e =  �  �  ∂v3  ∂x2 − ∂v2  ∂x3  ∂v1  ∂x3 − ∂v3  ∂x1  ∂v2  ∂x1 − ∂v1  ∂x2  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 Let Ω(t, pt) = d⃗v(t,pt)−d⃗v(t,pt)T  2  , and let ⃗ωe(t, pt) be the associated vector relative to  the Euclidean basis (⃗ei), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  ⃗ωe = 1  2  ⃗  curle⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) gives [Ω]|⃗e =  1  2  �  �  0  ∂v1  ∂x2 − ∂v2  ∂x1  ∂v1  ∂x3 − ∂v3  ∂x1 0  ∂v2  ∂x3 − ∂v3  ∂x2 0  �  �, with [Ω]|⃗e antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24) gives (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Pseudo-cross product, and pseudo-vector  Framework: M31 the space of 3∗1 matrices, so we leave the vector framework to enter the matrix world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 A column matrice is also called a pseudo-vector, or a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19  �  �  x1  x2  x3  �  � noted  =  [⃗x] and  �  �  y1  y2  y3  �  � noted  =  [⃗y] being two matrices in M31, their pseudo-cross  product is  �  �  x1  x2  x3  �  � ⟲∧  �  �  y1  y2  y3  �  � :=  �  �  x2y3 − x3y2  x3y1 − x1y3  x1y2 − x2y1  �  � noted  =  [⃗x]  ⟲∧[⃗y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Thus the pseudo-cross product of two pseudo-vectors is a pseudo-vector (is a matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  109  110  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Antisymmetric matrix represented by a pseudo-vector  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 Let A = [Aij] =  �  �  0  −c  b  c  0  −a  −b  a  0  �  � be an antisymmetric matrix (Aji = −Aij for all  i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The pseudo-vecteur  ⟲ω associated to A is the column matrix  ⟲ω :=  �  �  a  b  c  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y] =  ⟲ω  ⟲∧[⃗y] ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  y1  y2  y3  �  � =  ⟲ω  ⟲∧  �  �  y1  y2  y3  �  � ,  for all matrix [⃗y] =  �  �  y1  y2  y3  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Antisymmetric endomorphism and its pseudo-vectors representations  Let R3 be our usual affine space, (·, ·)g be a Euclidean dot product, and (⃗ei) be a (·, ·)g-Euclidean  associated basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Ω be an antisymmetric endomorphism relative to (·, ·)g, so ΩT = −Ω, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus [Ω]|⃗e is an antisymmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Call  ⟲ω the associated pseudo-vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27), for all ⃗y ∈ ⃗R3,  [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y]|⃗e =  ⟲ω  ⟲∧[⃗y]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  This formula is widely used in mechanics, and unfortunately sometimes noted Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗y = ⃗ω ∧ ⃗y (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ):  Be careful: (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28) is not a vectorial formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This is just a formula for matrix calculations which  gives false result if a change of basis is considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', with (⃗a1,⃗a2,⃗a3) be a (·, ·)g-Euclidean basis, and  (⃗b1,⃗b2,⃗b3) = (−⃗a1,⃗a2,⃗a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So (⃗bi) is also a (·, ·)g-Euclidean basis, but with a different orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- Vector approach: Let P be the transition matrix from (⃗ai) to (⃗bi), so P =  �  �  −1  0  0  0  1  0  0  0  1  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  [Ω]|⃗a =  �  �  0  −c  b  c  0  −a  −b  a  0  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, Ω being an endomorphism, the change of basis formula gives  [Ω]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ω]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P =  �  �  −1  0  0  0  1  0  0  0  1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  0  −c  b  c  0  −a  −b  a  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  −1  0  0  0  1  0  0  0  1  �  � =  �  �  0  c  −b  −c  0  −a  b  a  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  Thus the vectors ⃗ωa and ⃗ωb are given by (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22):  [⃗ωa]|⃗a =  �  �  a  b  c  �  � ,  [⃗ωb]|⃗b =  �  �  a  −b  −c  �  � ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  ⃗ωa = a⃗a1 + b⃗a2 + c⃗a3,  ⃗ωb = a⃗b1 − b⃗b2 − c⃗b3,  �  thus  ⃗ωb = −⃗ωa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  2- Matrix approach (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27) gives [Ω]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y] =  ⟲ωa  ⟲∧[⃗y] and [Ω]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗y] =  ⟲ωb  ⟲∧[⃗y], with  ⟲ωa =  �  �  a  b  c  �  �  and  ⟲ωb =  �  �  a  −b  −c  �  � ,  so  ⟲ωa ̸= −  ⟲ωb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  And  ⟲ω does not represent a single vector either, since it does not satisfy the vector change of basis formula  ⟲ωb ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⟲ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  ⟲ω is not a vector (is not tensorial): It is just a matrix (called a “pseudo-vector”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Examples  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Rectilinear motion  Let �Φ : [t1, t2] × Obj → Rn be a C1 motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t0 ∈]t1, t2[ and PObj ∈ Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  110  111  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 The motion of PObj is rectilinear iff, for all t0, t ∈ [t1, t2],  �ΦPObj (t) − �ΦPObj (t0)  t−t0  ∥ �ΦPObj  ′(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  And the motion is rectilinear uniform iff, for all t0, t ∈ [t1, t2],  �ΦPObj (t) = �ΦPObj (t0) + (t−t0) �ΦPObj  ′(t0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  p(t) = p(t0) + (t−t0) ⃗V t0(t0, p(t0))  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  when p(t) = �Φ(t, PObj), that is, the trajectory is traveled at constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Circular motion  Let ( ⃗E1, ⃗E2) be a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t0 ∈ [t1, t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A motion Φt0 is a circular motion iff  −−−−−→  OΦt0  P (t) = x(t) ⃗E1 + y(t) ⃗E2,  [−−−−−→  OΦt0  P (t)]| ⃗E =  �  x(t) = a + R cos(θ(t))  y(t) = b + R sin(θ(t))  �  ,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  for some R > 0 (called the radius), some a, b ∈ R, and some function θ : R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  �  a  b  �  = OC ∈ R2  is the center of the circle and θ(t) is the angle at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the particle PObj (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �Φ(t0, PObj) = P) stays on  the circle with center OC and radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The circular motion is uniforme iff, for all t, θ′′(t) = 0, that is, ∃ω0 ∈ R, ∀t ∈ [t1, t2], θ(t) = ω0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation:  ⃗ϕt0  P (t) = R cos(θ(t) ⃗E1 + R sin(θ(t)) ⃗E2,  so  [⃗ϕt0  P (t)]| ⃗E =  �  R cos(θ(t))  R sin(θ(t))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Thus the Lagrangian velocity of a circular motion is  ⃗V t0  P (t) = (Φt0  t )′(t) = (⃗ϕt0  P )′(t),  so  [⃗V t0  P (t)]| ⃗E = Rθ′(t)  �  − sin(θ(t))  cos(θ(t))  �  ,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  and ⃗V t0  P (t) is orthogonal to ⃗ϕt0  P (t) (the radius vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the Lagrangian acceleration is  ⃗Γ t0  P (t) = Rθ′′(t)  �  − sin(θ(t))  cos(θ(t))  �  + R(θ′(t))2  �  − cos(θ(t))  − sin(θ(t))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Consider  ⃗er(t) =  ⃗ϕt0  P (t)  ||⃗ϕt0  P (t)|| =  �  cos(θ(t))  sin(θ(t))  �  ,  and  ⃗eθ(t) =  �  − sin(θ(t))  cos(θ(t))  �  ,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  thus (⃗er(t),⃗eθ(t)) is an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then :  ⃗V t0  P (t) = Rθ′(t)⃗eθ(t),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  and :  ⃗Γ t0  P (t) = −R(θ′(t))2 ⃗er(t) + Rθ′′(t)⃗eθ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', in R3 and a motion in he “horizontal” plane given by (⃗e1,⃗e2), the vertical line being given by ⃗E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here  ⃗V t0  P (t) = ⃗ω(t) ∧ ⃗ϕt0  P (t),  where  ⃗ω(t) = ω(t)⃗e3  and  ω(t) = θ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  And  ⃗Γ t0(t) = d⃗ω  dt (t) ∧ ⃗ϕt0  P (t) + ⃗ω(t) ∧ ⃗V t0  P (t)  (= Rdω  dt (t)⃗eθ(t) − ω2(t)R⃗er(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  111  112  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Motion of a planet (centripetal acceleration)  Illustration: Obj is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a planet from the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (⃗e1,⃗e2,⃗e3) be a Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' fixed relative to stars an (⃗e1,⃗e2) define the ecliptic plane),  (·, ·)g be the Euclidean associated dot product, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='|| the Euclidean associated norm, O an origin in R3  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the center of the Sun), and R = (O, (⃗ei)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Consider a motion �Φ of Obj in R, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t0 ∈ [t1, t1], and consider Φt0 =noted Φ or ⃗ϕ t0 =noted ⃗ϕ,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 The motion of a particle PObj is a centripetal acceleration motion iff the particle is not  static and, at all time, its acceleration vector ⃗A(t) points to a fixed point F (focus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We will take the focus F as the origin of the referential, that is, O := F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, for all t ∈ [t1, t2], −−−−−→  OΦP (t) ∥ ⃗AP (t), that is,  −−−−−→  OΦP (t) ∧ ⃗AP (t) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  Remark E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23 A rectilinear motion is a centripetal acceleration motion, but such a motion is usually  excluded in the definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24 The motion of a planet from the solar system is a centripetal acceleration motion: An  elliptical motion of focus the center of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25 The second Newton’s law of motion � ⃗f = m⃗γ (Galilean referential) gives: If � ⃗f is,  at all time, directed to a unique point F, then the motion is a centripetal acceleration motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let Φ be a centripetal acceleration motion, let O be the focus, and let ⃗ϕP (t) := −−−−−→  OΦP (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So the  Lagrangian velocity and acceleration are  ⃗VP (t) = dΦP  dt (t) = d⃗ϕP  dt (t),  and  ⃗AP (t) = d2ΦP  dt2 (t) = d2⃗ϕP  dt2 (t),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  and ⃗ϕP (t) ∧ ⃗AP (t) = ⃗0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26 The areolar velocity at t is the vector  ⃗Z(t) = 1  2 ⃗ϕP (t) ∧ ⃗VP (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27 If Φ is a centripetal acceleration motion, then the areolar velocity is contant, that is,  d⃗Z  dt (t) = ⃗0 pour tout t, so  ⃗Z(t) = ⃗Z(t0),  ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  That is, the position vectors sweep equal areas in equal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗Z(t0) = ⃗0 iff Φ is a rectilinear motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If ⃗Z(t0) ̸= ⃗0 then :  ⃗ϕP (t) and ⃗VP (t) are orthogonal to ⃗Z(t0) at all time t,  The motion of the particle PObj takes place in the affine plane orthogonal to ⃗Z(t0) passing through O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗VP (t) never vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43) give 2 d⃗Z  dt (t) = d⃗ϕP  dt (t)∧⃗VP (t)+⃗ϕ(t)∧ d⃗VP  dt (t) = ⃗VP (t)∧⃗VP (t)+⃗ϕ(t)∧ ⃗AP (t) = ⃗0+⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus ⃗Z is constant, ⃗Z(t) = ⃗Z(t0) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, if ⃗Z(t0) ̸= ⃗0 then ⃗Z(t) ̸= ⃗0 pour tout t, and  ⃗Z(t) = 1  2 ⃗ϕP (t) ∧ ⃗VP (t) gives that ⃗ϕP (t) et ⃗VP (t) are orthogonal to ⃗Z(t0) for all t, thus ⃗AP (t) is  orthogonal to ⃗Z(t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Taylor expansion reads ⃗ϕP (t) = ⃗ϕP (t0) + ⃗VP (t0)(t−t0) +  � t  τ=t0 ⃗AP (τ)(t−τ)2 dτ, with ⃗VP (t0)  and ⃗AP (τ) ⊥ ⃗Z(t0) for all τ, thus ⃗ϕP (t) − ⃗ϕP (t0) ⊥ ⃗Z(t0) for all τ, that is −−−→  Op(t) − −−→  OP = −−−→  Pp(t) ⊥ ⃗Z(t0)  for all τ, Thus p(t) belongs to the affine plane containing P orthogonal to ⃗Z(t0), for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And −−→  OP =  ⃗ϕP (t0) ⊥ ⃗Z(t0), thus O belong to the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗Z(t) = ⃗Z(t0) ̸= ⃗0 implies ⃗VP (t) ̸= ⃗0 for all t, and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45) gives: (⃗ϕP (t), ⃗VP (t), ⃗Z(t0)) is a positively-  oriented basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since ⃗ϕP and ⃗V are continuous and do not vanish, since ⃗Z(t0) ̸= ⃗0, we get: PObj “turns  around ⃗Z(t0)” and keeps its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  112  113  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  If ⃗Z(t) = ⃗0 then ⃗ϕP (t) ∥ ⃗VP (t) for all t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45), so ⃗VP (t) = f(t)⃗ϕP (t) where f is some scalar  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗VP (t) = ⃗ϕP ′(t) gives ⃗ϕP ′(t) = f(t)⃗ϕP (t), thus ⃗ϕP (t) = ⃗ϕP (t0)eF (t) where F is a primitive  of f s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F(t0) = 0, thus ⃗ϕP (t) ∥ ⃗ϕP (t0), so −−−−−→  OΦP (t) ∥ −−−−−−→  OΦP (t0), for all t: The motion is rectilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Non rectilinear motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  The area swept by ⃗ϕP (t) is, at first order, the area of the  triangle whose sides are ⃗ϕP (t) and ⃗ϕP (t + τ) (“anglular sector”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, with τ close to 0, let  ⃗St(τ) = 1  2 ⃗ϕP (t) ∧ ⃗ϕP (t + τ),  and  St(τ) = ||⃗St(τ)||,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  the vectorial an scalar area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With ⃗ϕP (t+τ) = ⃗ϕP (t) + ⃗VP (t)τ + o(τ) we get  ⃗St(τ) = 1  2 ⃗ϕP (t) ∧ (⃗VP (t)τ + o(τ)),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  Since ⃗St(0) = 0 we get  ⃗St(τ)−⃗S(0)  τ  = 1  2 ⃗ϕP (t) ∧ ⃗VP (t) + o(1), then  d⃗St  dτ (0) = 1  2 ⃗ϕP (t) ∧ ⃗VP (t) = ⃗Z(t) = ⃗Z(t0),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  thanks to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46), thus  d⃗St  dτ (0) = d⃗St0  dτ (0),  ∀t ∈ [t0, T],  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)  that is, the rate of variation of ⃗St is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And with ||⃗St(∆τ)||2 = (⃗St(∆τ), ⃗St(∆τ)) we get  d||⃗St||2  dτ  (∆τ) = 2(d⃗St  dτ (∆τ), ⃗St(∆τ)),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51)  so, since ⃗St(0) = 0,  d||⃗St||2  dτ  (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52)  Therefore the function t → ||⃗St(0)||2 = St(0)2 is constant, thus t → St(0) est constant, and dSt  dτ (0) is  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28 Give a parametrization of the swept area, and redo the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  r(t) = ||⃗ϕP (t)||,  θ(t) = �  p(t)OP  (angle),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53)  then  ⃗ϕP (t) =  �  �  r(t) cos(θ(t))  r(t) sin(θ(t))  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  Thus  ⃗VP (t) =  �  �  r′(t) cos(θ(t) − r(t))θ′(t) sin(θ(t))  r′(t) sin(θ(t) + r(t))θ′(t) cos(θ(t))  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55)  With (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45) we get  ⃗Z(t) = 1  2  �  �  0  0  r2(t)θ′(t)  �  � ,  with  r2(t)θ′(t) = r2(t0)θ′(t0)  (constant),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A parametrization of the swept area is then  ⃗A :  �  [0, 1] × [t0, T] → R3  (ρ, t) → ⃗A(ρ, t)  �  ,  ⃗A(ρ, t) =  �  �  ρ r(t) cos(θ(t))  ρ r(t) sin(θ(t))  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='57)  Therefore, the tangent associated vectors are  ∂ ⃗A  ∂ρ (ρ, t) =  �  �  r(t) cos(θ(t))  r(t) sin(θ(t))  0  �  � ,  ∂ ⃗A  ∂t (ρ, t) =  �  �  ρr′(t) cos(θ(t) − ρr(t))θ′(t) sin(θ(t))  ρr′(t) sin(θ(t) + ρr(t))θ′(t) cos(θ(t))  0  �  � ,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58)  113  114  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Examples  hence the vectorial and scalare element areas are  d⃗σ = (∂ ⃗A  ∂ρ ∧ ∂ ⃗A  ∂t )dρdt =  �  �  0  0  ρr2θ′ dρdt  �  � ,  dσ = ρr2θ′ dρdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)  Therefore the area between t0 and t is  A(t) = A(t0) +  � 1  ρ=0  � t  τ=t0  ρr2(τ)θ′(τ) dρdτ = 1  2  � t  τ=t0  r(τ)2θ′(τ) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60)  Hence  A′(t) = r(t)2θ′(t) = r(t0)2θ′(t0)  (= constant = ||⃗Z(t0)||),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29 Prove the Binet formulas (non rectilinear central motion):  VP (t)2 = Z2  0  � 1  r2 + (d 1  r  dθ )2�  (t),  ⃗ΓP (t) = −Z2  0  r2  �1  r + d2 1  r  dθ2  �  (t)⃗er(t),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='62)  for the energy and the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27 tells that Φ is a planar motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53) and ⃗er(t) =  � cos(θ(t))  sin(θ(t))  �  we have  ⃗ϕ(t) = r(t)⃗er(t) (in the plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗eθ(t) =  � − sin(θ(t))  cos(θ(t))  �  , thus  ⃗V (t) = dr  dt (t)⃗er(t) + r(t)d⃗er  dt (t) = r′(t)⃗er(t) + r(t)θ′(t)⃗eθ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And ⃗er(t) ⊥ ⃗eθ(t) gives  V 2(t) = (r′(t))2 + (r(t)θ′(t))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Since θ′(t) ̸= 0 for all t (non rectilinear central motion) Let s(θ(t)) = r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let us suppose that θ is C1, thus  θ′ > 0 or θ′ < 0, and θ : t → θ(t) defines a change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  r′(t) = s′(θ(t))θ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61) and θ′(t) =  Z0  r2(t) give  V 2(t(θ)) = (s′(θ))2 Z2  0  r4(t) + r2(t) Z2  0  r4(t) = Z2  0((s′(θ))2  s4(θ)  +  1  s2(θ)) = Z2  0[  �d 1  s  dθ (θ)  �2  +  1  s2(θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus r(t) = s(θ) and dr  dθ := ds  dθ give the first Binet formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  ⃗Γ(t) = r′′(t)⃗er(t) + r′(t)d⃗er  dt (t) + (r′(t)θ′(t) + r(t)θ′′(t))⃗eθ(t) + r(t)θ′(t)d⃗eθ  dt (t),  with d⃗er  dt ∥ ⃗eθ, and d⃗eθ  dt (t) = −θ′(t)⃗er(t), and ��eθ ⊥ ⃗Γ (central motion), we get  ⃗Γ(t) = (r′′(t) − r(t)(θ′(t))2)⃗er(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And  r′(t) = s′(θ)θ′(t) = s′(θ) Z0  r2(t) = Z0 s′(θ)  s2(θ) = −Z0  d 1  s  dθ (θ),  thus  r′′(t) = −Z0  d2 1  s  dθ2 (θ) θ′(t) = − Z2  0  r2(t)  d2 1  s  dθ2 (θ),  which is the second Binet formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  114  115  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Riesz representation theorem  F  Riesz representation theorem  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  The Riesz representation theorem  Framework: (E, (·, ·)g) is Hilbert space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E is a vector space equipped with an inner dot product (·, ·)g  such that, with the associated norm defined by||⃗v||g :=  �  (⃗v,⃗v)g, (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g) is a complete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is the space of the linear and continuous forms on E (the space of linear “measuring tools”)  equipped with its norm ||ℓ||E∗ :=  sup  ||⃗x||g=1  |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We have the easy statement:  ∀⃗v ∈ E (vector), ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vg ∈ E∗ (linear continuous form)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗v, ⃗x)g, ∀⃗x ∈ E,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  and ||vg||E∗ = ||⃗v||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Usual notation in finite dimension: vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ⃗v •g ⃗x, or simply v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ⃗v • ⃗x if a  chosen (·, ·)g is imposed to all observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Indeed: Define vg : E → R by vg(⃗x) = (⃗v, ⃗x)g for all ⃗x ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The definition domain of vg is E and  vg is trivially linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the Cauchy–Schwarz inequality gives |vg(⃗x)| = |(⃗v, ⃗x)g| ≤ ||⃗v||g ||⃗x||g for all  ⃗x ∈ E, thus ||vg||E∗ ≤ ||⃗v||g < ∞, thus vg is continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And |vg(⃗v)| = |(⃗v,⃗v)g| = ||⃗v||g ||⃗v||g, thus  ||vg||E∗ ≥ ||⃗v||g, thus ||vg||E∗ = ||⃗v||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Riesz representation theorem concerns the converse: If you choose an inner dot product (·, ·)g  in E (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' English of French), then you can represent a “measuring instrument” ℓ ∈ E∗ by a vector ⃗ℓg ∈ E:  Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 (Riesz representation theorem, and definition) (E, (·, ·)g) being a Hilbert space,  ∀ℓ ∈ E∗ (linear continuous form), ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ℓg ∈ E (vector)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗ℓg, ⃗x)g, ∀⃗x ∈ E,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  and ||⃗ℓg||g = ||ℓ||E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗ℓg is called the (·, ·)g-Riesz representation vector of ℓ (depends on g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Usual notation in finite dimension: vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ⃗v •g ⃗x, or simply v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ⃗v • ⃗x if a chosen (·, ·)g is imposed  to all observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Easy in finite dimension: With a basis (⃗ei), if [ℓ]|⃗e = ( ℓ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓn ) (row matrix since ℓ is a linear  form) then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗e = [⃗ℓg]T  ⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗e, thus [⃗ℓg]⃗e = [g]−1  |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ℓ]T  |⃗e (column matrix), thus ⃗ℓg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  General case (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with E = L2(Ω) and the finite element method): If ℓ = 0 then ⃗ℓg = ⃗0 (trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Suppose ℓ ̸= 0: Thus Kerℓ = ℓ−1({0}) ̸= {⃗0} (the kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If dim E = 1, it is trivial (exercise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Suppose  dim E ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since ℓ is continuous, its kernel Kerℓ = ℓ−1({0}) is closed in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, if ⃗x ∈ E, then its (·, ·)g-  orthogonal projection ⃗x0 ∈ Kerℓ on Kerℓ exists, is unique, and is given by: ∀⃗y0 ∈ Kerℓ, (⃗x −⃗x0, ⃗y0)g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (So ⃗x − ⃗x0 ⊥g Kerℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Choose a ⃗x /∈ Kerℓ (possible since ℓ ̸= 0), and let ⃗n :=  ⃗x−⃗x0  ||⃗x−⃗x0||g ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So ⃗n is a (·, ·)g-  orthonormal vector to Kerℓ, and (Kerℓ)⊥ = Vect{⃗n} since dim(Kerℓ)⊥ = 1 (in finite dimension cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the  Dimension Formula which states that the dimension of the domain of a linear map is the sum of the  dimension of its range and the dimension of its kernel, and in infinite dimension see next exercise F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And E = Kerℓ ⊕ (Kerℓ)⊥ since both vector spaces are closed (an orthogonal is always closed in a Hilbert  space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus if ⃗x ∈ E then ⃗x = ⃗x0 + λ⃗n ∈ Kerℓ ⊕ (Kerℓ)⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (⃗x,⃗n)g = λ and ℓ(⃗x) = 0 + λℓ(⃗n) =  (⃗x,⃗n)gℓ(⃗n) = (⃗x, ℓ(⃗n)⃗n)g (bilinearity of (·, ·)g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗ℓg := ℓ(⃗n)⃗n satisfies (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And if ⃗ℓg1 and ⃗ℓg2  satisfy (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) then (⃗ℓg1 − ⃗ℓg2, ⃗x)g = 0 for all ⃗x ∈ E, thus ⃗ℓg1 − ⃗ℓg2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗ℓg is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And the Cauchy–Schwarz theorem give ||ℓ||E∗ := sup||⃗x||g=1 |ℓ(⃗x)| = sup||⃗x||g=1 |(⃗ℓg, ⃗x)g| = ||⃗ℓg||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗Rg is an isomorphism between Banach spaces: linearity since (⃗Rg(ℓ + λm), ⃗x)g = (ℓ + λm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x +  λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗Rg(ℓ), ⃗x)g +λ(⃗Rg(m), ⃗x)g = (⃗Rg(ℓ)+λ⃗Rg(m), ⃗x)g for all ⃗x gives ⃗Rg(ℓ+λm) = ⃗Rg(ℓ)+λ⃗Rg(m),  bijectivity thanks to (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2), and the norm is kept since ||⃗ℓg||g = ||ℓ||E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Prove: If ℓ ∈ E∗−{0} then dim(Kerℓ)⊥ = 1 (= dim(Im(ℓ)) = dim R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the restriction ℓ|Kerℓ⊥ :  �  (Kerℓ)⊥ → R  ⃗x → ℓ|Kerℓ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is linear (since ℓ is), and thus one to  one: Indeed it is onto since ℓ ̸= 0, and it is one to one since if ℓ|Kerℓ⊥(⃗x) = 0 = ℓ(⃗x) then ⃗x ∈ (Kerℓ)⊥ � Kerℓ = {⃗0},  thus ⃗x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus dim(Kerℓ)⊥ ≤ dim(Im(ℓ)) = 1: Indeed, if ⃗z1, ⃗z2 ∈ (Kerℓ)⊥−{⃗0} then ℓ|Kerℓ⊥(⃗z1) ∈ R and  ℓ|Kerℓ⊥(⃗z2) ∈ R, thus ∃λ ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ℓ|Kerℓ⊥(⃗z2) = λℓ|Kerℓ⊥(⃗z1), thus ℓ|Kerℓ⊥(⃗z2 − λ⃗z1) = 0, thus ⃗z2 − λ⃗z1 = ⃗0 since  ℓ|Kerℓ⊥ is one to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗n ∈ (Kerℓ)⊥ gives dim(Kerℓ)⊥ ≥ 1 (above proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus dim(Kerℓ)⊥ = 1 = Vect{⃗n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  115  116  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Riesz representation operator  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The Riesz representation operator  The Riesz representation theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 gives the (·, ·)g-Riesz representation operator  ⃗Rg :  �  E∗ → E  ℓ → ⃗Rg(ℓ) := ⃗ℓg,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (⃗Rg(ℓ),⃗v)g = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v, ∀⃗v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  So ⃗Rg transforms a « covariant ℓ » into a « contravariant ⃗ℓg » thanks to the tool (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB (fundamental): ⃗Rg is a isomorphism between the Banach spaces (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g) and (E∗, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||E∗), but ⃗Rg  is not canonical since it requires a man made tool (an inner dot product chosen by some observer) to be  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (An isomorphism E ↔ E∗ can never be canonical, see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  And with G the set of inner dot products in E, we have thus defined the Riesz representation mapping  ⃗R :  �  G × E∗ → E  (g, ℓ) → ⃗R(g, ℓ) := ⃗ℓg = ⃗Rg(ℓ) = ⃗ℓ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  So ⃗R has two inputs: A choice (·, ·)g by an observer for the first slot, a linear form for the second slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Quantification with a basis  Here E is finite dimensional, dim E = n, ℓ ∈ E∗ (a linear form), (·, ·)g is an inner dot product, (⃗ei) is a  basis, (πei) is the dual basis (classical notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  gij = g(⃗ei,⃗ej),  ℓ =  n  �  i=1  ℓiπei,  ⃗ℓg =  n  �  i=1  (⃗ℓg)i⃗ei,  ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πej =  n  �  i=1  Rij⃗ei,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e = [gij], [ℓ]|πe = ( ℓ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓn ) (row matrix), [⃗ℓg]|⃗e =  �  �  �  (⃗ℓg)1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (⃗ℓg)n  �  �  � (column matrix), [⃗Rg]πe,⃗e = [Rij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Duality notations: ℓ = �n  i=1ℓiei, ⃗ℓg = �n  i=1ℓi  g⃗ei, ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ej = �n  i=1Rij⃗ei, [⃗Rg]e,⃗e = [Rij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  [⃗ℓg] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ℓ]T  and  [⃗Rg] = [g]−1 ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (⃗ℓg)i =  n  �  j=1  ([g]−1)ij(ℓ)j =  n  �  j=1  (⃗Rg)ij(ℓ)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Full matrix notation: [⃗ℓg]|⃗e = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([ℓ]|πe)T , and [⃗Rg]|πe,⃗e = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notation to see the change of variance induced by (·, ·)g (bottom index for ℓ, top index for ⃗ℓg):  ℓi  g =  n  �  j=1  Rijℓj,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ℓi  g = �n  j=1gijℓj when ([g]−1)ij =noted [gij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (In particular, if (⃗ei) is a (·, ·)g-orthonormal basis, then [⃗Rg] = [g]−1 = I and ℓi  g = ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗e = [⃗ℓg]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗e for all ⃗x, thus [ℓ]|⃗e = [⃗ℓg]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e, thus [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|⃗e = [ℓ]T  |⃗e (since  [g]|⃗e = [g]T  |⃗e), thus [⃗ℓg] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ℓ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) ⃗ℓg gives �n  j=1(ℓ)j ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πj = �n  i=1(⃗ℓg)i⃗ei, thus �n  i,j=1(ℓ)jRij⃗ei = �n  i=1(⃗ℓg)i⃗ei, thus  �n  j=1Rij(ℓ)j = (⃗ℓg)i for all i, thus [⃗Rg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ℓ]T = [⃗ℓg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus [⃗Rg] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 If a chosen inner dot product (·, ·)g is imposed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Euclidean foot based) and if duality  notations are used, then a usual notation for ⃗ℓg is ℓ♯, since ⃗ℓg = ⃗Rg(ℓ) = �n  i=1ℓi⃗ei with a top index for  ℓi: the index i has been raised through ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) read (isometric framework)  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = ℓ♯ • ⃗x  and  [ℓ♯]|⃗e = [g]−1  |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ℓ]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  We won’t use this notation (we deal with objectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  116  117  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Change of Riesz representation vector, and Euclidean case  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Change of Riesz representation vector, and Euclidean case  For one linear form ℓ ∈ E∗, two observers with their inner dot products (·, ·)g and (·, ·)h get two Riesz  representation vectors ⃗ℓg = ⃗Rg(ℓ) and ⃗ℓh = ⃗Rh(ℓ) given by, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2):  ∀⃗x ∈ E,  (⃗ℓg, ⃗x)g = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = (⃗ℓh, ⃗x)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 For any basis (⃗ei) in E, we have the change of representation vector formula:  [h]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓh]|⃗e = [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|⃗e,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [⃗ℓh]|⃗e = [h]−1  |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  In particular (for the Euclidean case), with λ > 0:  If (·, ·)g = λ2(·, ·)h  then  ⃗ℓh = λ2⃗ℓg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Conversely, if ⃗ℓh = λ2⃗ℓg for all linear forms ℓ ∈ E∗, then (·, ·)g = λ2(·, ·)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, a linear form ℓ cannot be identified with a Riesz representation vector (which one: ⃗ℓg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗ℓh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In other words, a Riesz representation vector ⃗Rg(ℓ) is not objective, is not intrinsic to a linear form ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11) is a “change of vector formula” (one linear form gives two vectors relative to two  inner dot products);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is not a “change of basis formula” (for one vector and its two sets of components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) gives [⃗x]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|⃗e = [⃗x]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[h]|��e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓh]|⃗e for all ⃗x, hence [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|⃗e = [h]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓh]|⃗e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular λ2(·, ·)h = (·, ·)g give λ2(⃗ℓg, ⃗x)h = (⃗ℓg, ⃗x)g =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)(⃗ℓh, ⃗x)h for all ⃗x, hence λ2⃗ℓg = ⃗ℓh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Converse: λ2⃗ℓg = ⃗ℓh for all ℓ gives λ2(⃗ℓg, ⃗x)h = (⃗ℓh, ⃗x)h  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  = (⃗ℓg, ⃗x)g, for all ⃗x and for all ℓ, thus for  all ⃗ℓg thanks to the isomorphism ⃗Rg : E∗ → E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3), thus λ2(·, ·)h = (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 If (·, ·)g and (·, ·)h are the Euclidean dot products made with the foot and the metre then,  with (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9),  (·, ·)g = λ2(·, ·)h  =⇒  ⃗ℓh = λ2⃗ℓg,  with  λ2 > 10 :  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  ⃗ℓg (English) and ⃗ℓh (French) are quite different!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A Riesz representation vector is subjective, and certainly  not “canonical” (a word that you may find in books where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' nothing is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', aviation: If you do want to use a Riesz representation vector to represent a ℓ ∈ Rn∗, it is vital to  know which Euclidean dot product is in use, see also remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 (Mars Climate Orbiter Crash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Recall:  The foot is the international unit of altitude for aviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 If f ∈ C1(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and p ∈ Rn, the differential of f at p is the linear form df(p) ∈ Rn∗  defined by, for all ⃗w ∈ ⃗Rn,  df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w := lim  h→0  f(p + h⃗w) − f(p)  h  (definition independent of any inner dot product),  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If you can choose an inner dot product (·, ·)g then the gradient  ⃗  gradgf(p) is the (·, ·)g-Riesz  representation vector of df(p):  ⃗  gradgf(p) := ⃗Rg(df(p)),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w =  ⃗  gradgf(p) •g ⃗w, ∀⃗w ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  And (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12) gives  ⃗  gradhf(p) = λ2 ⃗  gradgf(p)  with  λ2 > 10 (English vs French) :  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  The gradient is very dependent on the observer (the gradient is subjective, the differential is objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 The “gradient” is observer dependent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We already had this observer dependence for the  usual derivative in the 1-D case f : x ∈ R → f(x) ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Question: What does f ′(x) = 3 mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 11- For one observer, it means f ′(x) = limh→0  f(x+h)−f(x)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' but.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' where in the departure  space this observer has chosen a basis vector ⃗a of length 1 for him (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' length 1 foot) which he calls 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, with no abusive notations, his derivative f ′(x) is in fact f ′  a(x) = limh→0  f(x+h⃗a)−f(x)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  117  118  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A Riesz representation vector is contravariant  12- For some other observer, it means f ′(x) = limh→0  f(x+h)−f(x)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' but.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' where in the departure  space this observer has chosen a basis vector ⃗b of length 1 for him (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' length 1 metre) which he calls 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, with no abusive notations, his derivative f ′(x) is in fact f ′  b(x) = limh→0  f(x+h⃗b)−f(x)  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  13- Both observer use the same formula f ′(x) = limh→0  f(x+h)−f(x)  h  but get different results!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In-  deed, if ⃗b = λ⃗a, then = lim  h→0  f(x + h⃗b) − f(x)  h  = lim  h→0  f(x + hλ⃗a) − f(x)  h  = λ lim  h→0  f(x + (hλ)⃗a) − f(x)  (hλ)  =  λ lim  k→0  f(x + k⃗a) − f(x)  k  thus  f ′  b(x) = λf ′  a(x),  with  λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  with foot and metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Quite different results!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (In fact f ′(x) = opposite side  adjacent side depends on the length of  the adjacent side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 We insist on the subjectivity of the gradient:  20- The differential of f at a point x along a vector ⃗w ∈ ⃗R is df(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = limh→0  f(x+h⃗w)−f(x)  h  and is  objective: The observers all use this same formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  21- An observer chooses a Euclidean dot product (·, ·)g (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' based on the foot), then represent df(x)  by its (·, ·)g-Riesz representation vector ⃗Rg(df(x)) =noted  ⃗  gradgf(x) called the gradient of f at x relative  to (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  22- Another observer chooses a Euclidean dot product (·, ·)h (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' based on the metre), then represent  df(x) by its (·, ·)h-Riesz representation vector ⃗Rh(df(x)) =noted  ⃗  gradhf(x) called the gradient of f at x  relative to (·, ·)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  23- Both observer use the same formula df(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = limh→0  f(x+h⃗w)−f(x)  h  to get a different result:  ⃗  gradhf = λ2 ⃗  gradgf, because they use different measuring tools (one based on the foot, the other on the  metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  24- Recall: The gradient depends on a choice of a Euclidean unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 In (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) we have f ′  b(x) = λf ′  a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the 1-D gradient gives gradbf(x) = λ2gradaf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To define a gradient gradaf we need a Euclidean dots products (·, ·)a built from a basis (⃗a) in ⃗R, while  to define f ′  a we need a unit of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Details: (⃗a) and (⃗b) are two bases in ⃗R with ⃗b = λ⃗a, thus (·, ·)a = λ2(·, ·)b  (since 1 = (⃗a,⃗a)a = (⃗b,⃗b)b = (λ⃗a, λ⃗a)b = λ2(⃗a,⃗a)b gives (⃗a,⃗a)a = λ2(⃗a,⃗a)b and (⃗a) is a basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And we  have f ′  b(x) =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16) λf ′  a(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' df(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b = λdf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a, thus (gradfb(x),⃗b)b = λ(gradfa(x),⃗a)a, thus (gradfb(x),⃗b)b =  λλ2(gradfa(x),  ⃗b  λ)b = (λ2gradfa(x),⃗b)b, thus gradfb(x) = λ2gradfa(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 1- Prove that (·, ·)g = λ2(·, ·)h gives ||⃗ℓh||g = λ||⃗ℓh||h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 2- Does it contradict the Riesz  representation theorem which gives ||ℓ||Rn∗ = ||⃗ℓg||Rn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- ⃗ℓh =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11) λ2⃗ℓg gives ||⃗ℓh||h = λ2||⃗ℓg||h = λ||⃗ℓg||g since ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||h = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- No, since ||ℓ||Rn∗ := sup||⃗x||Rn =1 |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x| depends on the norm ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||Rn chosen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Here ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||Rn is either ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g or ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus if you write ||ℓ||Rn∗ =noted ||ℓ||g∗ if you use the norme ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g, then ||ℓ||h∗ = sup⃗v∈⃗Rn  |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v|  ||⃗v||h = sup⃗v∈⃗Rn  |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v|  1  λ ||⃗v||g =  λ sup⃗v∈⃗Rn  |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v|  ||⃗v||g = λ||ℓ||g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  A Riesz representation vector is contravariant  ⃗ℓg is a vector in E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2), so it is contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To be convinced:  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Check:  [⃗ℓg]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|old  (contravariance formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider two bases (⃗eold,i) and (⃗enew,i) in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With the change of basis formulas [⃗x]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|old  and [g]|new = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives (with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='94)), for all ⃗x,  [⃗x]T  |old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|old = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x = [⃗x]T  |new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|new  = ([⃗x]T  |old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|new = [⃗x]T  |old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ℓg]|new),  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  thus [⃗ℓg]|old = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗ℓg]|new since [g] is invertible (an inner dot product is positive definite), thus (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  118  119  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  What is a vector versus a (·, ·)g-vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 • Dont forget: A representation vector ⃗ℓg is not intrinsic to the linear form ℓ because it  depends on a (·, ·)g (depends on a observer: foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Reminder: there is no natural canonical  isomorphism between E and E∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' it is impossible to identify a linear form with a vector, see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗ℓg is incompatible with the use of push-forwards, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗ℓg is incompatible with the use of Lie derivatives, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  What is a vector versus a (·, ·)g-vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- Originally, a vector was a bipoint vector ⃗v = −−→  AB in ⃗R3 used to represent of a “material object”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the height of a child is represented on a wall by a vertical bipoint vector ⃗x starting from the ground  up to a pencil line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The vector ⃗x is objective: Any observer uses this same vector to get the height of the  child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and then use “their subjective unit” (foot, metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') to give a value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Then (mid 19th century), the concept of vector space was introduced: It is a quadruplet (E, +, K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  where + is an inner law, (E, +) is a group, K is a field, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is a external law on E (called a scalar  multiplication) compatible with + (see any math book).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And then the concept of scalar inner dot product (in a vector space) was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- We can then get non “material” vectors (“subjectively built vectors”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : start with our usual  vector space ⃗Rn of bi-point vectors, then consider its dual (Rn∗, +, R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') =noted Rn∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, for a given  ℓ ∈ Rn∗ (a given measuring device), consider two observers: An English observer with his foot built  Euclidean dot product (·, ·)g, and a French observer with with his metre built Euclidean dot product (·, ·)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  These observers build their own artificial Riesz representation vectors ⃗ℓg = ⃗Rg(ℓ) ∈ ⃗Rn and ⃗ℓh = ⃗Rh(ℓ),  cf (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' They remark that ⃗ℓg ̸= ⃗ℓh: These constructions are very subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- Then, with differential geometry, a vector ⃗v has been redefined: It is a “tangent vector”, which  means that there exists a C1 curve c : s ∈ [a, b] → c(s) ∈ E such that ⃗v is defined at a p = c(s) ∈ Im(c)  by ⃗v(p) := ⃗c ′(s) (so a vector is part of a vector field, here defined along the range of c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (This definition  of a tangent vector is applicable to “tangent vectors to a surface” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' tangent vectors to a manifold,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1,2-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Then it is shown that ⃗v is equivalent to  ∂  ∂⃗v = the directional derivative in the  direction ⃗v (natural canonical isomorphism between E and E∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For other equivalent definitions of  vectors, see Abraham–Marsden [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  The “(·, ·)g-dual vectorial bases” of one basis (and warnings)  Framework: E is a finite dimensional vector space, dim E = n (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E = ⃗R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' An observer chooses  an inner dot product (·, ·)g (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', in ⃗R3, a foot-built Euclidean dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence the results will be  subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (⃗ei) is some basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  A basis and its many associated “dual vectorial basis”  Definition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 The (·, ·)g-dual vectorial basis (or (·, ·)g-vectorial dual basis, or (·, ·)g-dual basis) of the  basis (⃗ei) is the basis (⃗eig) in E defined by  ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  (⃗eig,⃗ej)g = δij,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗eig •g ⃗ej = δij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  NB: A vectorial dual basis is not unique: It depends on the chosen inner dot product, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: Pay attention to the notations: ⃗eig is a contravariant vector (⃗eig ∈ E), so, even if you use the  Einstein convention, the index i in ⃗eig must be a bottom index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (πei) be the (covariant) dual basis of the basis (⃗ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the πei ∈ E∗ are the objective (the same  for all observers) linear forms defined by πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = δij for all j, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 (Equivalent definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The (·, ·)g-dual vectorial basis of the basis (⃗ei) is the basis  (⃗eig) in E made of the (·, ·)g-Riesz representative vectors of the πei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗eig := ⃗Rg(πei) ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗eig •g ⃗v = πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v, ∀⃗v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  where ⃗Rg is the (·, ·)g-Riesz operator, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With duality notations, (ei) is the dual basis and ⃗eig := ⃗Rg(ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗eig,⃗v)g = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v for all ⃗v ∈ E where  here the position of the index i is bottom on the left and up on the right, since ⃗Rg changes a covariant  vector (a linear form) into a contravariant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  119  120  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The “(·, ·)g-dual vectorial bases” of one basis (and warnings)  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 Prove that the vectors ⃗eig satisfy the contravariant change of basis formula  [⃗eig]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|old  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  (the ⃗ejg are indeed “contravariant vectors”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' • First answer: ⃗eig is a vector in E, thus it is contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Second answer: Apply (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17) since ⃗eig is a Riesz-representation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Third answer = direct computation: Consider two bases (⃗ai) and (⃗bi), and the transition matrix P from  (⃗ai) to (⃗bi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗bj = �n  i=1Pij⃗ai for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) and the change of basis formulas for the vectors ⃗ei and the  bilinear form (·, ·)g give [⃗ej]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|⃗a = (⃗eig,⃗ej)g = [⃗ej]T  |⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|⃗b = (P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ej]|⃗a)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|⃗a =  [⃗ej]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|⃗a for all i, j, thus [⃗eig]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗eig]|⃗b, thus (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 Consider two inner dot products (·, ·)a and (·, ·)b (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a foot-built and a metre-built  Euclidean dot product), and a basis (⃗ei) in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Call (⃗eia) and (⃗eib) the (·, ·)a and (·, ·)b-dual vectorial  bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  (·, ·)a = λ2(·, ·)b  =⇒  ⃗eib = λ2⃗eia,  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λ2 > 10 with foot and metre built Euclidean bases: ⃗eib is very different from ⃗eia !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A dual vectorial  basis highly depends on an observer: A vectorial dual basis is not intrinsic to (⃗ei) (not objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) gives (⃗eib,⃗ej)b = δij = (⃗eia,⃗ej)a = λ2(⃗eia,⃗ej)b, thus (⃗eib − λ2⃗eia,⃗ej)b = δij, for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 If (⃗ei) is a (·, ·)g-orthonormal basis we trivially get ⃗eig = ⃗ei for all i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (⃗eig) = (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='This  particular case is not compatible with joint work by an English (foot) and French (metre) observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Components of ⃗ejg in the basis (⃗ei)  Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19 The components of ⃗ejg in the basis (⃗ei) are given by, for any j ∈ [1, n]N,  [⃗ejg]|⃗e = [⃗Rg]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ej]|⃗e = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ej]|⃗e = the j-th column of ([g]|⃗e)−1,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the i-th component of ⃗ejg is ([g]−1  |⃗e )ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus the matrix of g(·, ·) in the basis (⃗eig) is the inverse of the matrix of g(·, ·) in the basis (⃗ei):  ([g(⃗eig,⃗ejg)] =)  [g]|(⃗eig) = [g]|(⃗ei)  −1  (= ([g(⃗ei,⃗ej)])−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' First proof of (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) (straight forward calculation): (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) gives, for all i, j,  [⃗ej]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|⃗e = δij = [⃗ej]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ei]|⃗e,  thus  [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗eig]|⃗e = [⃗ei]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  Second proof of (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23): Apply (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) (generic Riesz representation result) to get (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, [g]|⃗e being symmetric we have [g]|⃗e−1 symmetric, and g(⃗eig,⃗ejg) = [⃗eig]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ejg]|⃗e =  [⃗ei]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ej]|⃗e = [⃗ei]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ej]|⃗e = ([g]|⃗e−1)ij, thus (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 ⃗R2, [g]|⃗e =  �  1  0  0  2  �  , thus [g]−1  |⃗e =  �  1  0  0  1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗e1g = ⃗e1, ⃗e2g = 1  2⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 Warning: When ([g]−1  |⃗e )ij =noted gij then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) reads  ⃗ejg =  n  �  i=1  gij⃗ei,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  where the Einstein convention is not satisfied: The Einstein convention is satisfied with  ⃗ejg =  n  �  i=1  (⃗ejg)i⃗ei  noted  =  n  �  i=1  (Pj)i⃗ei  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  (the components of vectors have up indices), and this can be verified with (⃗eig,⃗ej)g =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) δij which gives  �n  k,ℓ=1(Pi)kgkj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And in (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) the scalars gij is just another name for (Pj)i, nothing more (nothing  to do with the Einstein convention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  120  121  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The “(·, ·)g-dual vectorial bases” of one basis (and warnings)  We insist:In other words: M = [g]|⃗e = [Mij] is a matrix, and its inverse is the matrix M −1 = [Mij]−1:  A matrix is just a collection of scalars (has nothing to do with the Einstein convention), and its inverse  is also a collection of scalars, and you do not change this fact by calling M −1 =noted [M ij] (the use of up  indices is irrelevant for matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And because (Pj)i equals ([g]−1  |⃗e )ij =noted gij, some people rename ⃗ejg as ⃗e j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' to get ⃗e j = �n  i=1gij⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  But doing so they despise Einstein’s convention, despite eventual claims: They confuse covariance and  contravariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and add confusion to the confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: Recall: If in trouble with a notation which comes as a surprise (the notation gij here), use classical  notations: Then no misuse of Einstein’s convention and no possible misinterpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular here  ⃗ejg is a (contravariant) vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Multiple admissible notations for the components of ⃗ejg  Let P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) be the change of basis endomorphism from (⃗ei) to (⃗eig): defined by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = ⃗ejg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  let P = [P]|⃗e (the associated transition matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It gives multiple admissible (non confusing) notations  for the components of ⃗ejg relative to the basis (⃗ei):  ⃗ejg = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  j=1  Pij⃗ei =  n  �  j=1  (Pj)i⃗ei  �  ��  �  clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  =  n  �  j=1  (Pj)i⃗ei =  n  �  j=1  P i  j⃗ei  �  ��  �  dual  ,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the i-th component of the vector ⃗ejg has the names Pij = (Pj)i = (Pj)i = P ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' P = [P]|⃗e =  [Pij] = [(Pj)i] = [(Pj)i] = [P ij] (four different notations for the same matrix), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∀j,  [⃗ejg]|⃗e = [P]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗ej]|⃗e =  �  �  �  P1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Pnj  �  �  � =  �  �  �  (Pj)1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Pj)n  �  �  � =  �  �  �  P 1j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  P nj  �  �  � =  �  �  �  (Pj)1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Pj)n  �  �  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  = the j-th column of [P]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' You can choose any notation, depending on your current need or mood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  (Huge) differences between “the (covariant) dual basis” and “a dual vectorial basis”  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A basis (⃗ei) has an infinite number of vectorial dual bases (⃗eig), as many as the number of inner  dot products (·, ·)g (as many as observers), see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  While a basis (⃗ei) has a unique intrinsic (covariant) dual basis (πei) noted  =  (ei), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7): Two  observers who consider the same basis (⃗ei) have the same (covariant) dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' πei = ei is covariant, while ⃗ei and ⃗eig are contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And there is no transition matrix between  (⃗ei) and (πei) = (ei), since ⃗ei ∈ E and πei = ei ∈ E∗ don’t live in the same vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If you fly, it is vital to use the dual basis (πei) = (ei): It is possibly fatal if you confuse foot and  metre at takeoff and at landing (if you survived takeoff) because of the choice of different Euclidean  dot product (·, ·)g or (·, ·)h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the Mars Climate Orbiter crash, remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Einstein’s  convention can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' only if it is really followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  About the notation gij = shorthand notation for (g♯)ij  Definition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 The Riesz associated inner dot product g♯ ∈ L(E∗, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is the bilinear form defined  by, for all ℓ, m ∈ E∗,  g♯(ℓ, m) := g(⃗ℓg, ⃗mg),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (ℓ, m)g♯ := (⃗ℓg, ⃗mg)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  where ⃗ℓg = ⃗Rg(ℓ) and ⃗mg = ⃗Rg(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus g♯(·, ·) =noted (·, ·)g♯ is indeed an inner dot product in E∗: trivial check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification:  (⃗ei) is a basis in E and (ei) is its dual basis (duality notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30) gives:  (g♯)ij := g♯(ei, ej) = g(⃗eig,⃗ejg),  thus  [g♯]|e = [(g♯)ij] = [gij]−1 = [g]−1  |⃗e ,  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  [(g♯)ij]  shorthand  =  notation [gij] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  Classical notations: [g♯]|e = [(g♯)ij] = [g♯(πei, πej)] = [g(⃗eig,⃗ejg)] = [gij]−1 = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  121  122  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Goal  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23 How do we compute g♯(ℓ, m) with matrix computations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓ = �n  i=1ℓiei and m = �n  j=1mjej give g♯(ℓ, m) = �n  i,j=1ℓimjg♯(ei, ej) = �n  i,j=1ℓi(g♯)ijmj =  [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g♯]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [m]T  |⃗e = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]−1  |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [m]T  |⃗e (a linear form is represented by a row matrix,).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30) tells that the  �2  0  �  tensor g♯ ∈ L(E∗, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) was created from the  �0  2  �  tensor g =  (·, ·)g ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) using twice the (·, ·)g-Riesz representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- Show that if you use the (·, ·)g-Riesz representation theorem just once you get the  �1  1  �  tensor  g♮ ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) given by  g♮ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  2- Reciprocal: What is the  �0  2  �  tensor g♭ ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) that you create from the identity I ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E)  when using the (·, ·)g-Riesz representation theorem once?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- Summary: �I = g♮ gives (�I)♭ = g♭ = g and (�I)♯ = g♯  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- g♮ ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is defined by g♮(ℓ, ⃗w) = (⃗ℓg, ⃗w)g for all (ℓ, ⃗w) ∈ E∗ × E, where ⃗ℓg is the (·, ·)g-Riesz  representation vector of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus g♮(ℓ, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, for all (ℓ, ⃗w) ∈ E∗×E, hence g♮ ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is naturally  canonically associated with the identity I ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- The identity operator I ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) (observer independent) is naturally canonically associated with the  �1  1  �  tensor �I ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) defined by �I(ℓ, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w for all (ℓ, ⃗w) ∈ E∗ × E, thus �I = g♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G  Cauchy–Green deformation tensor C = F T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F  Framework: �Φ :  �  R × Obj → Rn  (t, PObj) → �Φ(t, PObj)  �  is a motion of Obj, Ωτ = �Φ(τ, PObj) is the configuration of Obj  at any τ, t0 and t are fixed, Φ := Φt0  t :  �  Ωt0 → Ωt  P → p = Φ(P)  �  is the associated motion between t0 and t,  and F(P) := dΦ(P) :  �  �  �  �  �  ⃗  Rn  t0 → ⃗Rn  t  ⃗W → ⃗w = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W := lim  h→0  Φ(P+h ⃗W) − Φ(P)  h  �  �  �  �  �  is the deformation gradient  at P between t0 and t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='0  Goal  Construction of C (summary of Cauchy’s approach):  1- At t0, consider two vectors ⃗W1 and ⃗W2,  2- at t, they are distorted by the motion and become the vectors F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- Then choose a Euclidean dot product (·, ·)g =noted ·  ·, the same at all t (to simplify);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- Then, by definition of the transposed, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2) = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • ⃗W2: You have got the  Cauchy strain tensor C := F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5- Then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2) − ⃗W1 • ⃗W2 = ((C−I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • ⃗W2 gives a measure of the deformation with ⃗W2  as a reference, measure that is used to build first order constitutive laws for the stress (Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Transposed F T: Inner dot products required  We first give the functional definition of F T (qualitative);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then we get the usual matrix representation  of F T relative to observers (quantification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of the function F T  At t0, a past observer chose an inner dot product (·, ·)G in ⃗Rn  t0, and at t the present observer chooses an  inner dot product (·, ·)g in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  By definition, the transposed of the linear map F(P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) relative to (·, ·)G and (·, ·)g is the  linear map F(P)T  Gg ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) defined by, for all ⃗UP ∈ ⃗Rn  t0 (vector at P) and ⃗wp ∈ ⃗Rn  t (vector at p),  (F(P)T  Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp, ⃗UP )G = (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗UP , ⃗wp)g,  in short  (F T  Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, ⃗U)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U, ⃗w)g ,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  122  123  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Transposed F T : Inner dot products required  see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines F T  Gg(p) := F(P)T  Gg when p = Φ(P):  F T  Gg :  �  �  �  Ωt → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0)  p → F T  Gg(p) := F(P)T  Gg  �  �  � ,  so in short  (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) •G ⃗U = ⃗w •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U) ,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  without forgetting that F T := F T  Gg depends on (·, ·)G and (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = ⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗z = ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗z, which dots are inner dot products?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' What does F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2 = ⃗W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2 mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' No choice: ( ⃗W, ⃗z) ∈ ⃗Rn  t0 × ⃗Rn  t , so (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗z) •G ⃗W = ⃗z •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W) = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W) •g ⃗z = ⃗W •G (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' No choice: ⃗W1, ⃗W2 ∈ ⃗Rn  t0, so (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2) = ⃗W1 •G (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 More generally, on a surface Ω (a manifold), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) is defined for all (⃗UP , ⃗wp) ∈ TP Ωt0 ×  TpΩt, where Tpτ Ωτ is the tangent space at Ωτ at pτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification with bases (matrix representation)  Classical notations: (⃗ai) is a basis in ⃗Rn  t0, and (⃗bi) is a basis in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Marsden–Hughes duality notations:  ( ⃗EI) is a basis in ⃗Rn  t0 and (⃗ei) is a basis in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the reference to the points P and p is omitted to  lighten the writings (use the full notation of § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 if in doubt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let [G] := [(⃗ai,⃗aj)G], [g] := [(⃗bi,⃗bj)g], [F]|⃗a,⃗b = [Fij] =noted [F], [F T ]|⃗b,⃗a = [(F T )ij] =noted [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) gives [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w] = [F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w], thus [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w] = [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w], for all ⃗U, ⃗w,  thus  [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F T ] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F T ] = [G]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 If (⃗ai) and (⃗bi) are (·, ·)G and (·, ·)g-orthonormal bases, then [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But recall: If  you need to work with a coordinate system, then the bases in use are the coordinate system bases which  are not orthonormal in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [G]−1 ̸= I and [g]−1 ̸= I in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Use classical notation, then Marsden duality notations, to express (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Classical notations:  Gij = G(⃗ai,⃗aj),  gij = g(⃗bi,⃗bj),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [G]⃗a = [Gij],  [g]|⃗b = [gij],  and  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  Fij⃗bi,  F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj =  n  �  i=1  (F T )ij⃗ai,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F]|⃗a,⃗b = [Fij],  [F T ]|⃗b,⃗a = [(F T )ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  Then (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj,⃗ai)G =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)(⃗bj, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai)g gives (�n  k=1(F T )kj⃗ak,⃗ai)G = (⃗bj, �n  k=1Fki⃗bk)g, thus �n  k=1(F T )kj(⃗ak,⃗ai)G =  �n  k=1Fki(⃗bj,⃗bk)g with Fki = ([F]T )ik, thus  n  �  k=1  Gik(F T )kj =  n  �  k=1  ([F]T )ikgkj,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F T )ij =  n  �  k,ℓ=1  ([G]−1)ikFℓkgℓj,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  for all i, j, thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Marsden notations: GIJ = G( ⃗EI, ⃗Ej), gij = g(⃗ei,⃗ej), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ = �n  i=1F i  J⃗ei, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  I=1(F T )I  j ⃗EI, thus  n  �  K=1  GIK(F T )K  j =  n  �  k=1  F k  Igkj,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (F T )I  j =  n  �  K,k=1  GIKF k  Kgkj  where  [GIJ] := [GIJ]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Remark: F ∗  (For mathematicians: F ∗ doesn’t seem to be very useful in mechanics, apart from making simple things  difficult, and playing games with components and duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The adjoint of the linear map F ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) (acting on vectors) is the linear map  F ∗ ∈ L(⃗Rn∗  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn∗  t0 ) (acting on functions) canonically defined by, for all m ∈ ⃗Rn∗  t ,  F ∗(m) := m ◦ F,  written  F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F (∈ ⃗Rn∗  t0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  123  124  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cauchy–Green deformation tensor C  So, for all (m, ⃗W) ∈ ⃗Rn∗  t  × ⃗Rn  t0,  (F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W (∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Quantification (matrix representation): (πai) and (πbi) are the covariant dual bases of (⃗ai) and (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (F ∗)ij be the components of F ∗ relative to these dual bases:  F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbj =  n  �  I=1  (F ∗)ijπai,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F ∗]|πb,πa = [(F ∗)ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) gives (F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai = πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai, thus  ∀i, j, (F ∗)ij = Fji ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F ∗]|πb,πa = ([F]|⃗a,⃗b)T ,  in short  [F ∗] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Marsden duality notations: F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ej = �n  I=1(F ∗)IjEI gives (F ∗)Ij = F jI for all I, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation of F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' As usual in classical mechanics, we use Euclidean dot products, here (·, ·)G in ⃗Rn  t0  and (·, ·)g in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then we use the (·, ·)G-Riesz representation vector ⃗RG(F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m) ∈ ⃗Rn  t0 of F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m ∈ ⃗Rn∗  t0 ,  and the (·, ·)g-Riesz representation vector ⃗Rg(m) ∈ ⃗Rn  t of m ∈ ⃗Rn∗  t , so, for all m ∈ ⃗Rn∗  t  and ⃗W ∈ ⃗Rn  t0,  (F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = ⃗RG(F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m) •G ⃗W,  and  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W) = ⃗Rg(m) •g F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Rg(m)) •G ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) gives ⃗RG(F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='m) = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Rg(m), thus  ⃗RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ∗ = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Rg,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  F ∗ = ⃗RG  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 The definition of F ∗ is intrinsic to F (objective), while the definition of F T is not intrinsic  to F (not objective) since it needs inner dot products (observer choices) to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Cauchy–Green deformation tensor C  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of C  Consider vectors ⃗Wi ∈ ⃗Rn  t0 at P, i = 1, 2, and their push forwards ⃗wi toward p = Φ(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  short notation for ⃗wi(p) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With the chosen inner dot products (·, ·)G in ⃗Rn  t0 and (·, ·)g  in ⃗Rn  t , we get (⃗w1(p), ⃗w2(p))g = (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1(P), F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2(P))g=(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)(F T  Gg(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1(P), ⃗W2(P))G when  p = Φ(P), written in short:  (⃗w1, ⃗w2)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F  � �� �  C  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 The (right) Cauchy–Green deformation tensor at P ∈ Ωt0 relative to (·, ·)G and (·, ·)g,  is the endomorphism CGg(P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) defined by  CGg(P) := F T  Gg(p) ◦ F(P),  in short  C := F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  So  C = F T ◦ F : ⃗W  F  −→ F( ⃗W)  F T  −→ F T (F( ⃗W)) = C( ⃗W),  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  with F and F T linear, thus C is linear and C( ⃗W) is written C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) tells that C  is characterized by, for all ⃗W1, ⃗W2 ∈ ⃗Rn  t0,  ⃗w1 •g ⃗w2 = (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) •G ⃗W2 = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  Moreover, (·, ·)g being symmetric (inner dot product), C is a (·, ·)G-symmetric endomorphism in ⃗Rn  t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  for all ⃗W1, ⃗W2 ∈ ⃗Rn  t0,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = ( ⃗W1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)G,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) •G ⃗W2 = ⃗W1 •G (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2),  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  since (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g = ( ⃗W1, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  124  125  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Time Taylor expansion of C  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14) gives [C] = [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F], with [F T ] =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)[G]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g], thus  [C] = [G]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F] ,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  short notation for [CGg]|⃗a = [G]−1  |⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([F]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Use classical notation, then duality notations, to express (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Classical notations:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  Fij⃗bi  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  n  �  i=1  Cij⃗ai,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F]|⃗a,⃗b = [Fij]  and  [C]|⃗a = [Cij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)-(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  give  (⃗ai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj)G  =  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj)g,  so  (⃗ai, �  k Ckj⃗ak)G  =  (�  k Fki⃗bk, �  ℓ Fℓj⃗bℓ)g,  thus  �  k Ckj(⃗ai,⃗ak)G = �  kℓ Fki(⃗bk,⃗bℓ)gFℓj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  n  �  k=1  GikCkj =  n  �  k,ℓ=1  Fki gkℓFℓj =  n  �  k,ℓ=1  ([F]T )ik gkℓFℓj,  so  [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F] ,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  so Cij = �n  k,ℓ,m=1([G]−1)imFkm gkℓFℓj = �n  k,ℓ,m=1([G]−1)im([F]T )mk gkℓFℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ =  n  �  i=1  F i  J⃗ei  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ =  n  �  I=1  CI  J ⃗EI,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [F]| ⃗  E,⃗e = [F i  J]  and  [C]| ⃗  E = [CI  J], and  n  �  K=1  GIKCK  J =  n  �  k,ℓ=1  F k  I gkℓF ℓ  J,  and  CI  J =  n  �  k,ℓ,M=1  GIMF k  M gkℓF ℓ  J  when  [GIJ] := [GIJ]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 (·, ·)G is a Euclidean dot product in foot, (·, ·)g is a Euclidean dot product in metre, so  (·, ·)g = µ2(·, ·)G with µ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3048;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (⃗ai) = (⃗bi) is a (·, ·)G-orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove  [C] = µ2[F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [C]|⃗a =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) [G]−1  |⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]T  |⃗a,⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]|⃗a,⃗a gives [C]|⃗a = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]T  |⃗a,⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='µ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[F]|⃗a,⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Shorten notation = (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Time Taylor expansion of C  Here we use a unique inner dot product (·, ·)G = (·, ·)g at all time (to compare results in the vicinity  of t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Moreover we use an orthonormal basis (to lighten the notations), thus, in short, [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  P is fixed, Ct0  t (P) =noted C(t), and [C(t)] = [F(t)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t)] (since [G] = [g] = I here), and  ⃗V t0  t (P) =noted ⃗V (t) and ⃗At0  t (P) =noted ⃗A(t) are the Lagrangian velocities and accelerations, and ⃗v(t, p)  and ⃗γ(t, p) are the Eulerian velocities and accelerations at t at p = Φt0  t (t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With Lagrangian variables (used to define C): F(t+h) = F(t) + h d⃗V (t) + h2  2 d ⃗A(t) + o(h2) gives  [C(t+h)] = [F(t+h)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F(t+h)]  = [F T + h d⃗V T + h2  2 d ⃗AT + o(h2)](t)[F + h d⃗V + h2  2 d ⃗A + o(h2)](t)  = [C(t) + h ([F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗V ] + [d⃗V ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F])(t)  �  ��  �  =[(Ct0  P )′(t)] =noted [C′(t)]  +h2  2 ([F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d ⃗A] + 2[d⃗V ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗V ] + [d ⃗A]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F])(t)  �  ��  �  =[(Ct0  P )′′(t)] =noted [C′′(t)]  )(t) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  (As usual with Lagrangian variables, we have three times involved: t0, t and t+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') In particular  [Ct0  P (t0+h)] = I + ([d⃗V ] + [d⃗V ]T )(t0) + h2  2 ([d ⃗A] + 2[d⃗V ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗V ] + [d ⃗A]T )(t0) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Abusively written Ct0  P (t0+h) = I + (d⃗V + d⃗V T )(t0) + h2  2 (d ⃗A + 2d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗V + d ⃗AT )(t0) + o(h2), but don’t  forget it is a matrix meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  125  126  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark: C♭  With Eulerian variables: With p(t) = Φt0(t, P), we have d⃗V t0(t, P) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t) and d ⃗At0(t, P) =  d⃗γ(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t), thus writing d⃗v := d⃗v(t, p(t)) and d⃗γ := d⃗γ(t, p(t)) (for short),  Ct0  P (t+h) = Ct0  P (t) + h (F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗v + d⃗vT )(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t))  + h2  2 (F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗γ + 2d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗γT )(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  abusive notation of [Ct0  P (t+h)] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (matrices relative to a basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 F ′′ = d ⃗A is easy to interpret, but C′′ = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗A + 2d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗V + d ⃗AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗A +  d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗V ) + (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗A + d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗V )T is not that easy to interpret (and in not linear in ⃗V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We already had a problem with the composition of flows: The formula F t0  t2 = F t1  t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t1 is simple  (determinism), but the formula Ct0  t2 = (F t0  t2 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t2 = (F t0  t1 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t1  t2 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t1  t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t1 = (F t0  t1 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ct1  t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t1 is “not that  simple” (̸= Ct1  t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ct0  t1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Indeed, to consider C instead of F amounts to consider the “motion squared”, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, ⃗W)g = ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W||2  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Since C′(t0) = d⃗V (t0) + d⃗V (t0)T this may have little consequences for linear approximation near t0,  but ultimately not small consequences for second-order approximations (and large deformations) if C′′ is  used to make constitutive laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The consideration of Lie derivatives may be an interesting alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Remark: C♭  For mathematicians: May produce errors, misuses, covariance-contravariance confusion, see next § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For the general ♭ notation see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of C♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 At P ∈ Ωt0, the bilinear form C♭  Gg(P) =noted C♭ ∈ L(⃗Rn  t0, ⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) associated with the  linear map CGg(P) =noted C ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) is defined by, for all ⃗W1, ⃗W2 ∈ ⃗Rn  t0 vectors at P,  C♭( ⃗W1, ⃗W2) := ( ⃗W1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)G  (= (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Then C♭ is a bilinear symmetric form (trivial) and is a metric in ⃗Rn  t0 when F t0  t  =noted F is a diffeo-  morphism (usual hypothesis), but not a Euclidean one (it is iff C = I i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' for rigid body motions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) gives [ ⃗W2]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [C♭].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W1] = [ ⃗W2]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W1] for all ⃗W1, ⃗W2 since C♭ and (·, ·)G  are symmetric, thus  [C♭] = [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [C]  (= [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Use duality notations to express (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27) with components, and explain the flat ♭ notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) gives C♭( ⃗EJ, ⃗EI) := (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ, ⃗EI)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus with C♭( ⃗EI, ⃗EJ) = CIJ and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ = �  I CI  J ⃗EI we get  CJI = �  K CK  J( ⃗EK, ⃗EI)G = �  K CK  JGKI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And C♭ and (·, ·)G are symmetric, thus  CIJ =  �  K  GIKCK  J,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [C♭]| ⃗  E = [G]| ⃗  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[C]| ⃗  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  The flat notation C♭ is due to: The top index I in CI  J has been transformed into a bottom index in CIJ in C♭,  which characterizes a change of variance because of the use of an inner dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27) also gives CIJ = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗EJ)g = �  kℓ F k  IF ℓ  J(⃗bk,⃗bℓ)g, thus  CIJ =  �  kℓ  F k  IgkℓF ℓ  J =  �  kℓ  (F T )I  kgkℓF ℓ  J,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [C♭]| ⃗  E = ([F]| ⃗  E,⃗e)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F]| ⃗  E,⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and remarks about C♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and Jaumann  C♭ can also be defined only with (·, ·)g by, for all ⃗W1, ⃗W2 ∈ ⃗Rn  t0,  C♭  g( ⃗W1, ⃗W2) := (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', C♭  g := g∗ =noted C♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So we can also say that C♭  g is the pull-back of the metric (·, ·)g by Φ, see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  126  127  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Stretch ratio and deformed angle  However C♭ = C♭  g is useless in itself: C♭ is not a Euclidean dot product (it is a metric defined at  each P by C♭  g(P)( ⃗W1, ⃗W2) := (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g for all ⃗W1, ⃗W2 ∈ ⃗Rn  t0 vectors at P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In fact, C♭ is  only useful to characterize a deformation if the value C♭( ⃗W1, ⃗W2) can be compared with the initial value  ( ⃗W1, ⃗W2)G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if a Euclidean dot product (·, ·)G was introduced in ⃗Rn  t0: This is why C♭ is classically  defined from C, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  You may want to use the infinitesimal strain tensor ε = F +F T  2  − I, or the Green–Lagrange defor-  mation tensor E = 1  2(C − I), obtained from F T := F T  Gg (essential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  There is no objective “trace” for a  �0  2  �  tensor like C♭, while Tr(C) is objective since C is an endo-  morphism (≃ a  �1  1  �  tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The Lie derivatives of a second order tensor depends on the type of the tensor, and the Lie derivative  of the  �1  1  �  tensor like C gives the Jaumann derivative, which is usually preferred to the Lie derivative of  the  �0  2  �  tensor like C♭ which is the lower convected Lie derivative, see remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So the introduction and use of C♭ in mechanics mostly complicate things unnecessarily, and interferes  with basic understandings like the distinction between covariance and contravariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 Interpretation issue (with Jaumann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2D = d⃗v + d⃗vT gives 2 DD  Dt = D(d⃗v)  Dt  + D(d⃗v)T  Dt  = d⃗γ + d⃗γT − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v − d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vT , thus, with (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) and  keeping in mind the matrix meaning,  C′′(t) = F(t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (2DD  Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vT + 2d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)  = 2F(t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (DD  Dt + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D)(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  The DD  Dt + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D term looks like a lower-convected Lie derivative, but with d⃗vT instead of d⃗v∗,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So you may find (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31) written as C′′ = 2F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗vD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But you get disappointing results when  using the the lower convected Lie derivative (Jaumann is usually preferred).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In fact, it is L⃗vD♭ (lower  convected Lie derivative) that should be used, where D♭  g :=  d⃗v♭  g+(d⃗v♭  g)T  2  , to get (C♭)′′ = 2F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗vD♭  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Stretch ratio and deformed angle  Here (·, ·)g = (·, ·)G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' at t0 and t we use the same Euclidean dot product, to be able to compare the  lengths relative to the same unit of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (If (·, ·)g ̸= (·, ·)G then use (·, ·)g = µ2(·, ·)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Stretch ratio  The stretch ratio at P ∈ ⃗Rn  t0 between t0 and t for a ⃗WP ∈ ⃗Rn  t0 is defined by  λ( ⃗WP ) := ||⃗wp||G  || ⃗WP ||G  = ||FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗WP ||G  || ⃗WP ||G  (= ||FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (  ⃗WP  || ⃗WP ||G  )||G)  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  where ⃗wp = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗WP is the deformed vector by the motion at p = Φ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', in short  ∀ ⃗W ∈ ⃗Rn  t0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' || ⃗W|| = 1,  λ( ⃗W) := ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  (You may find: λ(d ⃗X) = ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d ⃗X|| with d ⃗X a unit vector(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This notation should be avoided, see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Deformed angle  Recall: The angle θt0 =  �  ( ⃗W1, ⃗W2) formed by two vectors ⃗W1 and ⃗W2 in ⃗  Rn  t0−{⃗0} at P ∈ Ωt0 is given by  cos(θt0) =  ⃗  W1  || ⃗  W1||G ⃗  W2  || ⃗  W2||G (= (  ⃗  W1  || ⃗  W1||G ,  ⃗  W2  || ⃗  W2||G )G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With the deformed vectors ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi at p = Φt0  t (P), the deformed angle is θt defined by  cos(θt) :=  �  (⃗w1, ⃗w2) =  ⃗w1  ||⃗w1|| ⃗w2  ||⃗w2|| =  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1  ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1|| F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2  ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2||  (= (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1) • ⃗W2  ||⃗w1|| ||⃗w2|| ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  127  128  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Decompositions of C  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Decompositions of C  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Spherical and deviatoric tensors  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 The deformation spheric tensor is  Csph = 1  nTr(C) I,  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  with Tr(C) = the trace of the endomorphism C (there is no “trace” for the  �0  2  �  tensor C♭).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 The deviatoric tensor is  Cdev = C − Csph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  (So Tr(Cdev) = 0 , and C = Csph + Cdev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Rigid motion  The deformation is rigid iff, for all t0, t,  (F t0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t  = I,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Ct0  t = I,  written  C = I = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Thus, after a rigid body motion, lengths and angles are left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Diagonalization of C  Proposition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F being symmetric positive, C is diagonalizable, its eigenvalues are positive,  and ⃗  Rn  t0 has an orthonormal basis made of eigenvectors of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (C(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g = ( ⃗W1, C(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)G, thus C is (·, ·)G-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W1)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1)g = ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1||2  g > 0 when ⃗W1 ̸= ⃗0, since F invertible (Φt0  t is supposed to  be a diffeomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus C est (·, ·)G-symmetric definite positive real endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 Let λi be the eigenvalues of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the √λi are called the principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  the associated eigenvectors give the principal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Mohr circle  This § deals with general properties of 3 ∗ 3 symmetric positive endomorphism, like Ct0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Consider ⃗R3 with a Euclidean dot product (·, ·)R3 and a (·, ·)R3-orthonormal basis (⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let M : ⃗R3 → ⃗R3 be a symmetric positive endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus M is diagonalizable in a (·, ·)R3-  orthonormal basis (⃗e1,⃗e2,⃗e3), that is, ∃λ1, λ2, λ3 ∈ R, ∃⃗e1,⃗e2,⃗e3 ∈ ⃗R3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei = λi⃗ei  and  (⃗ei,⃗ej)R3 = δij,  so  [M]|⃗e = diag(λ1, λ2, λ3) =  �  �  λ1  0  0  0  λ2  0  0  0  λ3  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  And the orthonormal basis (⃗e1,⃗e2,⃗e3) is ordered s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' λ1 ≥ λ2 ≥ λ3 (> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let S be the unit sphere in R3, that is the set {(x, y, z) : x2 + y2 + z2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its image M(S) by M  is the ellipsoid {(x, y, z) : x2  λ2  1 + y2  λ2  2 + z2  λ2  3 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then consider ⃗n = �  i ni⃗ei s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ||⃗n||R3 = 1:  [⃗n]|⃗e =  �  �  n1  n2  n3  �  �  with  n2  1 + n2  2 + n2  3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  Thus its image ⃗A = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n ∈ M(S) satisfies  ⃗A = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n,  [ ⃗A]|⃗e =  �  �  λ1n1  λ2n2  λ3n3  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  Then define  An = ( ⃗A,⃗n)R3,  ⃗A⊥ = ⃗A − An⃗n,  A⊥ := || ⃗A⊥||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  So ⃗A = An⃗n + ⃗A⊥ ∈ Vect{⃗n} ⊗ Vect{⃗n}⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Remark: ⃗A⊥ is not orthonormal to the ellipsoid M(S), but  is orthonormal to the initial sphere S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  128  129  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Green–Lagrange deformation tensor E  Mohr Circle purpose: To find a relation:  A⊥ = f(An),  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  relation between “the normal force An” (to the initial sphere) and the “tangent forceA⊥” (to the initial  sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40) and An = (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n,⃗n)R3 give  �  �  �  �  �  n2  1 + n2  2 + n2  3 = 1,  λ1n2  1 + λ2n2  2 + λ3n2  3 = An  λ2  1n2  1 + λ2  2n2  2 + λ2  3n2  3 = || ⃗A||2 = A2  n + A2  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  This is linear system with the unknowns n2  1, n2  2, n2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The solution is  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  n2  1 = A2  ⊥ + (An − λ2)(An − λ3)  (λ1 − λ2)(λ1 − λ3)  ,  n2  2 = A2  ⊥ + (An − λ3)(An − λ1)  (λ2 − λ3)(λ2 − λ1)  ,  n2  3 = A2  ⊥ + (An − λ1)(An − λ2)  (λ3 − λ1)(λ3 − λ2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  The n2  i being non negative, and with λ1 > λ2 > λ3 ≥ 0, we get  �  �  �  �  �  A2  ⊥ + (An − λ2)(An − λ3) ≥ 0,  A2  ⊥ + (An − λ3)(An − λ1) ≤ 0,  A2  ⊥ + (An − λ1)(An − λ2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  Then let x = An and y = A⊥, and consider, for some a, b ∈ R, the equation  y2 + (x − a)(x − b) = 0,  so  (x − a+b  2 )2 + y2 = (a−b)2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  This is the equation of a circle centered at ( a+b  2 , 0) with radius |a−b|  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)2 tells that An and A⊥ are inside the circle centered at ( λ1+λ3  2  , 0) with radius λ1−λ3  2  ,  and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)1,3 tell that An and A⊥ are outside the other circles (adjacent and included in the first,  drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 What happens if λ1 = λ2 = λ3 > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  �  �  �  �  �  �  �  �  �  �  �  �  �  n2  1 + n2  2 + n2  3 = 1,  n2  1 + n2  2 + n2  3 = An  λ1 ,  n2  1 + n2  2 + n2  3 = A2  n + A2  ⊥  λ2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  �  �  �  �  Thus An = λ1 and A2  n + A2  ⊥ = λ2  1, thus A⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Here C = λ1I,  and we deal with a dilation: A⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19 What happens if λ1 = λ2 > λ3 > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  �  �  �  �  �  �  �  n2  1 + n2  2 + n2  3 = 1,  λ1(1 − n2  3) + λ3n2  3 = An,  λ2  1(1 − n2  3) + λ2  3n2  3 = A2  n + A2  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  �  �  �  �  Thus An = λ1 − (λ1 − λ3)n2  3 ∈ [λ3, λ1], and A⊥ = ±(λ2  1 −  (λ2  1 − λ2  3)n2  3 − A2  n)  1  2 , with A2  n + A2  ⊥ a point on the circle with radius λ2  1(1 − n2  3) + λ2  3n2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Green–Lagrange deformation tensor E  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) gives (⃗w1, ⃗w2)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g = (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, ⃗W)G at p = Φ(P), thus  (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = ((C − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  129  130  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Small deformations (linearization): The infinitesimal strain tensor ε  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 The Green–Lagrange tensor (or Green–Saint Venant tensor) at P relative to t0 and t  is the endomorphism Et0  t (P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) defined by  Et0  t (P) := Ct0  t (P) − It0  2  ,  in short  E = C − I  2  (= F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F − I  2  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  (In particular E = 0 for rigid body motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') And Et0  t : Ωt0 → L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) is the Green–Lagrange tensor  relative to t0 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The 1  2 because (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') corresponds to the “motion squared”, see the following linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And we get the time Taylor expansion of Et0  P (t) = 1  2(Ct0  P (t) − It0) with p(t) = Φt0  P (t) and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25):  Et0  P (t+h) = F t0  P (t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  h d⃗v + d⃗vT  2  + h2  2 (d⃗γ + d⃗γT  2  + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v)  �  (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  P (t) + o(h2)  = F t0  P (t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  h D + h2 ((DD  Dt + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D)(t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  P (t) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Small deformations (linearization): The infinitesimal strain tensor ε  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Landau notations big-O and little-o  Reminder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let f, g : R → R and x0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  f = O(g) near x0  ⇐⇒  ∃C > 0, ∃η > 0, ∀x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' |x − x0| < η, |f(x)| < C|g(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  and f is said to be “comparable with g” near x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If |g| > 0 then the conclusion reads |f(x)|  |g(x)| < C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And f = O(xn) near x=0 iff |f(x)|  |xn| < C near x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And  f = o(g) near x0  ⇐⇒  ∀ε > 0, ∃η > 0, ∀x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' |x − x0| < η, |f(x)| < ε|g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)  and f is said to be “negligible compared with g near x0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If |g| > 0 then the conclusion reads |f(x)|  |g(x)| −→x→x0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And f = o(xn) near x=0 iff |f(x)|  |xn| −→x→0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Definition of the infinitesimal strain tensor ε  The motion is supposed to be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Along a trajectory, with F t0  P (t0) = I we have, near t0,  F t0  P (t0+h) = I + O(h),  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51)  thus F t0  P (t0+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = ⃗W + O(h) for all ⃗W ∈ ⃗Rn  t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', near t0,  ||⃗w − ⃗W|| = O(h)  when  ⃗w = F t0  P (t0+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52)  This supposes the use of a unique inner dot product (·, ·)G = (·, ·)g at all time, and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52) means  ||⃗w − ⃗W||g = O(h) near t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 If (·, ·)g is an inner dot product, the same at all time, and if (⃗ei) is a (·, ·)g-orthonormal  basis, the same at all time, then the infinitesimal strain tensor at P is the matrix defined by  [ε(P)]|⃗e =  [F(P)]|⃗e + [F(P)]T  |⃗e  2  − [I],  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53)  abusively written in short,  ε := F + F T  2  − I  (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  (And more precisely, at P ∈ Ωt0 and between t0 and t, [εt0  t (P)]|⃗e =  [F t0  t (P )]|⃗e+[F t0  t (P )]T  |⃗e  2  − [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  So ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  W +F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  W  2  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W means [ε]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W]|⃗e =  [F ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗  W ]|⃗e+[F ]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗  W ]|⃗e  2  − [ ⃗W]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  130  131  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition  Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 ε in (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54) cannot be a tensor (cannot be a function) since F t0  t (P) :  ⃗  Rn  t0 → ⃗  Rn  t and  F t0  t (P)T : ⃗  Rn  t → ⃗  Rn  t0 and It0 : ⃗  Rn  t0 → ⃗  Rn  t0 don’t have the same definition domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So ε is not a function, is not a tensor: It is a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But is called “the infinitesimal strain tensor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23 The Green–Lagrange tensor E = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −I  2  ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) satisfies near t0:  E = ε + o(t−t0)  (= F + F T  2  − I + o(t−t0))  (matrix meaning),  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55)  which means [E] = [ε] + o(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, “for small deformations” we write E ≃ ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E ≃ F +F T  2  − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation: (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55) is a linearization of E, since we keep the linear part of the “quadratic” E =  1  2(F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F − I) given by (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, ⃗U)g = 1  2  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U)g − ( ⃗W, ⃗U)g  �  for all ⃗U, ⃗W ∈ ⃗Rn  t0 (“motion squared” cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  the (F·, F·)g term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A (·, ·)g-orthonormal basis being chosen, [F T ] =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)[F]T , thus [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F], thus  2[E] = [C] − [I] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F] − [I] = ([F]T − [I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([F] − [I) + [F]T + [F] − 2[I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56)  Then, near t0 and with h = t−t0, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51) gives ([F]T − [I]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ([F] − I]) = O(h)O(h) = O(h2), thus  2[E] = [F]T + [F] − 2[I] + O(h), thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  H  Finger tensor F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T (left Cauchy–Green tensor)  Finger’s approach is consistent with the foundations of relativity (Galileo classical relativity or Einstein  general relativity): We can only do measurements at the current time t, and we can refer to the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  There is a lot of misunderstandings, as was the case for the Cauchy–Green deformation tensor C, due  to the lack of precise definitions: Definition domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Value domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Points at stake (p or P)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Euclidean  dot product (English?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' French?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Covariance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Contravariance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='.  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Let �Φ be motion, t0 ∈ R, Φt0 the associated motion, P ∈ Ωt0, t ∈ R, and F t0  t (P) := dΦt0  t (P) ∈ L( ⃗  Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And let (·, ·)G and (·, ·)g be Euclidean dot products in ⃗  Rn  t0 and ⃗  Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The Finger tensor bt0  t (pt), or left Cauchy–Green deformation tensor, at t at pt relative  to t0 is the endomorphism ∈ L( ⃗  Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t ) defined by, with P = Φt0  t  −1(pt),  bt0  t (pt) := F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t0  t )T  Gg(pt)  written in short  b = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T ,  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' is defined by (bt0  t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g = (F t0  t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, F t0  t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)G = ((F t0  t )T (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, (F t0  t )T (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)G, for  all ⃗w1, ⃗w2 vectors at pt ∈ Ωt, written in short  (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  (To compare with C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  And the Finger tensor relative to t0 is  bt0 :  �  �  �  �  �  C =  �  t  ({t} × Ωt) → L( ⃗  Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t )  (t, pt) → bt0(t, pt) := bt0  t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  NB: bt0 looks like a Eulerian function, but isn’t, since it depends on a t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Other definition found:  Bt0  t := bt0  t ◦ (Φt0  t )−1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Bt0  t (P) := bt0  t (pt) = F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P)T ,  written  B = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  Pay attention: Bt0  t (P) ∈ L( ⃗  Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t ) is an endomorphism at t at pt, not at t0 at P: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Bt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt(pt) =  bt0  t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt(pt) is meaningful, while Bt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wt0(P) is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The push-forward by Φ := Φt0  t  of the Cauchy–Green deformation tensor C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F is  Φ∗(C) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T = b, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15): It is the Finger tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So the endomorphism C in ⃗Rn  t0 is the  pull-back of the endomorphism b in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (However a push-forward and a pull-back don’t depend on any  inner dot product while the transposed F T does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  131  132  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  b−1  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  b−1  With pull-backs (towards the virtual power principle at t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With pt  = Φt0  t (P) and  ⃗Wi(P) =  (F t0  t (P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wi(pt):  ( ⃗W1, ⃗W2)G = (F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)G = (F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g = (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  So b−1 := (bt0  t )−1 is useful:  (bt0  t )−1 :  �  Ωt → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  pt → (bt0  t )−1(pt) = F t0  t (P)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P)−1 = Ht0  t (pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Ht0  t (pt)  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  with pt = Φt0  t (P) and Ht0  t (pt) = (F t0  t (P))−1 cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus we can define  (bt0)−1 :  �  �  �  �  �  �  t  ({t} × Ωt) → L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  (t, pt) → (bt0)−1(t, pt) := (bt0  t )−1(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Remark: (bt0)−1 looks like a Eulerian function, but isn’t, since it depends on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In short:  b−1 = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H,  to compare with  C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F,  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  and with ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W,  b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W,  to compare with  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w,  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  and with ⃗Wi = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi,  (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g = ( ⃗W1, ⃗W2)G,  to compare with  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = (⃗w1, ⃗w2)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Remark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 pt = Φt0  t (P) and b(pt) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(P)T and C(P) = F(P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(P) give  b(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(P) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='C(P),  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  written b = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus b−1 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1, so  Φt0∗  t  b−1 = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −T = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F)−1 = C−1,  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the pull-back of b−1 is C−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' b−1 is the push-forward of C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Time derivatives of b−1  With (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) let (bt0)−1 =noted b−1 = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, along a trajectory, and with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45), we get  Db−1  Dt  = DHT  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H + HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='DH  Dt = −d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H − HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v  = − b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v − d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Exercice H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Prove (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) with (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) gives  D  Dt(b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g  = 0 = (  Db−1  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g + (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' D ⃗w1  Dt , ⃗w2)g + (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, D ⃗w2  Dt )g, and  ⃗wi(t, p(t)) = F t0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wt0(P) gives D ⃗wi  Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wi, thus (  Db−1  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g +(b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g +(b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w2)g = 0,  thus (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Prove (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) with F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = It0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' b−1 = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T )−1 = F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1 gives F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = It0, thus (F T )′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Db−1  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ′ = 0,  thus F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Db−1  Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = 0, thus (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  132  133  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Euler–Almansi tensor a  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Euler–Almansi tensor a  Euler–Almansi approach is consistent with the foundations of relativity (Galileo relativity or Einstein  general relativity): We can only do measurements at the current time t, and we can refer to the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  At t in Ωt, consider the Finger tensor b = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T and its inverse b−1 = F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F T = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 Euler–Almansi tenor at pt ∈ Ωt is the endomorphism at0  t (pt) ∈ L( ⃗  Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t ) defined by  at0  t (pt) = 1  2(It − bt0  t (pt)−1) = 1  2(It − H(pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H(pt)),  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  written  a = 1  2(I − b−1) = 1  2(I − HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H),  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  to compare with the Green–Lagrange tensor E = 1  2(C − I) = 1  2(F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F − I) ∈ L( ⃗  Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark: at0 looks like a Eulerian function, but isn’t, since it depends on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) gives (⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi)  (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = 2(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g,  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  to compare with (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = 2(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (This also gives (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w1, ⃗w2)g = (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  And (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15) gives  F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = E,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  a = F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1,  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  standing for F t0  t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='at0  t (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P) = Et0  t (P) when p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 at0  t is not the push-forward of Et0  t  by Φt0  t (the push-forward is F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Time Taylor expansion for a  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) gives  Da  Dt = b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='b−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Almansi modified Infinitesimal strain tensor �ε  We are at t (present time) and remember the past: We prefer a definition of a infinitesimal strain tensor  �ε from the Euler–Almansi tensor a, instead of ε from Euler–Lagrange tensor E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Same Euclidean framework as in § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2, and matrix meaning again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We have I − b−1 = I − HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H = −(I − HT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I − H) + 2I − HT − H where H stands for Ht0  t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, for small displacement we get I − b−1 = 2I − HT − H + O(h), so  a(t, p(t)) = �ε(t, p(t)) + O(h)  where  �ε := I − H + HT  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  And, with t = t0 + h we have F t0(t, P) = I + (t−t0) d⃗v(t, P) + o(t−t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35), thus we have  Ht0(t, p(t)) = F t0(t, P)−1 = I − (t−t0) d⃗v(t, P) + o(t−t0) when p(t) = Φt0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  F t0(t, P) − I = I − Ht0(t, p(t)) + O(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Therefore, for small displacements (|t − t0| << 1):  a(t, p(t)) ≃ �ε(t, p(t)) ≃ εt0(t, P)  (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  I  Polar decomposition, elasticity and objectivity  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Polar decompositions of F (“isometric objectivity”)  The motion is supposed regular, t0, t ∈ R, pt0 ∈ Ωt0, F := F t0  t (pt0) (= dΦt0  t (pt0)), (·, ·)G and (·, ·)g are  Euclidean dot products in ⃗Rn  t0 and ⃗Rn  t , and C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Here the covariant objectivity is  abandoned due to the need for inner dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  133  134  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Polar decompositions of F (“isometric objectivity”)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U (right polar decomposition)  The endomorphism C being (·, ·)G-symmetric definite positive (the motion is supposed to be regular),  ∃α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', αn ∈ R∗  + (the eigenvalues), ∃⃗c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗cn ∈ ⃗Rn  t0 (associated eigenvectors), such that, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci = αi⃗ci  and  (⃗ci) is a (·, ·)G-orthonormal basis in ⃗Rn  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  So, if (⃗ai) is a (·, ·)G-Euclidean basis then D = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [C]⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P, where D = diag(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', αn) = [C]⃗c and  P −1 = P T , P = [Pij] being the transition matrix from (⃗ai) to (⃗ci), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' defined by ⃗cj = �  i Pij⃗ai for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then, define the endomorphism U ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0), called the right stretch tensor, by, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci = √αi ⃗ci,  and  U noted  =  √  C,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  the √αi being called the principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, define the linear map R ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ), called the  rotation map, by  R := F ◦ U −1 noted  =  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  so that  F = R ◦ U  noted  =  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U,  called the right polar decomposition of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 We have:  C = U ◦ U noted  =  U 2,  U is symmetric definite positive,  R−1 = RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  And  the right polar decomposition  F = R ◦ U  is unique :  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  If F = �R◦ �U where �U ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) is symmetric definite positive and �R ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) satisfies �R−1 = �RT ,  then �U = U and �R = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) yields (U ◦ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗cj  = λ⃗cj  = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗cj for all j, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), thus U ◦ U  = C ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  (U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci,⃗cj)G = (⃗ci, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗cj)G = (⃗ci, √αj⃗cj)G = √αj(⃗ci,⃗cj)G = √αjδij = √αiδij = √αi(⃗ci,⃗cj)G =  (√αi⃗ci,⃗cj)G = (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci,⃗cj)G for all i, j, thus U T = U (symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then RT ◦ R = U −T ◦ F T ◦ F ◦ U −1 = U −T ◦ C ◦ U −1 = U −1 ◦ (U ◦ U) ◦ U −1 = It0 identity  in ⃗Rn  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Details: (RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, ⃗Z)G = (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z)G =  (U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z)G = (U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, ⃗Z)G = ( ⃗W, ⃗Z)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Thus R−1 = RT ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0), thus R ◦ RT =  R ◦ R−1 = It identity in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And F = �R ◦ �U = R ◦ U gives F T ◦ F = �U T ◦ �RT ◦ �R ◦ �U = U T ◦ �RT ◦ �R ◦ U, thus F T ◦ F = �U T ◦ �U,  with F T ◦ F = U T ◦ U, thus �U ◦ �U = U ◦ U =  √  C, thus �U = U (uniqueness of the positive square root  eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence �R = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  F = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U (shifted right polar decomposition for covariant objectivity)  In fact we need to be more specific if the gift of ubiquity is prohibited: Since we work with the affine  space Rn, consider the Marsden’s shifter  S := St0  t (pt0) :  �  Tpt0(Ωt0) noted  =  ⃗Rn  t0 → Tpt(Ωt) noted  =  ⃗Rn  t  ⃗wt0,pt0 → (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0,pt0 )(t, pt) = ⃗wt0,pt0  where  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  NB: 1- S looks like the algebraic identity if you have time and space ubiquity gift (otherwise it is not),  2- S is not a topological identity since it changes the norms in the general case: You consider ||⃗wt0,pt0 ||G  at t0 and ||S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wt0,pt0 ||g = ||⃗wt0,pt0 ||g at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then, let R0 ∈ L(Tpt0(Ωt0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Tpt0(Ωt0)) =noted L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) be the endomorphism defined by, in short,  R0 := S−1 ◦ R noted  =  S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R,  so  R = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R0  (= S ◦ R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Full notations: (R0)t0  t,Gg(pt0) := (St0  t )−1(Rt0  t,Gg(pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  134  135  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Polar decompositions of F (“isometric objectivity”)  Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The endomorphism R0 = S−1 ◦ R is a rotation operator in (⃗Rn  t0, (·, ·)G):  R−1  0  = RT  0  in (⃗Rn  t0, (·, ·)G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  And  F = S ◦ R0 ◦ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Interpretation: F is composed of: The pure deformation U (endomorphism in ⃗Rn  t0), the rotation R0  (endomorphism in ⃗Rn  t0), and the shift operator S : ⃗Rn  t0 → ⃗Rn  t (from past to present time and position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (RT  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2, ⃗W1)G = (R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G  (definition of the transposed)  = (S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G  (definition of R0)  = (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G  (S is the algebraic identity)  = (RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2, ⃗W1)g  (definition of RT )  = (R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2, ⃗W1)g  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) )  = (R−1  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W2, ⃗W1)G  (S is the algebraic identity),  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  true for all ⃗W1, ⃗W2 ∈ ⃗Rn  t0, thus RT  0 = R−1  0  in (⃗Rn  t0, (·, ·)G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) and (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4) give (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Let D = diag(αi), let (⃗ai) be a Euclidean basis in ⃗Rn  t0, let P be the transition matrix  from (⃗ai) to (⃗ci), so [C]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove [U]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  √  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Case (⃗ai) = ( ⃗Ei) is a (·, ·)g-orthonormal  basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The n equations (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) (for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n), read as the matrix equation [C]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D since [⃗cj]⃗a is the  j-th column of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And he n equations (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) (for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n), read as the matrix equation [U]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  √  D since  [⃗cj]⃗a is the j-th column of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Instead of R0 ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8), you may prefer to consider �R0 ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) defined  by R = �R0 ◦ S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', �R0 = R ◦ S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  F = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R (left polar decomposition)  Same steps than for the right polar decomposition, but with pull-backs (with F −1 instead of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let pt = Φt0  t (pt0) ∈ Ωt, let bt0  t (pt) := F t0  t (pt0) ◦ (F t0  t )T (pt) ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ), written b = F ◦ F T (the left  Cauchy–Green deformation tensor also called the Finger tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The endomorphism b being symmetric  definite positive: ∃β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', βn ∈ R∗  + (the eigenvalues), ∃⃗d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗dn ∈ ⃗Rn  t (associated eigenvectors), such that,  for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗di = βi ⃗di,  and  (⃗di) is a (·, ·)g-orthonormal basis in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Then, define the unique endomorphism V ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ), called the left stretch tensor, by, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗di =  �  βi ⃗di,  and  V noted  =  �  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  (Full notation: V t0  t,Gg(pt) =  �  bt0  t (pt)Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Then define the linear map Rℓ ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) by  Rℓ := V −1 ◦ F noted  =  V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  so that  F = V ◦ Rℓ  noted  =  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Rℓ,  called the left polar decomposition of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 We have: 1-  b = V ◦ V noted  =  V 2,  V is symmetric definite positive,  R−1  ℓ  = RT  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  And the left polar decomposition F = V ◦ R is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Rℓ = R and V = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1 (so U and V are similar), thus U and V have the same eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  αi = βi for all i, and ⃗di = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci for all i gives a relation between eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  135  136  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Linear elasticity: A Classical “tensorial” approach  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- Use F −1 and b−1 = (F −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F −1), instead of F and C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F, to get F −1 = R−1  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1  ℓ  ,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus F = Uℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Rℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then name Uℓ = V to get (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15) and (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Rℓ = F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U = (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R, thus, by uniqueness of the right polar decomposition, V =  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1 (so U and V are similar) and Rℓ = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, with (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12), βi ⃗di = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗di = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗di), thus with  ⃗ci = R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗di, then (⃗ci) is an orthonormal basis in ⃗Rn  t0 and βiR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci = αiR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ci gives βi = αi and the  ⃗ci are eigenvectors of U, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Linear elasticity: A Classical “tensorial” approach  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Classical approach (“isometric objectivity”), and an issue  With the infinitesimal strain “tensor” (which is not a tensor but a matrix)  ε = F + F T  2  − I,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  the homogeneous isotropic elasticity constitutive law reads (matrix equation for the stress)  (σ(Φ) =)  σ = λTr(ε)I + 2µε,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  where λ, µ are the Lamé coefficients and σ is the Cauchy stress “tensor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Issue: Recall: Adding F and F T to make ε is functionally a mathematical nonsense since F : ⃗Rn  t0 →  ⃗Rn  t and F T : ⃗Rn  t → ⃗Rn  t0 and I is some identity operator: σ is not a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular the meaning of  Tr(ε) is questionable (since ε is not an endomorphism and Tr(ε) means Tr([ε]) = Tr([F ])+Tr([F T ])  2  − n), as  well as the meaning of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n = 1  2(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n), or the meaning of  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n = λTr(ε)⃗n + 2µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  since ⃗n has to be defined at (t0, pt0) for F and at (t, pt) for F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Cauchy’s approach: ⃗n is defined  at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  So, despite the eventual claims, neither ε nor σ are tensors (they don’t have a functional meaning):  They only have a questionable matrix meaning (observer dependent) [ε] := [F ]+[F ]T  2  − [I] and  [σ] = λTr([ε])[I] + 2µ[ε],  and  [σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗n] = λTr([ε])[⃗n] + 2µ[ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 To justify the name “tensor” applied to ε, you may read: “For small displacements the  Eulerian variable pt and the Lagrangian variable pt0 can be confused”: pt ≃ pt0 (so Ωt0 and Ωt are  “almost equal”, so F(pt0) + F T (pt) can be considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Which means that you use the zero-th order  Taylor expansions pt = Φt0  pt0 (t) = pt0 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But then, you cannot also use the first (or higher) order  Taylor expansion in following calculations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' you cannot use velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  A functional (tensorial) formulation (“isometric objectivity”)  Question: Can the constitutive law (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) be modified into a tensorial expression (a functional expression)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposal for a yes:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the “right polar decomposition” F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U where U ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Green  Lagrange tensor E = C−I  2  (endomorphism in ⃗Rn  t0) then reads, with (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5),  E = U 2−It0  2  = (U−It0)2 + 2(U − It0)  2  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  (the Green–Lagrange tensor is independent of the rotation R), thus, with U − It0 = O(h) (small defor-  mation approximation), we get the modified infinitesimal strain tensor at pt0 ∈ Ωt0  �ε = U−It0 ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0),  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  endomorphism in ⃗Rn  t0 (to compare with ε which is not a function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the previous §).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Full notation  �εt0  t,Gg(pt0) = U t0  t,Gg(pt0)−It0(pt0) in L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') And, for all ⃗W ∈ ⃗Rn  t0 we get  �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W − ⃗W = R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w − ⃗W  ∈ ⃗Rn  t0,  when  ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  Interpretation: From ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W ∈ ⃗Rn  t (the deformed by the motion), remove the “rigid  body rotation” to get R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W ∈ ⃗Rn  t0, to which you remove the initial ⃗W to obtain �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W ∈ ⃗Rn  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  136  137  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Linear elasticity: A Classical “tensorial” approach  In particular ||�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W||G = ||(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W||G measures the relative elongation undergone by ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And you can  then apply R to get back into ⃗Rn  t at pt:  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = ⃗w − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W ∈ ⃗Rn  t ,  when  ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, at pt0 ∈ Ωt0, consider the stress tensor �Σ(Φ) =noted �Σ ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) (functionally well)  defined by  �Σ = λTr(�ε)It0 + 2µ�ε = λTr(U−It0)It0 + 2µ(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  (The trace Tr(�ε) is well defined since �ε is an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Then for any ⃗W ∈ ⃗Rn  t0 you get in ⃗Rn  t0, at  pt0 ∈ Ωt0,  �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = λTr(�ε) ⃗W + 2µ�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = λTr(U−It0) ⃗W + 2µ(U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W− ⃗W)  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  (functionally well defined in ⃗Rn  t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then rotate and shift with R to get into ⃗Rn  t at pt,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = λTr(�ε)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = λTr(U−It0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W  = λTr(U−It0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W + 2µ(F − R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W,  = λTr(U−It0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W + 2µ(⃗w − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W),  where  ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  You have defined the two point tensor (functionally well defined)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�Σ = λTr(�ε)R + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then you can propose the constitutive law with the stress tensor (the symmetric endomorphism)  in ⃗Rn  t given by  (�σ(Φ) =)  �σ = R ◦ �Σ ◦ R−1  noted  =  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1 ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  (Functionally well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') And, for all vector fields ⃗w defined in Ωt, you get the (functionally well  defined) vector field  �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w  ∈ ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  Interpretation of (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30): Shift and rigid rotate backward by applying R−1, apply the elastic stress  law with Σ which corresponds to a rotation free motion (Noll’s frame indifference principle), then shift  and rigid rotate forward by applying R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Detailed expression for (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30): With Tr(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1) = Tr(�ε) (see exercise I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8), we get, at (t, pt),  �σ = λTr(�ε) It + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1 = λTr(U−It0) It + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1  = λTr(U−It0) It + 2µ(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1−It).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  And for any ⃗w ∈ ⃗Rn  t , and with ⃗w = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W, you get  �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = λTr(�ε) ⃗w + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = λTr(U−It0) ⃗w + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W  = λTr(U−It0) ⃗w + 2µ(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w−⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  To compare with the classical functionally meaningless (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 Doing so, you avoid the use of the Piola–Kirchhoff tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Prove: Tr(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1) = Tr(�ε) = �  i(αi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (NB: �ε is an endomorphism in ⃗Rn  t0 while  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1 is an endomorphism in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' det|⃗e(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1 − λIt) = det|⃗e(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(�ε−λIt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1) = det| ⃗  E,⃗e(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' det| ⃗  E(�ε−λI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' det|⃗e, ⃗  E(R−1) = det| ⃗  E(�ε−λI)  for all Euclidean bases ( ⃗Ei) and (⃗ei) in ⃗Rn  t0 and ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (With L = �ε and components, Tr(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1) =  �  i(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R−1)i  i = �  ijk Ri  jLj  k(R−1)k  i = �  jk(R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='R)k  j Lj  k = �  jk δk  j Lj  k = �  j Lj  j = Tr(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  137  138  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Elasticity with a covariant objective approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Elongation in R2 along the first axis : origin O, same Euclidean basis ( ⃗E1, ⃗E2) and Eu-  clidean dot product at all time, ξ > 0, t ≥ t0, L, H > 0, P ∈ [0, L] × [0, H], [−−→  OP]| ⃗E =  �  X0  Y0  �  , and  [−−−−−−→  OΦt0  t (P)]| ⃗E =  �  X0 + ξ(t−t0)X0  Y0  �  =  �  X0(κ+1)  Y0  �  =  �  x  y  �  = [−→  Op]| ⃗E, where κ = ξ(t−t0) > 0 for t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- Give F, C, U =  √  C and R = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Relation with the classical expression ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Spring −−→  OP = −−→  Oct0(s) = X0 ⃗E1+Y0 ⃗E2+s ⃗W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [−−→  OP]| ⃗E = [−−→  Oct0]| ⃗E =  �  X0+sW1  Y0+sW2  �  | ⃗E  with s ∈ [0, L]  and ⃗W = W1 ⃗E1 + W2 ⃗E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Give the deformed spring, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' give p = ct(s) = Φt0  t (ct0(s)), and ⃗ct′, and the  stretch ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- [F] = [dΦ] =  � κ+1  0  0  1  �  , same Euclidean dot product and basis at all time, thus [F T ] = [F]T = [F],  then [C] = [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [F] = [F]2 =  � (κ+1)2  0  0  1  �  , thus [U] = [F] =  � κ+1  0  0  1  �  , thus [R] = [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' All the matrices are  given relative to the basis ( ⃗Ei), thus F, C, U, R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1 = (κ+1)2 ⃗E1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E2 = ⃗E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Since R = I and [ε] = [�ε], (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31) gives the usual result [σ] = λTr([ε])I + 2µ[ε], cf (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18) (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- −−−−→  Oct(s)  =  −−−−−−−−−→  OΦt0  t (ct0(s))  =  � (X0+sW1)(κ+1)  Y0+sW2  �  | ⃗  E  ,  thus ⃗ct  ′(s)  =  � W1(κ+1)  W2  �  | ⃗  E  ,  stretch ration  W 2  1 (κ+1)2+W 2  2  W 2  1 +W 2  2  at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Simple shear in R2 : [−−−−−−→  OΦt0  t (P)]| ⃗E =  �  X + ξ(t−t0)Y  Y  �  =noted  �  X + κY  Y  �  =  �  x  y  �  =  [−→  Op]| ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Same questions, and moreover give the eigenvalues of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- [F] =  � 1  κ  0  1  �  , [C] =  � 1  0  κ  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  � 1  κ  0  1  �  =  � 1  κ  κ  κ2+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Eigenvalues: det(C − λI) = λ2 −  (2+κ2)λ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Discriminant ∆ = (2+κ2)2 − 4 = κ2(κ2+4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Eigenvalues α± = 1  2(2+κ2 ± κ  √  κ2+4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (We check that  α± > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Eigenvectors ⃗v±(main directions of deformations) given by (1−α±)x+κy = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', y = 1  2(κ±  √  κ2+4)x,  thus, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗v± =  �  2  κ ±  √  κ2+4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (We check that ⃗v+ ⊥ ⃗v−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') With P the transition matrix from ( ⃗E1, ⃗E2) to  (  ⃗v+  ||⃗v+||,  ⃗v−  ||⃗v−||) and D = diag(α+, α−) we get C = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1 (with P −1 = P T since here (  ⃗v+  ||⃗v+||,  ⃗v−  ||⃗v−||) is an  orthonormal basis), thus U = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  √  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1 (we check that U T = U and U 2 = C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And R = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- −−−−→  Oct(s) = −−−−−−−−−→  OΦt0  t (ct0(s)) =  � (X0+sW1) + κ(Y0+sW2)  Y0+sW2  �  , thus [⃗ct  ′(s)] =  � W1 + κW2  W2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Stretch ratio  (W1+κW2)2+W 2  2  W 2  1 +W 2  2  at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Second functional formulation: With the Finger tensor  The above approach uses the push-forward: It uses F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' you arrive with your memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' You may prefer  to use the pull-back, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' use F −1 (you remember the past which is Cauchy’s point of view): Then you  use F −1 = R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='V −1 the right polar decomposition of F −1, and you consider the tensor  ��εt = V −1−It ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ),  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  and  (σt(Φ) =)  σt = λTr(��εt)It + 2µ��εt,  and  σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗nt = λTr(��εt)⃗nt + 2µ��εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  (Quantities functionally well defined: Give a tensorial approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Elasticity with a covariant objective approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In § I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 you need to start with Euclidean dot products, so from the start the result can’t be covariant  objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Can you start without Euclidean dot products to set up general laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Proposal:  Hypothesis: The Cauchy stress ⃗w is a Eulerian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then we could use the (covariant objective) Lie derivative which characterizes the rate of stress,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5: With a particle PObj ∈ Obj, with ⃗v(τ, pτ) = ∂�Φ  ∂τ (τ, PObj) its Eulerian velocity at τ at  138  139  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The displacement vector ⃗U  pτ = �Φ(τ, PObj), the Lie derivative of a Eulerian vector field ⃗w along ⃗v is, at (t, pt),  L⃗v ⃗w(t, pt) = lim  τ→t  ⃗w(τ, pτ) − ⃗wt∗(τ, pτ)  τ − t  = (∂ ⃗w  ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Hence  the  proposal,  with  the  virtual  power  principle  to  measure  the  rate  of  stress  (see  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10780v1 for a full description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- Hypotheses: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1- Suppose that n Eulerian vector fields ⃗wj (“force fields”), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, enable to  characterize a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (In fact, for elasticity problems it could be better to replace vector fields ⃗wj with  1-forms αi to characterize the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2- With a basis (⃗ei) chosen in ⃗Rn  t , with (ei) its (covariant) dual basis in ⃗Rn∗  t , assume that the internal  power density at (t, pt) is given by (at first order):  pint(⃗v) =  n  �  j=1  ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗v ⃗wj =  n  �  j=1  ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (∂ ⃗wj  ∂t + d⃗wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  (At second order you can add second order Lie derivatives as L⃗v(L⃗v ⃗wj), similarly for higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  2- Then, so that this pint satisfies the frame invariance hypothesis, choose a Euclidean dot prod-  uct (·, ·)g in ⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, first, the internal power has to vanish if d⃗v = 0, thus we are left with  pint(⃗v) = −  n  �  j=1  ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wj = −τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v,  where  τ =  n  �  j=1  ⃗wj ⊗ ej,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  defined at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (The j-th column of [τ]|⃗e is [⃗wj]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') And, second, the internal power vanishes if d⃗v +d⃗vT = 0  (rotation), thus we are left with  pint(⃗v) = −τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v + d⃗vT  2  = −σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v  where  σ = τ + τ T  2  ,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  this pint(⃗v) = −σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v being the usual expression of the internal power at first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36) may be applied to orthotropic elasticity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' for a material which fibers at some  time t0 are along ⃗e1, in a 2-D case for simplicity: 1- With an elongation type motion (Φe) given by  [(Fe)(pt0)] = [d(Φe)(pt0)] =  �  1+α11(pt0)  0  0  1−α22(pt0)  �  you measure the Young moduli in the direc-  tions ⃗e1 and ⃗e2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 2- And with a shear type motion given by [(Fs)(pt0)] = [d(Φs)(pt0)] =  �  1  γ12  0  1  �  you  measure the shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For more complex material, you may need more vectors ⃗wj to describe the constitutive law, that is,  (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36) may be considered with �m  i=1ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L⃗v ⃗wj with m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 The Lie approach is different from the usual classic approach:  1- The classic approach looks for an order two stress tensor [σ] as a function of the deformation  gradient [F], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- The Lie approach begins with the internal power (which measures forces), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36), which then  enable to build τ and the σ (the stress tensor), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', application to visco-elasticity: With the Lie approach, you automatically use Lie derivative of  vector fields (and/or of differential forms), instead of Lie derivative of order 2 tensor fields (which does  not seem to give good result, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the Maxwell visco-elastic type laws, as well as footnote1 page 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  J  Displacement  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  The displacement vector ⃗U  In Rn, let pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the bi-point vector  ⃗Ut0  t (pt0) = Φt0  t (pt0) − It0(pt0) = pt − pt0 = −−→  pt0pt  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  is called the displacement vector at pt0 relative to t0 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' This defines the map  ⃗Ut0  t  :  �  Ωt0 → ⃗Rn  pt0 → ⃗Ut0  t (pt0) := pt − pt0 = −−→  pt0pt  when  pt = Φt0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  139  140  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The differential of the displacement vector  Remark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 ⃗Ut0  t (pt0) doesn’t define a vector field (it is not tensorial), because ⃗Ut0  t (pt0) = pt−pt0 = −−→  pt0pt  is a bi-point vector which is neither in ⃗Rn  t0 or in ⃗Rn  t since pt0 ∈ Ωt0 and pt ∈ Ωt (it requires time and  space ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, it makes no sense on a non-plane surface (manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' More at § J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 For elastic solids in Rn, the function ⃗Ut0 is often considered to be the unknown (to be  computed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But the “real” unknown is the motion Φt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And it is sometimes confused with the extension  of a spring 1-D case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 where ||⃗wt0(pt0)|| represents the initial length and ||⃗wt0∗(t, pt)||  represents the current length of the spring, while the length of the displacement vector ⃗Ut0  t  = pt − pt0  can be very long for a very small elongation ||⃗wt0∗(t, pt)|| − ||⃗wt0(pt0)|| of the spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The differential of the displacement vector  The differential of ⃗Ut0  t  at pt0 is  d⃗Ut0  t (pt0) = dΦt0  t (pt0) − It0 = F t0  t (pt0) − It0,  written  d⃗U = F − I,  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  thus isn’t defined as a function, because F t0  t (pt0) : ⃗Rn  t0 → R while It0 : ⃗Rn  t0 → ⃗Rn  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So d⃗Ut0  t (pt0) as to be understood as a matrix: With  [⃗Ut0  t (pt0)] = [−−→  pt0pt] = [−−−−−−→  OΦt0  t (pt0)] − [−−→  Opt0],  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  relative to an origin O and a unique basis at all time, compute [d⃗Ut0  t (pt0)] = [dΦt0  t (pt0)] − I, abusively  written d⃗U = dΦ − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, with ⃗W ∈ ⃗Rn  t0 ,  d⃗U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W − ⃗W,  which means  = [F t0  t (pt0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗W] − [ ⃗W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Thus we have defined (matrix meaning)  ⃗Ut0 :  �  [t0, T] × Ωt0 → ⃗Rn  (t, pt0) → ⃗Ut0(t, pt0) := ⃗Ut0  t (pt0),  and  ⃗Ut0  pt0 :  �  [t0, T] → ⃗Rn  t → ⃗Ut0  pt0 (t) := ⃗Ut0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Deformation “tensor” ε (matrix), bis  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) gives (matrix meaning)  F t0  t (pt0) = It0 + d⃗Ut0  t (pt0),  written  F = I + d⃗U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Therefore, Cauchy–Green deformation tensor C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F reads, in short, (matrix meaning)  C = I + d⃗U + d⃗UT + d⃗UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗U  (matrix meaning),  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [Ct0  t (pt0)] = [It0] + [d⃗Ut0  t (pt0)] + [d⃗Ut0  t (pt0)]T + [d⃗Ut0  t (pt0)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗Ut0  t (pt0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus the Green–Lagrange deformation tensor E = C−I  2 , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47), reads, in short, (matrix meaning)  E = d⃗U + d⃗UT  2  + 1  2d⃗UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗U  (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Thus the deformation tensor ε, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54), reads (matrix meaning)  ε = E − 1  2(d⃗U)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗U,  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  with ε the “linear part” of E (small displacements: we only used the first order derivative dΦt0  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  140  141  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Small displacement hypothesis, bis  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Small displacement hypothesis, bis  (Usual introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Let pt = Φt0  t (pt0), ⃗Wi ∈ ⃗  Rn  t0, ⃗wi(pt) = F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi(pt0) ∈ ⃗Rn  t (the push-forwards),  written ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then define (matrix meaning)  ⃗∆i := ⃗wi − ⃗Wi = dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Wi,  and  ||⃗∆||∞ = max(||⃗∆1||Rn, ||⃗∆2||Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Then the small displacement hypothesis reads (matrix meaning):  ||⃗∆||∞ = o(|| ⃗W||∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Thus ⃗wi = ⃗Wi + ⃗∆i (with ⃗∆i “small”) and the hypothesis (·, ·)g = (·, ·)G (same inner dot product at t0  and t) give  (⃗w1, ⃗w2)G − ( ⃗W1, ⃗W2)G = (⃗∆1, ⃗W2)G + (⃗∆2, ⃗W1)G + (⃗∆1, ⃗∆2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) gives 2(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = 2(ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G + (d⃗UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G, And (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12) gives  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G = (ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗W1, ⃗W2)G + O(||⃗∆||2  ∞),  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  so Et0  t  is approximated by εt0  t , that is, Et0  t ≃ εt0  t (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Displacement vector with differential geometry  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  The shifter  We give the steps, see Marsden–Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The complexity introduced is due to the small displacement  hypothesis applied to the Green–Lagrange tensor E = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −I  2  which linearization gives ε = F +F T  2  − I  (the classical approach “squares the motion” to get E, then “linearizes” E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' to get back to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with a  spurious F T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let P ∈ Ωt0, ⃗WP ∈ ⃗Rn  t0, pt = Φt0  t (P) ∈ Ωt, and ⃗wpt = F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗WP ∈ ⃗Rn  t (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Affine case Rn (continuum mechanics): With pt = Φt0  t (P), the shifter is:  �  St0  t :  � Ωt0 × ⃗Rn  t0 → Ωt × ⃗Rn  t  (P, ⃗ZP ) → �  St0  t (P, ⃗ZP ) = (pt, St0  t (⃗ZP ))  with  St0  t (⃗ZP ) = ⃗ZP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  (The vector is unchanged but the time and the application point have changed: A real observer has no  ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So:  St0  t ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  and  [St0  t ]|⃗e = I identity matrix,  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  the matrix equality being possible after the choice of a unique basis at t0 and at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (simplified  notation) �  St0  t (P, ⃗ZP ) =noted St0  t (⃗ZP ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the deformation tensor ε at P can be defined by  εt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z(P) = (St0  t )−1(F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z(P)) + F t0  t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (St0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z(P))  2  − ⃗Z(P),  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  in short: ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z = (St0  t )−1(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z)+F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (St0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Z)  2  − ⃗Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a manifold: Ω is a manifold (like a surface in R3 from which we cannot take off).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let TP Ωt0  be the tangent space à P (the fiber at P), and TptΩt be the tangent space à pt (the fiber at pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In general TP Ωt0 ̸= TptΩt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' on a sphere “the Earth”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The bundle (the union of fibers) at t0 is  TΩt0 = �  P ∈Ωt0 ({P} × TP Ωt0), and the bundle at t is TΩt = �  pt∈Ωt({pt} × TptΩt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the shifter  �  St0  t :  �  TΩt0 → TΩt  (P, ⃗ZP ) → �  St0  t (P, ⃗ZP ) = (pt, St0  t (⃗ZP )),  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  is defined such that ⃗ZP ∈ TP Ωt0 “as little distorted as possible” along a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', on a sphere, if the  path is a geodesic, if θt0 is the angle between ⃗ZP and the tangent vector to the geodesic at P, then  θt0 is also the angle between St0  t (⃗ZP ) and the tangent vector to the geodesic at pt, and St0  t (⃗ZP ) has  the same length than ⃗ZP (at constant speed in a car you think the geodesic is a straight line, although  St0  t (⃗ZP ) ̸= ⃗ZP : the Earth is not flat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  141  142  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Alternating multilinear form  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The displacement vector  (Affine space framework, Ωt0 open set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let P ∈ Ωt0, ⃗WP ∈ ⃗Rn  t0, pt = Φt0  t (P) ∈ Ωt, and  dΦt0  t = F t0  t  ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Define  δ�  ⃗Ut0  t  :  �  �  �  Ωt0 × ⃗Rn  t0 → Ωt × L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t )  (P, ⃗ZP ) → δ�  ⃗Ut0  t (P, ⃗ZP ) = (pt, δ ⃗Ut0  t (⃗ZP ))  with  δ ⃗Ut0  t (⃗ZP ) = (F t0  t  − St0  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ZP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  Then δ�  ⃗Ut0  t  = F t0  t  − St0  t  is a two-point tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  Ct0  t = (F t0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t  = (δUt0  t + St0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (δUt0  t + St0  t )  = I + (St0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δUt0  t + (δUt0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='St0  t + (δUt0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δUt0  t ,  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  since (St0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='St0  t  = I identity in TΩt0: Indeed, ((St0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='St0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, ⃗B)Rn = (St0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗A, St0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗B)Rn = ( ⃗A, ⃗B)Rn,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14), for all ⃗A, ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the Green–Lagrange tensor is defined on Ωt0 by  Et0  t = 1  2(Ct0  t − It0) = (St0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δUt0  t + St0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (δUt0  t )T  2  + 1  2(δUt0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δUt0  t ,  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  to compare with (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K  Determinants  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Alternating multilinear form  Let E be a vector space, and let L(E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) =noted L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) be the set of multilinear forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  m ∈ L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) iff  m(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗x + λ⃗y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = m(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') + λm(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  for all ⃗x, ⃗y  ∈  E  and all λ  ∈  R and for all “slot”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular,  m(λ1⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λn⃗xn)  =  (�  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n λi) m(⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗xn), for all λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λn ∈ R and all ⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗xn ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If n = 1 then a 1-alternating multilinear function is a linear form, also called a 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If n ≥ 2 then Aℓ :  �  En → R  (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) → Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn)  �  ∈ L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is a n-alternating multilinear form iff, for  all ⃗u,⃗v ∈ E,  Aℓ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = −Aℓ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='),  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  the other elements being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If n = 1, the set of 1-forms is Ω1(E) = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If n ≥ 2, the set of  n-alternating multilinear forms is  Ωn(E) = {m ∈ L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) : m = Aℓ is alternating}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  If Aℓ, Bℓ ∈ Ωn(E) and λ ∈ R then Aℓ + λBℓ ∈ Ωn(E) thanks to the linearity for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  Ωn(E) is a vector space, sub-space in (F(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Leibniz formula  Particular case dim E=n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Aℓ ∈ Ωn(E) (a n-alternating multilinear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Recall (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Cartan [5]):  1- A permutation σ : [1, n]N → [1, n]N is a bijective map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' one-to-one and onto);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Sn be the set  of permutations of [1, n]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- A transposition τ : [1, n]N → [1, n]N is a permutation that exchanges two elements, that is, ∃i, j s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  τ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='), the other elements being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- A permutation is a composition of transpositions (theorem left as an exercise, of see Cartan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  a permutation is even iff the number of transpositions is even, and a permutation is odd iff the number  of transpositions is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Based on: The parity (even or odd) of a permutation is an invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- The signature ε(σ) = ±1 of a permutation σ is +1 if σ is even, and is −1 if σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  142  143  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Determinant of vectors  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (Leibniz formula) Let Aℓ ∈ Ωn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n =noted (⃗ei) be a basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' For  all vectors ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn ∈ E, with ⃗vj = �n  i=1vi  j⃗ei for all j,  Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = c  �  σ∈Sn  ε(σ)  n  �  i=1  vσ(i)  i  = c  �  τ∈Sn  ε(τ)  n  �  i=1  vi  τ(i)  (with c := Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  Thus if c = Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) is known, then Aℓ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus dim(Ωn(E)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Classic not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : ⃗vj = �n  i=1vij⃗ei, Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = c �  σ∈Sn ε(σ) �n  i=1 vσ(i),i = c �  τ∈Sn ε(τ) �n  i=1 vi,τ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let F := F([1, n]N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [1, n]N) =noted [1, n][1,n]N  N  be the set of functions i :  �  [1, n]N → [1, n]N  k → ik = i(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Aℓ being multilinear, Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = �n  j1=1 vj1  1 Aℓ(⃗ej1,⃗v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) (“the first column” development).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' By  recurrence we get Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = �n  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',jn=1 vj1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='vjn  n Aℓ(⃗ej1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗ejn) = �  j∈F  �n  k=1 vj(k)  k  Aℓ(⃗ej(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗ej(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And Aℓ(⃗ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗ein) ̸= 0 iff i : k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n} → i(k) = ik ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n} is one-to-one (thus bijective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = �  σ∈Sn  �n  i=1 vσ(i)  i  Aℓ(⃗eσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗eσ(n)) = �  σ∈Sn ε(σ) �n  i=1 vσ(i)  i  Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en), which is the  first equality in (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then �  σ∈Sn ε(σ) �n  i=1 vσ(i)  i  = �  σ∈Sn ε(σ) �n  i=1 vσ(σ−1(i))  σ−1(i)  since σ is bijectif, thus  �  σ∈Sn ε(σ) �n  i=1 vσ(i)  i  = �  τ∈Sn ε(τ −1) �n  i=1 vi  τ(i), thus the second equality in (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4) since ε(τ)−1 = ε(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (See Cartan [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Determinant of vectors  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n being a basis in E, the alternating multilinear form det|⃗e ∈ Ωn(E) defined  by  det  |⃗e (⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) = 1  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  is called the determinant relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And, with prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (here c = 1),  det  |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) =  �  σ∈Sn  ε(σ)  n  �  i=1  vσ(i)  i  =  �  τ∈Sn  ε(τ)  n  �  i=1  vi  τ(i)  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  is called the determinant of the vectors ⃗vi relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And we write  Ωn(E) = Vect{det  |⃗e }  (the 1-D vector space spanned by det|⃗e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Thus, if Aℓ ∈ Ωn(E) then  Aℓ = Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) det  |⃗e ,  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  thus if (⃗bi) is another basis then  ∃c ∈ R,  det  |⃗b  = c det  |⃗e ,  with  c = det  |⃗b  (⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Change of measuring unit: If (⃗ai) is a basis and ⃗bj = λ⃗aj for all j, prove  ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n,  ⃗bj = λ⃗aj  =⇒  det  |⃗a = λn det  |⃗b  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  (relation between volumes relative to a change of measuring unit in the Euclidean case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' det  |⃗a (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn) = det  |⃗a (λ⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λ⃗an)  multi  =  linear λn det  |⃗a (⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗an)  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  = λn (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  = λn det  |⃗b  (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) ̸= 0 iff (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) is a basis, or equivalently, det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = 0 iff  ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If one of the ⃗vi is = ⃗0 then det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = 0 (multilinearity), and if �n  i=1ci⃗vi = 0 and one  of the ci ̸= 0 and then a ⃗vi is a linear combination of the others thus det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = 0 (since det|⃗e  is alternate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) ̸= 0 ⇒ the ⃗vi are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if the ⃗vi are independent then  (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) is a basis, thus det|⃗v(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = 1 ̸= 0, with det|⃗v = c det|⃗e, thus det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  143  144  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Determinant of a matrix  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 In R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗v1 = �2  i=1 vi  1⃗ei and ⃗v2 = �2  j=1 vj  2⃗ej (duality notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  det  |⃗e (⃗v1,⃗v2) = v1  1v2  2 − v2  1v1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Development relative to the first column (linearity used for the first vector ⃗v1 = v1  1⃗e1 + v2  1⃗e2):  det|⃗e(⃗v1,⃗v2) = det|⃗e(v1  1⃗e1 + v2  1⃗e2,⃗v2) = v1  1 det|⃗e(⃗e1,⃗v2) + v2  1 det|⃗e(⃗e2,⃗v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus (linearity used for the second  vector ⃗v2 = v1  2⃗e1 + v2  2⃗e2): det|⃗e(⃗v1,⃗v2) = 0 + v1  1v2  2 det(⃗e1,⃗e2) + v2  1v1  2 det(⃗e2,⃗e1) + 0 = v1  1v2  2 − v2  1v1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 In R3, with ⃗vj = �3  i=1 vi  j⃗ei, prove:  det(⃗v1,⃗v2,⃗v3) =  3  �  i,j,k=1  εijkvi  1vj  2vk  3,  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  where εijk = 1  2(j−i)(k−j)(k−i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' εijk = 1 if (i, j, k) = (1, 2, 3), (3, 1, 2) or (2, 3, 1) (even signature),  εijk = −1 if (i, j, k) = (3, 2, 1), (1, 3, 2) and (2, 1, 3) (odd signature), and εijk = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Development relative to the first column (as in exercise K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Result = v1  1v2  2v3  3 + v1  2v2  3v3  1 + v1  3v2  1v3  2 − v3  1v2  2v1  3 − v3  2v2  3v1  1 − v3  3v2  1v1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Determinant of a matrix  Let M = [Mij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n be a n2 real matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × R (Cartesian product n-times) with its  canonical basis ( ⃗Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗vj ∈ ⃗Rn, ⃗vj = �n  i=1Mij ⃗Ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So M =  � [⃗v1]| ⃗E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', [⃗vn]| ⃗E  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 The determinant of the matrix M =  � [⃗v1]| ⃗E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', [⃗vn]| ⃗E  �  is  det(M) := det  | ⃗E  (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Let M T be the transposed matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (M T )ij = Mji for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  det(M T ) = det(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' det[Mij] = det  ⃗E  (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn)  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  =  �  σ∈Sn  ε(σ)  n  �  i=1  vσ(i)  i  =  �  τ∈Sn  ε(τ)  n  �  i=1  vi  τ(i) = det[Mji].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Volume  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Let (⃗ei) be a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider a parallelepiped in Rn which sides are vectors  ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its algebraic volume relative to (⃗ei) is  algebraic volume = det  |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  And its volume relative to (⃗ei) is (non negative)  volume =  ��det  |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if n = 1 and ⃗v = v1⃗e1, then det|⃗e(⃗v) = v1 is the algebraic length of ⃗v (relative to the unit  of measurement given by ⃗e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And | det|⃗e(⃗v)| = |v1| is the length of ⃗v (the norm of ⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (The volume  function (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) →  ��det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn)  �� is not a multilinear form, because the absolute value function is  not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if n = 2 or 3, see exercises K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ei) be a Cartesian basis and (ei) = (dxi) be the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Cartan [6],  det  |⃗e  noted  =  e1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∧ en = dx1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∧ dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  And, for integration, the volume element (non negative) uses a Euclidean basis (⃗ei) and is  dΩ(⃗x) = | det  |⃗e | = |dx1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∧ dxn| noted  =  dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  Thus  the  volume  of  a  parallelepiped  at  ⃗x  which  sides  are  given  by  δx1⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', δxn⃗un  is  dΩ(⃗x)(δx1⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', δxn⃗un) = |δx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δxn| | det|⃗e(⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗un)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the volume of a polygonal domain Ω =  144  145  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Determinant of an endomorphism  �N  i=1 Pi where Pi is a parallelepiped which sides are given by δxi,1⃗ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', δxi,n⃗ui,n is  |Ω| =  N  �  i=1  | det  |⃗e (⃗ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗ui,n)|δxi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δxi,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  And thus (Riemann approach), the volume of a regular domain Ω is written  |Ω| =  �  Ω  dΩ =  �  ⃗x∈Ω  | det  |⃗e (⃗ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗ui,n)| dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  In particular, since any regular volume Ω can be approximated with cubes as small as wished, |Ω| =  �N  i=1 |δxi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δxi,n det|⃗e(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en)| = �N  i=1 |δxi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='δxi,n| gives  |Ω| =  �  Ω  dΩ =  �  ⃗x∈Ω  dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Let Ψ : ⃗q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', qn) ∈ [a1, b1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × [an, bn] → ⃗x = (x1 = Ψ1(⃗q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn = Ψn(⃗q)) ∈ Ω  be a parametric description of a domain Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove  dΩ(⃗x) = |JΨ(⃗q)| dq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dqn  (= | det  |⃗e (⃗p1(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗pn(⃗x))| dq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dqn),  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  where (⃗pi(x)) = ( ∂Ψ  ∂qi (⃗q)) is the parametric basis at ⃗x = Ψ(⃗q) and JΨ(⃗q) = det|⃗e[dΨ(⃗q)]|⃗e is the Jacobian  matrix of Ψ at ⃗q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And thus |Ω| =  �  ⃗q |JΨ(⃗q)| dq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Polar coordinates for illustration purpose (immediate generalization): Consider the disk Ω parametrized  with the polar coordinate system Ψ : ⃗q = (ρ, θ) ∈ [0, R] × [0, 2π] → ⃗x = (x = ρ cos θ, y = ρ sin θ) ∈ R2  where a Euclidean basis (⃗e1,⃗e2) has been used in R2 (so ⃗x = ρ cos θ⃗e1 + ρ sin θ⃗e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The associated polar ba-  sis at ⃗x = Ψ(⃗q) is (⃗p1(⃗x) =  ∂Ψ  ∂ρ (ρ, θ), ⃗p2(⃗x) =  ∂Ψ  ∂θ (ρ, θ)), so [⃗p1(⃗x)]|⃗e =  � cos θ  sin θ  �  and [⃗p2(⃗x)]|⃗e =  � −ρ sin θ  ρ cos θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus det|⃗e(⃗p1(⃗x), ⃗p2(⃗x)) = ρ (> 0 here), thus dΩ = |ρ| dρdθ = ρ dρdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the volume is |Ω| =  �  ⃗x∈Ω dΩ =  � R  ρ=0  � 2π  θ=0 ρ dρdθ (= πR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 What is the “volume element” on a regular surface Σ in R3, called the “surface element”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗e1,⃗e2,⃗e3) be a Euclidean basis in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We need a regular parametric description Ψ : (u, v) ∈ [a1, b2]×  [a2, b2] → ⃗x = Ψ(u, v) = x1(u, v)⃗e1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' + x3(u, v)⃗e3 of the geometric surface Σ = Im(Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗t1(⃗x) = ∂Ψ  ∂u (u, v)  and ⃗t2(⃗x) = ∂Ψ  ∂v (u, v) are tangent vectors at Σ at ⃗x = Ψ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hence a normal unit vector is ⃗n(⃗x) =  ⃗t1(⃗x)∧⃗t2(⃗x)  ||⃗t1(⃗x)∧⃗t2(⃗x)||,  and thus det|⃗e(⃗t1,⃗t2,⃗n) = ||⃗t1(⃗x) ∧ ⃗t2(⃗x)|| is the area of the parallelogram which sides are given by ⃗t1 and ⃗t2  (volume with height 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the surface element at ⃗x = Ψ(u, v) is dΣ(⃗x) = || ∂Ψ  ∂u (u, v) ∧ ∂Ψ  ∂v (u, v)|| dudv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  |Σ| =  �  ⃗x∈Σ dΣ(⃗x) =  � b1  u=a1  � b2  v=a2 || ∂Ψ  ∂u (u, v) ∧ ∂Ψ  ∂v (u, v)|| dudv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Determinant of an endomorphism  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition and basic properties  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 The determinant of an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) relative to a basis (⃗ei) is  �  det  |⃗e (L) := det  |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  This define �  det|⃗e : L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (If the context is not ambiguous, then �  det|⃗e =noted det|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- If L = I the identity, then �  det|⃗e(I) = 1 for all basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- For all ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn ∈ E,  det  |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vn) = �  det  |⃗e (L) det  |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  3- If L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗e = [Lij], then  �  det  |⃗e (L) = det([L]|⃗e) = det([Lij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  4- For all M ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E), and with M ◦ L =noted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L (thanks to linearity),  �  det  |⃗e (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L) = �  det  |⃗e (M) �  det  |⃗e (L) = �  det  |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  5- L is invertible iff �  det|⃗e(L) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  145  146  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Determinant of an endomorphism  6- If L is invertible then  �  det  |⃗e (L−1) =  1  �  det|⃗e(L)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  7- If (·, ·)g is an inner dot product in E and LT  g is the (·, ·)g transposed of L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (LT  g ⃗w, ⃗u)g = (⃗w, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)g  for all ⃗u, ⃗w ∈ E) then  �  det  |⃗e (LT  g ) = �  det  |⃗e (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  8- If (⃗ei) and (⃗bi) are two (·, ·)g-orthonormal bases in ⃗Rn  t (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' two Euclidean basis for the same  measuring unit), then det|⃗b = ± det|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- �  det|⃗e(I) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) det|⃗e(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) det|⃗e(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) = 1, true for all basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Let m : (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) → m(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) := det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vn): It is a multilinear alternated form, since  L is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus m =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) m(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) det|⃗e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With m(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) �  det|⃗e(L), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- Apply (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) with M = [L]|⃗e to get (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- det|⃗e((M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) = det|⃗e(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en)) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24) �  det|⃗e(M) det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  5- If L is invertible, then 1 = �  det|⃗e(I) = �  det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L−1) = �  det|⃗e(L) �  det|⃗e(L−1), thus �  det|⃗e(L) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If �  det|⃗e(L) ̸= 0 then det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) ̸= 0, thus (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) is a basis, thus L is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  6- (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) gives 1 = �  det|⃗e(I) = �  det|⃗e(L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L) = �  det|⃗e(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  det|⃗e(L−1), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  7-  (LT  g ⃗w, ⃗u)g  =  (⃗w, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)g  gives  [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  g ]|⃗e  =  ([L]|⃗e)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [g]|⃗e,  thus  det([g]|⃗e) det([LT  g ]|⃗e)  =  det(([L]|⃗e)T ) det([g]|⃗e), and det([g]|⃗e) ̸= 0 (exercise), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  8- Let P be the change of basis endomorphism from (⃗ei) to (⃗bi), and P be the transition matrix  from (⃗ei) to (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Both basis being (·, ·)g-orthonormal, P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = I, thus det(P) = ±1 = �  det|⃗e(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And det|⃗e(⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn) = det|⃗e(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) = �  det|⃗e(P) det|⃗e(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗en) = �  det|⃗e(P) det|⃗b(⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn), thus  det|⃗e = �  det|⃗e(P) det|⃗b = ± det|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 Two (·, ·)g-orthonormal bases (⃗ei) and (⃗bi) have the same orientation iff det|⃗b = + det|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 Prove �  det|⃗e(λL) = λn �  det|⃗e(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  det  |⃗e (λL) = det  |⃗e (λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) = λn det  |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗en) = λn �  det  |⃗e (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The determinant of an endomorphism is objective  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 Let (⃗ai) and (⃗bi) be bases in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The determinant of an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E)  is objective (observer independent, here basis independent):  (det([L]|⃗a) =)  �  det  |⃗a (L) = �  det  |⃗b  (L)  (= det([L]|⃗b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  NB: But the determinant of n vectors is not objective, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) (compare the change of basis formula  for vectors [⃗w]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗a with the change of basis formula for endomorphisms [L]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ai) and (⃗bi) be bases in E, and P be the transition matrix from (⃗ai) to (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The change  of basis formula [L]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P and (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26) give det([L]|⃗b) = det(P −1) det([L]|⃗a) det(P) = det([L]|⃗a),  thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25) gives (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 Let (⃗ai) and (⃗bi) be bases in E, and P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) be the change of basis endomorphism  from (⃗ai) to (⃗bi) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = ⃗bj for all j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove  det  |⃗a (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn) = �  det  |⃗a (P),  thus  det  |⃗a = �  det  |⃗a (P) det  |⃗b  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  det  |⃗b  =  det|⃗a  �  det|⃗a(P)  ,  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  det  |⃗a (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn) = det  |⃗a (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗an)  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  =  �  det  |⃗a (P) det  |⃗a (⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗an) =  �  det  |⃗a (P) 1 =  �  det  |⃗a (P) det  |⃗b  (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗bn),  thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30) and det|⃗a = �  det|⃗a(P) det|⃗b and det|⃗a(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn) = �  det|⃗a(P) det|⃗b(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  146  147  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Determinant of a linear map  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Determinant of a linear map  (Needed for the deformation gradient F t0  t (P) = dΦt0  t (P) : ⃗Rn  t0 → ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let A and B be vector spaces, dim A = dim B = n, and (⃗ai) and (⃗bi) be bases in A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition and first properties  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19 The determinant of a linear map L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) relative to the bases (⃗ai) and (⃗bi) is  �  det  |⃗a,⃗b  (L) := det  |⃗b  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  (And �  det|⃗a,⃗b(L) =noted det(L) if the bases are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Thus, (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) gives, with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = �n  i=1Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [L]|⃗a,⃗b = [Lij]:  �  det  |⃗a,⃗b  (L) = det([L]|⃗a,⃗b) = det([Lij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 Let ⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗un ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  det  |⃗b  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗un) = �  det  |⃗a,⃗b  (L) det  |⃗a (⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' m : (⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗un) ∈ An → m(⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗un) := det|⃗b(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗un) ∈ R is a multilinear alternated form,  since L is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And m(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗an) = det|⃗b(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗an) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31) �  det|⃗a,⃗b(L) = �  det|⃗a,⃗b(L) det|⃗a(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',⃗an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus m = �  det|⃗a,⃗b(L) det|⃗a, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Corollary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 Let A, B, C be vector spaces such that dim A = dim B = dim C = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ai), (⃗bi), (⃗ci)  be bases in A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let L : A → B and M : B → C be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, with M ◦ L =noted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L (thanks  to linearity),  �  det  |⃗a,⃗c(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L) = �  det  |⃗a,⃗b  (L) �  det  |⃗b,⃗c  (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  det  |⃗a,⃗c(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L) = det  |⃗c (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗an)) = �  det  |⃗b,⃗c  (M) det  |⃗b  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗an) = �  det  |⃗b,⃗c  (M) �  det  |⃗a,⃗b  (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Jacobian of a motion, and dilatation  Let �Φ be a motion, let t0, t ∈ R, let Φt0  t be the associated motion, let F t0  t (pt0) := dΦt0  t (pt0) : ⃗Rn  t0 → ⃗Rn  t  the deformation gradient at pt0 ∈ Ωt0 relative to t0 and t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ( ⃗Ei) be a Euclidean basis in ⃗  Rn  t0  and (⃗ei) be a Euclidean basis in ⃗  Rn  t for all t ≥ t0, and [F t0  t (pt0)]| ⃗E,⃗e = [Fij(pt0)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej =  �n  i,j=1Fij(pt0)⃗ei for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 The “volume dilatation” at pt0, relative to the Euclidean bases ( ⃗Ei) in  ⃗  Rn  t0 and (⃗ei)  in ⃗  Rn  t , is  J| ⃗E,⃗e(Φt0  t )(pt0) := �  det  | ⃗E,⃗e  (F t0  t (pt0))  (= det  |⃗e (F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗En) = det([Fij(pt0)])),  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  usually written J| ⃗E,⃗e := det([F]| ⃗E,⃗e) (or simply J = det(F) when everything is implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, at t0 at pt0, (pt0, ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗En) is a unit parallelepiped which volume is 1 relative to the unit of mea-  surement chosen in ⃗Rn  t0, and, at t at pt = Φt0  t (pt0), J| ⃗E,⃗e(Φt0  t )(pt0) = det|⃗e(F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗En)  is the volume of the parallelepiped (pt, F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F t0  t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗En) relative to the unit of measurement  chosen in ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation: With t2 > t1 ≥ t0, and [⃗ei) is the basis at t1 and t2:  Dilatation if J| ⃗E,⃗e(Φt0  t2)(pt0) > J| ⃗E,⃗e(Φt0  t1)(pt0) (volume increase),  147  148  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Dilatation rate  contraction if J| ⃗E,⃗e(Φt0  t2)(pt0) < J| ⃗E,⃗e(Φt0  t1)(pt0) (volume decrease), and  incompressibility if J| ⃗E,⃗e(Φt0  t2)(pt0) = J| ⃗E,⃗e(Φt0  t1)(pt0) for all t (volume conservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular, if (⃗ei) = ( ⃗Ei) then J|⃗e,⃗e(Φt0  t0)(pt0) = 1, and if t > t0, then  Dilatation if J|⃗e,⃗e(Φt0  t )(pt0) > 1 (volume increase),  contraction if J|⃗e,⃗e(Φt0  t )(pt0) < 1 (volume decrease), and  incompressibility if J|⃗e,⃗e(Φt0  t )(pt0) = 1 for all t (volume conservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23 Let ( ⃗Ei) be a Euclidean basis in ⃗Rn  t0, and let (⃗ai) and (⃗bi) be two Euclidean bases in ⃗Rn  t  for the same Euclidean dot product (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  J| ⃗E,⃗a(Φt0  t (P)) = ±J| ⃗E,⃗b(Φt0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' P being the transition matrix from (⃗ai) to (⃗bi), det(P) = ±1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30) gives [F]| ⃗  E,⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[F]| ⃗  E,⃗b,  thus det([F]| ⃗  E,⃗a) = ± det([F]| ⃗  E,⃗b), thus det|⃗a(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗En) = ± det|⃗b(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Determinant of the transposed  Let (A, (·, ·)g) and (B, (·, ·)h) be finite dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) (a linear map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Recall: The transposed LT  gh ∈ L(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A) is defined by, for all ⃗u ∈ A and all ⃗w ∈ B, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='68)  (LT  gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, ⃗u)g := (⃗w, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Let (⃗ai) be a basis in A and (⃗bi) be a basis in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  �  det([LT  gh]|⃗b,⃗a) = det([L]|⃗a,⃗b)det([(·, ·)g]|⃗a)  det([(·, ·)h]|⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  Indeed, (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37) gives [(·, ·)g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [LT  gh]|⃗b,⃗a = ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [(·, ·)h]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Dilatation rate  A unique Euclidean basis (⃗ei) at all time is chosen, and (·, ·)g is the associated inner dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  ∂Jt0  ∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt)  A regular motion �Φ is considered, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5), and the Eulerian velocity is ⃗v(t, pt) = ∂�Φ  ∂t (t, PObj) at pt =  �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t0 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The associated motion Φt0 is given by Φt0(t, pt0) = �Φ(t, PObj) =noted pt when  pt0 = �Φ(t0, PObj),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), and is supposed to be at least C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lagrangian velocity is ⃗V (t, pt0) =  ∂Φt0  ∂t (t, pt0), and the Eulerian velocity satisfies ⃗v(t, pt) = ∂Φt0  ∂t (t, pt0) when pt = Φt0(t, pt0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  F t0(t, pt0) = dΦt0(t, pt0) = F t0  t (pt0) = dΦt0  t (pt0), and consider the Jacobian  Jt0  t (pt0) = det  |⃗e (F t0  t (pt0)) = Jt0(t, pt0),  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  Lemma K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24  ∂Jt0  ∂t (t, pt0) satisfies, with pt = Φt0  t (pt0),  ∂Jt0  ∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt)  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  (value to be considered at t at pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, �Φ is incompressible iff div⃗v(t, pt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let O be a origin in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let −−−→  OΦt0 = �n  i=1Φi⃗ei, ⃗V t0 = �n  i=1V i⃗ei, ⃗v = �n  i=1vi⃗ei, F t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej =  dΦt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej = �n  i=1  ∂Φi  ∂Xj ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let [F t0]| ⃗E,⃗e =noted F, Jt0 =noted J and [dΦi]| ⃗E =  � ∂Φi  ∂X1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂Φi  ∂Xn  �  =noted dΦi  148  149  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Dilatation rate  (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus J = det F = det  �  �  dΦ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn  �  �, thus (a determinant is multilinear)  ∂J  ∂t = det  �  �  �  �  �  �  ∂(dΦ1)  ∂t  dΦ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn  �  �  �  �  �  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' + det  �  �  �  �  �  dΦ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn−1  ∂(dΦn)  ∂t  )  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With Φt0 C2, thus ∂(dΦi)  ∂t  (t, pt0) Swhartz  =  d(∂Φi  ∂t )(t, pt0) = dV i(t, pt0) = dvi(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t, pt0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  det  �  �  �  �  �  �  ∂(dΦ1)  ∂t  dΦ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn  �  �  �  �  �  �  = det  �  �  �  �  �  �  �  n  �  i=1  ∂v1  ∂xi dΦi  dΦ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn  �  �  �  �  �  �  �  det is  =  alternating det  �  �  �  �  �  �  ∂v1  ∂x1 dΦ1  dΦ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn  �  �  �  �  �  �  = ∂v1  ∂x1 det  �  �  �  �  dΦ1  dΦ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  dΦn  �  �  �  � = ∂v1  ∂x1 J  Idem for the other terms, thus  ∂J  ∂t (t, pt0) = ∂v1  ∂x1 (t, pt) J(t, pt0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' + ∂vn  ∂xn (t, pt) J(t, pt0) = div⃗v(t, pt) J(t, pt0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25 div⃗v(t, pt) is the dilatation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Leibniz formula  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26 (Leibniz formula) Under regularity assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' hypotheses of the Lebesgue  theorem to be able to derive under  �  ) we have  d  dt  ��  pt∈Ωt  f(t, pt) dΩt  �  =  �  pt∈Ωt  �Df  Dt + f div⃗v  �  (t, pt) dΩt  =  �  pt∈Ωt  �∂f  ∂t + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + f div(⃗v)  �  (t, pt) dΩt  =  �  pt∈Ωt  �∂f  ∂t + div(f⃗v)  �  (t, pt) dΩt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  Z(t) :=  �  p∈Ωt  f(t, p) dΩt =  �  P ∈Ωt0  f(t, Φt0(t, P)) Jt0(t, P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (The Jacobian is positive for a regular motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Then (derivation under  �  )  Z′(t) =  �  P ∈Ωt0  Df  Dt (t, pt) Jt0(t, P) + f(t, pt)∂Jt0  ∂t (t, P) dΩt0  =  �  P ∈Ωt0  (Df  Dt (t, pt) + f(t, pt) div⃗v(t, pt))Jt0(t, P) dΩt0,  thanks to (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And div(f⃗v) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v + f div⃗v gives (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Corollary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27 With (⃗u, ⃗w)g =noted ⃗u • ⃗w (in the given Euclidean framework),  d  dt  �  Ωt  f(t, pt) dΩt =  �  Ωt  ∂f  ∂t (t, pt) dΩt +  �  ∂Ωt  (f⃗v • ⃗n)(t, pt) dΓt,  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  sum of the temporal variation within Ωt and the flux through the surface ∂Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Apply (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  149  150  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂J/∂F = J F −T  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  ∂J/∂F = J F −T  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Meaning of  ∂ det  ∂Mij ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let Mnn = {M = [Mij] ∈ Rn2} be the set of n ∗ n matrices, and consider the function  Z := det :  �  Mnn → R  M = [Mij] → Z(M) := det(M) = det([Mij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  Question: What does  ∂Z  ∂Mij (M) mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: It is the “standard meaning” of a directional derivative  ∂f  ∂xi (⃗x) = df(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  where here  f = Z, thus ⃗x =noted M is a matrix (a vector in Mnn), and (⃗ei) is the canonical basis (mij) in Mnn  (all the elements of the matrix mij vanish but the element at intersection of line i and column j which  equals 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So:  ∂Z  ∂Mij  (M) := dZ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='mij = lim  h→0  Z(M + hmij) − Z(M)  h  (∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Calculation of  ∂ det  ∂Mij  Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28  ∀i, j,  ∂Z  ∂Mij  (M) = Z(M) (M −T )ij,  written  ∂Z  ∂M = Z M −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂Z  ∂Mij (M) := limh→0  det(M+hmij)−det(M)  h  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The development of the determinant det(M + hmij)  relative to the column j gives  det(M + h[mij]) = det(M) + h cij  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  where cij is the (i, j)-th cofactor of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  ∂Z  ∂Mij (M) = limh→0  Z(M+hmij)−Z(M)  h  = cij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And since  M −1 =  1  det(M)[cij]T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [cij] = det(M)M −T , we get  ∂Z  ∂Mij (M) = det(M)(M −T )ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  ∂J/∂F = J F −T usually written [ ∂J  ∂Fij ] = J F −T  Setting of § K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8: With F := dΦ(pt0) we have F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej = �n  i=1Fij⃗ei where Fij = ∂Φi  ∂Xj (pt0), and  JΦ,pt0, ⃗E,⃗e  noted  =  J :  �  �  �  �  �  L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) → R  F → J(F) := det([Fij])  (= det([ ∂Φi  ∂Xj  (pt0)]),  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  so, J(F) is the Jacobian �  det| ⃗E,⃗e(dΦ(pt0)) of Φ at pt0 relative to ( ⃗Ei) and (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45) gives:  Corollary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29  ∀i, j,  ∂J  ∂Fij  (F) = J(F) ([F]−T )ij,  written  ∂J  ∂F = J F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Interpretation of  ∂J  ∂Fij ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The first derivations into play are along the directions ⃗Ej at t0: The Fij = ∂Φi  ∂Xj (pt0) := dΦi(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Question:  ∂J  ∂Fij is the usual notation for a directional derivative, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' § K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  ∂J  ∂Fij is the derivative  in which direction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: 1- “Identify” F ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) with the tensor �F ∈ L(⃗Rn∗  t , ⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) given by �F(ℓ, ⃗U) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So, if F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej = �n  i=1Fij⃗ei then �F = �n  i,j=1Fij⃗ei ⊗ πEj, relative to a basis ( ⃗Ei) and its covariant dual basis  (πEi) in ⃗Rn  t0 and a basis (⃗ei) and ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Define the function �  det ⃗E,⃗e = �J :  �  �  �  L(⃗Rn∗  t , ⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → R  �F → �J( �F) := J(F) = det  ⃗E,⃗e  (F) = det([Fij])  �  �  �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  150  151  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Transformed parallelepiped  3- Then it is meaningful to differentiate �J along the direction ⃗ei ⊗ πEj ∈ L(⃗Rn∗  t , ⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) to get  ∂ �J  ∂Fij  ( �F) := lim  h→0  �J|⃗e, ⃗E( �F + h⃗ei ⊗ πEj) − �J|⃗e, ⃗E( �F)  h  ( noted  =  ∂J  ∂Fij  (F));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  This is a derivation in both directions πEj in ⃗Rn  t0 (past at pt0) and ⃗ei in ⃗Rn  t (present at pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' What does  this derivative mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (The answer is unknown to the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  L  Transport of volumes and areas  Here Rn = R3 the usual affine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let t0, t ∈ R, and Φt0  t : R × Ωt0 → Ωt, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let FP = dΦt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (·, ·)g be a Euclidean dot product in ⃗Rn (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='), with ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||g the associated norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Transformed parallelepiped  The Jacobian of Φt0  t at P relative to a (·, ·)g-Euclidean bases is defined in (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35): With FP = F t0  t (P),  JP = J(P) := det  |⃗e (F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗En)),  and  JP > 0  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  the motion being supposed regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, if (⃗U1P , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗UnP ) is a parallelepiped at P at t0, if ⃗uip = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗UiP ,  then (⃗u1p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗unp) is a parallelepiped at p at t which volume is  det  |⃗e (⃗u1p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗unp) = JP det  |⃗e (⃗U1P , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗UnP ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Transformed volumes  Riemann integrals and (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) give the change of variable formula: For any regular function f : Ωt → R,  �  pt∈Ωt  f(pt) dΩt =  �  P ∈Ωt0  f(Φt0  t (P)) |J(P)| dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  (See (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18): dΩt is a positive measure: It is not a multilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') In particular,  |Ωt| =  �  pt∈Ωt  dΩt(pt) =  �  P ∈Ωt0  |J(P)| dΩt0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  (With J(P) > 0 for regular motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Transformed parallelogram  Consider two independent vectors ⃗U1P , ⃗U2P ∈ ⃗Rn  t0 at t0 at P, and the vectors ⃗u1p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U1P and ⃗u2p =  FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U2P at t at p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Since Φt0  t is a diffeomorphism, ⃗u1p and ⃗u2p are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then choose a Euclidean dot product (·, ·)g (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') to be able to use the vectorial product,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15), the same at all time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the areas of the parallelograms are  ||⃗U1P ∧ ⃗U2P ||g  and  ||⃗u1p ∧ ⃗u2p)||g,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  and unit normal vectors to the quadrilaterals are  ⃗NP =  ⃗U1P ∧ ⃗U2P  ||⃗U1P ∧ ⃗U2P ||g  ∈ ⃗Rn  t0,  and  ⃗np =  ⃗u1p ∧ ⃗u2p  ||⃗u1p ∧ ⃗u2p||g  ∈ ⃗Rn  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Proposition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If ⃗u1p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U1P and ⃗u2p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U2P , then  ⃗u1p ∧ ⃗u2p = JP F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �⃗U1P ∧ ⃗U2P  �  ,  and  ||⃗u1p ∧ ⃗u2p||g = JP ||F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗U1P ∧ ⃗U2P )||g,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  since JP > 0 (for regular motions), and  ⃗np =  F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗NP  ||F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗NP ||g  (̸= FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗NP in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  151  152  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Transformed surface  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗WP ∈ ⃗  Rn  t0 and ⃗wp = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗WP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the volume of the parallelepiped (⃗u1p, ⃗u2p, ⃗wp) is  (⃗u1p ∧ ⃗u2p, ⃗wp)g = det(⃗u1p, ⃗u2p, ⃗wp) = det(FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U1P , FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗U2P , FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗WP ) = det(FP ) det(⃗U1P , ⃗U2P , ⃗WP )  = JP (⃗U1P ∧ ⃗U2P , ⃗WP )g = JP (⃗U1P ∧ ⃗U2P , F −1  P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wp)g = JP (F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗U1P ∧ ⃗U2P ), ⃗wp)g,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  for all ⃗wp, thus (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7), thus  ⃗u1p∧⃗u2p  ||⃗u1p∧⃗u2p||g =  JP F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗U1P ∧⃗U2P )  JP ||F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗U1P ∧⃗U2P )||g , thus (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Transformed surface  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Deformation of a surface  A parametrized surface Ψt0 in Ωt0 and the associated geometric surface St0 are defined by  Ψt0 :  �  [a, b] × [c, d] → Ωt0  (u, v) → P = Ψt0(u, v)  �  and  St0 = Im(Ψt0) ⊂ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  (It is also represented after a choice of an origin O by the vector valued parametrized surface ⃗rt0 = −−−→  OΨt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  The transformed parametric surface is Ψt := Φt0  t ◦ Ψt0 and the associated geometric surface is St:  Ψt := Φt0  t ◦ Ψt0 :  �  [a, b] × [c, d] → Ωt0  (u, v) → p = Ψt(u, v) = Φt0  t (Ψt0(u, v)) = Φt0  t (P)  �  and  St = Φt0  t (St0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  (After a choice of an origin O, the associated vector valued parametrized surface is ⃗rt = −−→  OΨt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Let ( ⃗E1, ⃗E2) be the canonical basis in the space R × R ⊃ [a, b] × [c, d] = {(u, v)} of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The surface Ψt0 is supposed to be regular, that is, Ψt0 is C1 and, for all P = Ψt0(u, v) ∈ St0, the  tangents vectors ⃗T1P and ⃗T2P at P are independent, that is,  ⃗T1P := dΨt0(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1  noted  =  ∂Ψt0  ∂u (u, v),  ⃗T2P := dΨt0(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E2  noted  =  ∂Ψt0  ∂v (u, v),  �  �  �  �  �  and  ⃗T1P ∧ ⃗T2P ̸= ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  And the tangent vectors at St at p = Φt0  t (P) at t are  �  �  �  �  �  ⃗t1p := dΨt(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E1 = ∂Ψt  ∂u (u, v),  so  ⃗t1p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗T1P  (= dΦt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂Ψt0  ∂u (u, v)),  ⃗t2p := dΨt(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗E2 = ∂Ψt  ∂v (u, v),  so  ⃗t2p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗T2P  (= dΦt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂Ψt0  ∂v (u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  These vectors are independent since Φt0  t is a diffeomorphism and Ψt0 is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In facts, we used tangent  vectors to curves and their push-forwards, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 and § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Euclidean dot product and unit normal vectors  Then choose a Euclidean dot product (·, ·)g (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='), to be able to use the vectorial product,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15), the same at all time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the scalar area elements dΣP at P at St0 relative to Ψt0, and dσp  at p at St relative to Ψt, are  �  �  �  �  �  dΣP := ||∂Ψt0  ∂u (u, v) ∧ ∂Ψt0  ∂v (u, v)||g du dv  (= ||⃗T1P ∧ ⃗T2P ||g du dv),  dσp := ||∂Ψt  ∂u (u, v) ∧ ∂Ψt  ∂v (u, v)||g du dv  (= ||⃗t1p ∧ ⃗t2p||g du dv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  And the areas of St0 and St are  �  �  �  �  �  �  �  �  �  |St0| =  �  P ∈St0  dΣP :=  � b  u=a  � d  v=c  ||∂Ψt0  ∂u (u, v) ∧ ∂Ψt0  ∂v (u, v)||g du dv,  |St| =  �  p∈St  dσp :=  � b  u=a  � d  v=c  ||∂Ψt  ∂u (u, v) ∧ ∂Ψt  ∂v (u, v)||g du dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  (See (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18): dΣP and dσp are positive measures: They are not multilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  152  153  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Piola identity  And the unit normal vectors ⃗NP at St0 at P at t0 and ⃗np at St at p at t are  �  �  �  �  �  �  �  �  �  �  �  ⃗NP =  ∂Ψt0  ∂u (u, v) ∧ ∂Ψt0  ∂v (u, v)  || ∂Ψt0  ∂u (u, v) ∧ ∂Ψt0  ∂v (u, v)||g  (=  ⃗T1P ∧ ⃗T2P  ||⃗T1P ∧ ⃗T2P ||g  )  ⃗np =  ∂Ψt  ∂u (u, v) ∧ ∂Ψt  ∂v (u, v)  || ∂Ψt  ∂u (u, v) ∧ ∂Ψt  ∂v (u, v)||g  (=  ⃗t1p ∧ ⃗t2p  ||⃗t1p ∧ ⃗t2p||g  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  Then the vectorial area elements d⃗ΣP at P at St0 = Im(Ψt0) relative to ⃗rt0 and d⃗σp at p at St = Im(Ψt)  relative to Ψt are  �  �  �  �  �  d⃗ΣP := ⃗NP dΣP = ∂Ψt0  ∂u (u, v) ∧ ∂Ψt0  ∂v (u, v) du dv  (= ⃗T1P ∧ ⃗T2P du dv)  d⃗σp := ⃗np dσp = ∂Ψt  ∂u (u, v) ∧ ∂Ψt  ∂v (u, v) du dv  (= ⃗t1p ∧ ⃗t2p du dv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  (Useful to get the flux through a surface:  �  Γ ⃗f • ⃗n dσ =  �  Γ ⃗f • d⃗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (NB: d⃗ΣP and d⃗σp are not multilinear since dΣP and dσp are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Relations between surfaces  ⃗t1p ∧ ⃗t2p = JP F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗T1P ∧ ⃗T2P ), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7), gives  ∂Ψt  ∂u (u, v) ∧ ∂Ψt  ∂v (u, v) = JP F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (∂Ψt0  ∂u (u, v) ∧ ∂Ψt0  ∂v (u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  This gives the relation between vectorial and scalar area elements,  ⃗n dσp = d⃗σp = JP F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗ΣP = JP F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗NP dΣP ,  and  dσp = JP ||F −T  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗NP ||g dΣP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  (Check with (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Piola identity  Reminder: Let M = [M i  j] be a 3∗3 matrix function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We use the usual divergence in continuum mechanics  (non objective) given by divM :=  �  �  �  ∂M 1  1  ∂X1 + ∂M 1  2  ∂X2 + ∂M 1  3  ∂X3  ∂M 2  1  ∂X1 + ∂M 2  2  ∂X2 + ∂M 2  3  ∂X3  ∂M 3  1  ∂X1 + ∂M 3  2  ∂X2 + ∂M 3  3  ∂X3  �  �  � =  �  �  �  �  �n  j=1  ∂M 1  j  ∂Xj  �n  j=1  ∂M 2  j  ∂Xj  �n  j=1  ∂M 3  j  ∂Xj  �  �  �  �, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if Cof(M)  is the matrix of cofactors (in R3: Cof(M)i  j = M i+1  j+1M i+2  j+2 − M i+1  j+2M i+2  j+1), then M −1 =  1  det M Cof(M)T ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',  (det M)M −1 = Cof(M)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  The framework being Euclidean, we use a Euclidean basis and the associated matrix, and thus (matrix  meaning)  J(P)F(P)−T = Cof(F(P)) noted  =  Cof(F)(P),  written  JF −T = Cof(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Proposition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (Piola identity) In R3, we have  div(JF −T )(P) = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∀i,  n  �  j=1  ∂Cof(F)i  j  ∂Xj  (P) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Also written �n  j=1  ∂  ∂Xj (J ∂Xi  ∂xj ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22) is a just a matrix computation since we used the  divergence of a matrix (we used components relative to a given basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We are in R3, thus Cof(F)i  j = F i+1  j+1F i+2  j+2 − F i+1  j+2F i+2  j+1, and F = [dΦt] = [ ∂ϕi  ∂Xj ], that is, F i  j = ∂ϕi  ∂Xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  ∂Cof(F)i  j  ∂Xj  =  ∂2ϕi+1  ∂Xj∂Xj+1  ∂ϕi+2  ∂Xj+2 + ∂ϕi+1  ∂Xj+1  ∂2ϕi+2  ∂Xj∂Xj+2 −  ∂2ϕi+1  ∂Xj∂Xj+2  ∂ϕi+2  ∂Xj+1 − ∂ϕi+1  ∂Xj+2  ∂2ϕi+2  ∂Xj∂Xj+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, for all i = 1, 2, 3, we get �n  j=1  ∂Cof(F )i  j  ∂Xj  = 0 (the terms cancel each other out two by two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  153  154  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Piola transformation  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Piola transformation  Let ⃗u be a vector field in Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The goal is to find a vector field ⃗UPiola in Ωt0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' for all open subset ωt ⊂ Ωt  with ωt0 = Φt0  t  −1(ωt) ⊂ Ωt0,  �  ∂ωt0  ⃗UPiola • ⃗N dΣ =  �  ∂ωt  ⃗u • ⃗n dσ,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  or  �  ωt0  div(⃗UPiola) dΩt0 =  �  ωt  div(⃗u) dΩt,  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  P ���ωt0  div(⃗UPiola)(P) dΩt0 =  �  P ∈ωt0  div(⃗u)(Φt0  t (P)) J(P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  (The motion is supposed to be regular, so J(P) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus we want  div⃗UPiola(P) = J(P) div⃗u(p)  when  p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Definition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 The Piola transform is the map  �  �  �  C∞(Ωt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Rn) → C∞(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Rn)  ⃗u → ⃗UPiola,  ⃗UPiola(P) := J(P)F(P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(p)  when  p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  (So ⃗UPiola(P) = J(P)Φ∗(⃗u)(P) where Φ∗(⃗u)(P) = F(P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(p) = the pull-back with Φ = Φt0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 With p = Φt0  t (P), ⃗UPiola = �n  i=1U i  Piola⃗ei and ⃗u = �n  i=1ui⃗ei we get  div⃗UPiola(P) = J(P) div⃗u(p),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  n  �  i=1  ∂U i  Piola  ∂Xi  (P) = J(P)  n  �  i=1  ∂ui  ∂xi (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' div(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61) �  div(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗w gives div((JF −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u ◦ Φt0  t ))(P) = (div(JF −T )(P), ⃗u(p))g +  (J(P)F(P)−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. (d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(P))=(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)0+J(P)(F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(P)−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗u(p) = J(P) I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗u(p) = J(P)div⃗u(p),  which gives (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M  Work and power  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definitions  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Work  (Thermodynamic like approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The elementary work is a differential form α, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α = dU (internal  energy density), α = δW = (elementary work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider a regular curve c : t ∈ [t0, T] → c(t) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  let ⃗v(t, c(t)) := ⃗c ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The work of α along the curve is  �  c  α :=  � T  t=t0  α(t, c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗c ′(t) dt noted  =  � T  t=t0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗c  =  � T  t=t0  α(t, c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, c(t)) dt noted  =  � T  t=t0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', W t0  T (α, c) =  �  c δW = work along c of the differential form α = δW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then consider an object Obj and its motion �Φ : (t, PObj) → p(t) = �Φ(t, PObj) = �ΦPObj (t) ∈ Rn, the  curves cPObj = �ΦPObj : t ∈ [t0, T] → p(t) = �ΦPObj (t) ∈ Rn, and the Eulerian velocities ⃗v(t, p(t)) = �ΦPObj  ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The work for Obj and a Eulerian differential form α along �Φ is the sum of work of α of all particles,  formally �  PObj ∈Obj(  �  cPObj αPObj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So with the associated motion Φt0(t, pt0) = �Φ(t, PObj) = p(t) = Φt0  pt0 (t)  when pt0 = �Φ(t0, PObj), and with Ωt = �Φ(t, Obj),  W t0  T (�Φ) :=  �  pt0∈Ωt0  � T  t=t0  α(t, Φt0  pt0 (t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, Φt0  pt0 (t)) dt dΩt0 =  � T  t=t0  �  pt∈Ωt  α(t, pt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, pt) dΩt dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  (The last equality if Fubini theorem can be applied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' if α is C0 and Φt0 is C1, Obj being bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  154  155  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Piola–Kirchhoff tensors  Exercice M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If α is a stationary and exact differential form, α = dU, then prove that  �  c  dU = U(c(T)) − U(c(t0)) noted  =  ∆U  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  only depends on the extremities c(t0) and c(T) of the curve c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  c dU =  � T  t=t0 dU(c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗c ′(t) dt =  � T  t=t0  d(U◦c)  dt  (t) dt = [U ◦ c]T  t0 = U(c(T)) − U(c(t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark (continuum mechanics): An observer chooses a Euclidean dot product (·, ·)g = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' •g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' • .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (if  (·, ·)g is imposed and implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And if he chooses to represent a linear form αt(pt) with its (·, ·)g-Riesz  representation vector ⃗ft(pt) (observer dependent), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8), then  �  c  α =  � T  t=t0  α(t, c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗c ′(t) dt =  � T  t=t0  ⃗f • d⃗c =  � T  t=t0  ⃗f • ⃗v dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  And its associated power density  Definition: The power density of a differential form α along �Φ is the Eulerian function  ψ := α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v :  �  �  �  �  �  C =  �  t∈[t0,T ]  ({t} × Ωt) → R  (t, p) → ψ(t, p) = α(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  And the power at t is  Pt(�Φ) = P(t, �Φ) :=  �  p∈Ωt  ψ(t, p) dΩt =  �  p∈Ωt  αt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vt(p) dΩt  noted  =  P(t,⃗vt) = Pt(⃗vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Remark: With a Euclidean dot product (·, ·)g, then with the (·, ·)g-Riesz representation vector ⃗f of α  (observer dependent) we get  ψ = ⃗f • ⃗v,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ψ(t, p) = ⃗f(t, p) •g ⃗v(t, p)  (= (⃗f(t, p),⃗v(t, p))g),  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  which gives P(t, �Φ) :=  �  p∈Ωt ⃗f(t, p) • ⃗v(t, p) dΩt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Piola–Kirchhoff tensors  Consider a regular Eulerian velocity field ⃗v, so d⃗v is an endomorphism (identified with a  �1  1  �  tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then we need another endomorphism τ (identified with a  �1  1  �  to get the objective double contraction  τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v := Tr(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v),  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  which means τ(t, p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v(t, p) := Tr(τ(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v(t, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: With a basis (⃗ei) at t and τ = �  ij τ i  j⃗ei ⊗ ej and d⃗v = �  jk vj  |k⃗ej ⊗ ek (the endomor-  phisms have been written like  �1  1  �  tensors for calculation purpose), τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗v = �  ijk τ i  jvj  |k⃗ei ⊗ ek and  τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v =  n  �  i,j=1  τ i  jvj  |i  (objective value),  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  see (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then choose a Euclidean dot product (·, ·)g, to be able to use the double matrix product  τ : d⃗v :=  n  �  i,j=1  τ i  jvi  |j = [τ]|⃗e  T : [d⃗v]|⃗e  (subjective value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Objective internal power for the stress: function of d⃗v  Usual hypothesis for the internal stress in a material: At first order, the power density is of the  type  ψ = τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ψ(t, p) = τ(t, p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗v(t, p), ∀(t, p) ∈ C,  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  thus the power at t is  Pt(⃗vt) =  �  p∈Ωt  ψ(t, p) dΩt =  �  p∈Ωt  τ t(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗vt(p) dΩt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  155  156  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Piola–Kirchhoff tensors  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  The first Piola–Kirchhoff tensor  The Piola–Kirchhoff approach consists in transforming Eulerian quantities into Lagrangian quantities to  refer to the initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12) gives  Pt(⃗vt) =  �  P ∈Ωt0  ψt(Φt0  t (P)) |Jt0  t (P)| dΩt0 =  �  P ∈Ωt0  τ t(Φt0  t (P)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗vt(Φt0  t (P)) Jt0  t (P) dΩt0  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  (the Jacobian Jt0  t (P) = det(F t0  t (P)) of Φt0  t at P is positive for a regular motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Lagrangian velocity  ⃗V t0(t, P) = ⃗vt(Φt0  t (P)) satisfies d⃗V t0  t (P) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P) where pt = Φt0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  τ t(pt) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗vt(pt) = τ(pt) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. (d⃗V t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P)−1) = (F t0  t (P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='τ(pt)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗V t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  Quantification: Choose a basis and a Euclidean dot product (·, ·)g, thus  Pt(⃗vt) =  �  P ∈Ωt0  (Jt0  t (P)τ(pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P)−T  �  ��  �  PKt0  t (P )  ) : d⃗V t0  t (P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The first Piola–Kirchhoff (two point) tensor at P ∈ Ωt0, relative to t0, t and a basis (⃗ei),  is the linear map PKt0  t (P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) defined by  PKt0  t (P) = Jt0  t (P) σt(Φt0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P)−T ,  where  σ = τ T ,  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  abusively written  PK = J σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  Hence  Pt(⃗vt) =  �  Ωt0  PKt0  t (P) : d⃗V t0  t (P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 The Piola–Kirchhoff tensor is not that easy to master: Everything is quite simple in  a Eulerian framework (the configuration at t where the laws are expressed to begin with), but then  everything is made more complicated when expressed in an initial configuration (at t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, when the  Piola–Kirchhoff tensor is used to introduce the Lie derivatives (Eulerian type), it makes the Lie derivative  quite a mysterious mathematical object, see footnote page 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Continuation of the remark: With the pull-backs, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) reads (Jt0  t (P) being positive)  P(t, �Φ) =  �  pt∈Ωt  ψt(pt) dΩt =  �  P ∈Ωt0  �  (Φt0  t )∗ψt  �  (P)  �  (Φt0  t )∗dΩt  �  ,  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  since ((Φt0  t )∗dΩt) = Jt0  t (P) dΩt0 and ((Φt0  t )∗ψt)(P) = ψt(pt) (scalar valued functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  It gives the  Piola–Kirchhoff tensor (pull-back to the initial configuration) since (Φt0  t )∗(αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vt)(pt) = (αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vt)(Φt0  t (P)) =  αt(Φt0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗vt(Φt0  t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  The second Piola–Kirchhoff tensor  The first Piola–Kirchhoff tensor PK may confuse Eulerian and Lagrangian variables, linear maps and  endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And PK(pt0) is not symmetric: It can’t be since PK(pt0) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) is not an  endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' To get a symmetric tensor, the second Piola–Kirchhoff tensor is defined:  Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The second Piola–Kirchhoff tensor is the endomorphism SKt0  t (P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) defined  by, in short,  SK = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PK = JF −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Full notation: SKt0  t (P) = (F t0  t (P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PKt0  t (P) = Jt0  t (P)(F t0  t (P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='σt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F t0  t (P))−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So if σt(p) ∈ L(⃗Rn  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) is symmetric then SKt0  t (P) ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  156  157  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Classical hyper-elasticity: ∂W/∂F  Thus, with the pull-back of the endomorphism d⃗vt ∈ L(⃗Rn  t , ⃗Rn  t ):  ((Φt0  t )∗d⃗vt)(P) = F t0  t (P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P),  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  and with d⃗vt(pt) = d⃗V t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t (P)−1 and σt(p) symmetric (so SKt0  t  is symmetric),  Pt(⃗vt) =  �  Ωt0  PKt0  t : d⃗V t0  t  dΩt0 =  �  Ωt0  (F t0  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='SKt0  t ) : d⃗V t0  t  dΩt0 =  �  Ωt0  ([F t0  t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [SKt0  t ]) : [d⃗V t0  t ]T dΩt0  =  �  Ωt0  SKt0  t : ((d⃗V t0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t ) dΩt0 =  �  Ωt0  SKt0  t : (F t0  t  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗V t0  t  + d(⃗V t0  t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F t0  t  2  ) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 It is a “chosen time derivative” of SK(t) = J(t)F(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='σ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F(t)−T that leads to some  kind of Lie derivative as explain in books in continuum mechanics, as in footnote page 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Classical hyper-elasticity: ∂W/∂F  E and F are finite dimensional spaces, dim E = n, dim F = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition  Reminder: Consider a function  �  W :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → R  L → �  W(L)  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  Its differential d�  W :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → L(L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  L → d�  W(L)  �  is defined at L by, in a direction M, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3),  d�  W(L)(M) = lim  h→0  �  W(L + hM) − �  W(L)  h  noted  =  ∂�  W  ∂L (L)(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Also written d�  W(L)(M) = d�  W(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M since d�  W(L) is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 �  W : F ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) → �  W(F) ∈ R (real valued function), with F := F t0  t (pt0) the  deformation gradient at ∈ Ωt0 at t at pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus d�  W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M = limh→0  �  W (F +hM)−�  W (F )  h  =noted ∂�  W  ∂F (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M ∈  R is the derivative of �  W at F in a direction M ∈ L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 m = n, endomorphisms L ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn), and �  W(L) := Tr(L) (the trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Here  dTr(L)(M) = limh→0  Tr(L+hM)−Tr(L)  h  = Tr(M) (the trace is linear), thus dTr(L) = Tr for all L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Expression with bases (quantification): The ∂W/∂Lij  Let (⃗ai) and (⃗bi) be bases in E and F, with (πai) the (covariant) dual basis of (⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (Lij) i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n =  (⃗bi ⊗ πaj) be the associated basis in L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the Lij are defined by Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ = δjℓ⃗bi for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', m  and j, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41 (all the elements of the matrix [Lij]|⃗a,⃗b vanish except the element at the  intersection of row i and column j which equals 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The derivation of �  W at L in a  direction Lij is  d�  W(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Lij = lim  h→0  �  W(L + hLij) − �  W(L)  h  noted  =  ∂�  W  ∂Lij  (L)  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  (usual notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Associated matrix relative to the chosen bases:  [d�  W(L)]|Lij := [ ∂�  W  ∂Lij  ] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',m  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  noted  =  [d�  W(L)]|⃗a,⃗b  ( noted  =  [d�  W(L)ij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  So if M = �m  i=1  �n  j=1MijLij then d�  W(L)(M) = �  ij Mij d�  W(L)(Lij) since d�  W(L) is linear, so  d�  W(L)(M) =  n  �  i,j=1  ∂�  W  ∂Lij  (L)Mij = [d�  W(L)]|⃗a,⃗b : [M]|⃗a,⃗b,  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  double matrix contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: d�  W(L)(M) = �  ij  ∂�  W  ∂Li  j (L)M i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  157  158  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Classical hyper-elasticity: ∂W/∂F  Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 The notation [M]| ⃗E,⃗e : [d�  W(L)]| ⃗E,⃗e is just a matrix product, since M = L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rm) and  d�  W(L) ∈ L(L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗  Rm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) are different kinds of mathematical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Continuing example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 with (⃗ei) = ( ⃗Ei): Then �  W(L) = Tr(L) gives d�  W(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M =  Tr(M) = �  i Mii, thus  ∂�  W  ∂Lij (L) = δij for all i, j, thus [d�  W(L)]|⃗e = [I] = [ ∂Tr  ∂Lij (L)] (identity matrix), and  we recover dTr(L)(M) = [ ∂Tr  ∂Lij (L)] : [M] = [I] : [M] = �n  i=1Mii = Tr(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Continuing example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7: The meaning of the derivation  ∂�  W  ∂Fij is intriguing: It is a  derivation in the direction Lij =noted ⃗ei ⊗ πEj, where (⃗ei) is a basis at p = Φt0  t (P) in ⃗Rn  t and (πEj) is the  dual basis of a basis ( ⃗Ej) at P in ⃗Rn  t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂�  W  ∂Fij (F) = d�  W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='Lij =noted d�  W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗ei ⊗ πEj) is a derivation  “at the same time” in the directions ⃗ei (at (t, p)) and πEj (at (t0, P)), where F stands for F t0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Motions and ω-lemma  Generalization of (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) to C1 functions, with UE open subset in a affine space which associated vector  space is E,  �  W :  �  UE × L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → R  (P, L) → �  W(P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  At P, let �  WP (L) := �  W(P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The differential d�  WP (L) =noted ∂2�  W(P, L) in a direction M ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is  ∂2�  W(P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M := d(�  WP )(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M = lim  h→0  �  W(P, L + hM) − �  W(P, L)  h  noted  =  ∂�  W  ∂L (P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  With a motion Φ := Φt0  t : Ωt0 → Ωt define  f :  �  C1(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ωt) → C0(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  Φ → f(Φ) := �  W(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' )),  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  a function of Φ which only depends on its first (covariant) gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, for all P ∈ Ωt0,  f(Φ)(P) = �  W(P, dΦ(P)) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  (This kind of relation is generally deduced after application of the frame invariance principle, and the  hypothesis of dependence on only the first order derivative dΦ = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Lemma M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 (ω-lemma) For all Φ, Ψ ∈ C1(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ωt),  df(Φ)(Ψ) = ∂2�  W(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ)(dΨ) noted  =  ∂�  W  ∂F (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ)(dΨ) ,  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all P ∈ Ωt0, df(Φ)(Ψ)(P) = ∂�  W  ∂F (P, dΦ(P))(dΨ(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With bases ( ⃗Ei) and (⃗ei) in ⃗Rn  t0 and ⃗Rn  t and dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Ej = �n  i=1  ∂Ψi  ∂Xj ⃗ei, we get  df(Φ)(Ψ) =  n  �  i,j=1  ∂�  W  ∂Fij  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ) ∂Ψi  ∂Xj  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') noted  =  [ ∂�  W  ∂Fij  ] : [ ∂Ψi  ∂Xj  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Marsden notations: df(Φ)(Ψ) = �n  i,J=1  ∂�  W  ∂F i  J  ∂Ψi  ∂XJ = [ ∂�  W  ∂F i  J ] : [ ∂Ψi  ∂XJ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' C1(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ωt) is a vector space, so df(Φ)(Ψ) = limh→0  f(Φ+hΨ)−f(Φ)  h  ∈ C0(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Ωt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for any  P ∈ Ωt0 we have df(Φ)(Ψ)(P) = limh→0  f(Φ+hΨ)(P )−f(Φ)(P )  h  = limh→0  �  W (P,dΦ(P )+h dΨ(P ))−�  W (P,dΦ(P ))  h  =  d ⃗WP (dΨ(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  158  159  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Classical hyper-elasticity: ∂W/∂F  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Application to classical hyper-elasticity: PK = ∂W/∂F  Let (·, ·)g be a unique Euclidean dot product in ⃗Rn  t at all times t, and let ( ⃗Ei) and (⃗ei) be Euclidean  bases at t0 and at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let σt(p) = the Cauchy stress tensor at t at p = Φt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let PK(P) = J(P) σt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F −T (P) = the first Piola–Kirchhoff (two point) tensor at P, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Since PK depends on Φ, the full notation is PK = PK(Φ) given by  PK(Φ)(P) = J(P) σt(Φ(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦ(P)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 If there exists a function �  PK such that PK reads  PK(Φ)(P) = �  PK(P, dΦ(P))  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  then �  PK is called a constitutive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (First order hypothesis: �  PK only depends on dΦ = F the first  order derivative of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 The material is hyper-elastic iff there exists a function �  W :  �  Ωt0 × L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t ) → R  (P, L) → �  W(P, L)  �  such that  (PK(Φ) =)  �  PK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ) = ∂�  W  ∂F (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ),  written  �  PK = ∂�  W  ∂F ,  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  that is, �  PK(P, F(P)) = ∂�  W  ∂F (P, F(P)) for all P ∈ Ωt0, where F = dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With bases ( ⃗EI) and (⃗ei) in ⃗Rn  t0 and ⃗Rn  t , and (EI) the dual basis of ( ⃗EI), and PK = �n  i,J=1PKi  J⃗ei⊗EJ,  [PK(Φ)]| ⃗E,⃗e = [�  PK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F)]| ⃗E,���e = [∂�  W  ∂F (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F)]| ⃗E,⃗e,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [PKi  J] = [ ∂�  W  ∂F i  J  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', F)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Thus, for any (virtual) motion Ψ : Ωt0 → Ωt, with (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32) and (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27),  �  PK(dΦ)(dΨ) = ∂�  W  ∂F (dΦ)(dΨ)  =  �  iJ  ∂�  W  ∂F i  J  (F) ∂Ψi  ∂XJ  noted  =  [�  PK] : [dΨ],  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  that is, �  PK(dΦ)(dΨ)(P) = �  iJ  ∂�  W  ∂F i  J (P, F t0  t (P)) ∂Ψi  ∂XJ (P) for all P ∈ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 With a unique Euclidean dot product (·, ·)g both in ⃗Rn  t0 and ⃗Rn  t , with Euclidean bases  ( ⃗Ei) ∈ ⃗Rn  t0 and (⃗ei) ∈ ⃗Rn  t , and with (Ei) the dual basis of ( ⃗Ei), with C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F, prove (derivation in the  direction ⃗ei ⊗ EJ):  ∂C  ∂F i  J  (F) =  �  K  F i  K ⃗EJ ⊗ EK +  �  K  F i  K ⃗EK ⊗ EJ  (= dC(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗ei ⊗ EJ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  ∂  √  C  ∂F (F) = 1  2  ��  C(F)  �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂C  ∂F (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  ∂  √  C  ∂C  = 1  2(  √  C)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let F = �  iJ F i  J⃗ei ⊗ EJ, so F T = �  Ij(F T )I  j ⃗EI ⊗ ej = �  Ij F j  I ⃗EI ⊗ ej, and C = �  IJ CI  J ⃗Ei ⊗ EJ =  F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F = �  IJ  �  k(F T )I  kF k  J ⃗EI ⊗ EJ = �  IJ  �  k F k  I F k  J ⃗EI ⊗ EJ = C(F), so CI  J = �  k F k  I F k  J = CI  J(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  C(F + h⃗ei ⊗ EJ) = (F + h⃗ei ⊗ EJ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F + h⃗ei ⊗ EJ) = (F T + h ⃗EJ ⊗ ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F + h⃗ei ⊗ EJ)  = C(F) + h ( ⃗EJ ⊗ ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='F + h F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗ei ⊗ EJ) + h2 ⃗EJ ⊗ Ei  = C(F) + h (  �  K  F i  K ⃗EJ ⊗ EK +  ���  K  (F T )K  i ⃗EK ⊗ EJ) + h2 ⃗EJ ⊗ EJ  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  Thus (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And C(F + h⃗ei ⊗ EJ) − C(F) = (  √  C(F + h⃗ei ⊗ EJ) +  √  C(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (  √  C(F + h⃗ei ⊗ EJ) −  √  C(F)) gives  dC(F)(⃗ei ⊗ EJ) = 2  √  C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d  √  C(F)(⃗ei ⊗ EJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ∂  √  C  ∂F i  J (F) = 1  2(  �  C(F))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂C  ∂F i  J (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (C + h⃗ei ⊗ ej) − C = (  √  C + h⃗ei ⊗ ej +  √  C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (  √  C + h⃗ei ⊗ ej −  √  C), divided by h, gives ⃗ei ⊗ ej  =  2  √  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' limh→0  √  C+h⃗ei⊗ej−  √  C  h  = 2  √  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d  √  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(⃗ei ⊗ ej), thus L = 2  √  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(d  √  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L) for all L (linearity of d  √  C), thus  d  √  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L = 1  2(  √  C)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  159  160  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Classical hyper-elasticity: ∂W/∂F  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Corollary (hyper-elasticity): SK = ∂W/∂C  With the symmetry of the second Piola–Kirchhoff tensor SK = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='PK, we deduce SKt0  t (Φt0  t )(P) =  �  SKt0  t (P, F t0  t (P))  (constitutive  function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And  we  deduce  the  existence  of  a  function  �  W  :  �  Ωt0 × L(⃗Rn  t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn  t0) → R  (P, L) → �  W(P, L)  �  such that,  �  SKt0  t (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', C) = ∂�  W  ∂C (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  (See Marsden and Hughes [12] for details and the thermodynamical hypotheses required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  N  Conservation of mass  Let ρ(t, p) = ρt(p) be the (Eulerian) mass density at t at p ∈ Ωt, supposed to be > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The mass m(ωt) of  a subset ωt ⊂ Ωt is  m(ωt) =  �  p∈ωt  ρt(p) dωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  Conservation of mass principle (no loss nor production of particles): For all ωt0 ⊂ Ωt0 and all t,  m(ωt) = m(ωt0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  p∈ωt  ρt(p) dωt =  �  P ∈ωt0  ρt0(P) dωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Proposition N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) then, with Jt0  t (P) = det(dΦt0  t (P)) (positive Jacobian the motion being sup-  posed regular) and p = Φt0  t (P),  ρt(p) = ρt0(P)  Jt0  t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The change of variable formula gives  �  p∈ωt  ρt(p) dωt =  �  P ∈ωt0  ρt(Φt0  t (P)) Jt0  t (P) dωt0,  thus (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives ρt(Φt0  t (P))Jt0  t (P) = ρt0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 ⃗v = ⃗v(t, pt) being the Eulerian velocity at (t, pt) ∈ R × Ωt, (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives  Dρ  Dt + ρ div⃗v = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂ρ  ∂t + div(ρ⃗v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  Thus, for all ωt ⊂ Ωt,  �  ωt  ∂ρ  ∂t dωt = −  �  ∂ωt  ρ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n dσt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2) gives d  dt(  �  p(t)∈ωt ρ(t, p(t)) dωt) = 0, and Leibniz formula (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41) applied for all ωt gives (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then the Green formula  �  Ωt div(ρ⃗v) dΩt =  �  ∂Ωt ρ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n dσt gives (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Use (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) to prove (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' J(t, P)ρ(t, Φ(t, P)) = ρt0(P) give, with pt = Φ(t, P),  ∂J  ∂t (t, P) ρ(t, pt) + J(t, P)  �∂ρ  ∂t (t, pt) + dρ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΦ(t, P)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus ∂J  ∂t (t, P) = J(t, P) div⃗v(t, p), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40), gives (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  160  161  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Framework  O  Balance of momentum  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Framework  �Φ : [t0, T] × Obj → Rn is a regular motion, Ωt = �Φ(t, Obj), Γt = ∂Ωt (the boundary), ⃗v is the Eulerian  velocity field, ωt is a regular sub domain in Ωt and ∂ωt is its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  An observer chooses a Euclidean basis (⃗ei) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' made with the foot or the metre) and call (·, ·)g the  associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And ⃗n(t, p) = ⃗nt(p) is the outer unit normal at t at p ∈ ∂ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  All the functions are assumed to be regular enough to validate the following calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let ρ :  �  �  �  �  �  �  t∈[t0,T ]  ({t} × Ωt) → R  (t, pt) → ρ(t, pt)  �  �  �  �  �  (a mass density), let ⃗f :  �  �  �  �  �  �  t∈[t0,T ]  ({t} × Ωt) → ⃗Rn  (t, pt) → ⃗f(t, pt)  �  �  �  �  �  (a  body force density), and let ⃗T :  �  �  �  �  �  �  t∈[t0,T ]  ({t} × ∂ωt × ⃗  Rn  t ) → ⃗Rn  (t, pt,⃗n(pt)) → ⃗T(t, pt,⃗n(pt))  �  �  �  �  �  (a surface force density)  defined for any regular subset ωt ⊂ Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Master balance law  Definition O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The balance of momentum is satisfied by ρ, ⃗f and ⃗T iff, for all regular open subset ωt  in Ωt,  d  dt(  �  ωt  ρ⃗v dΩt) =  �  ωt  ⃗f dΩt +  �  ∂ωt  ⃗T∂ωt dΓt  (master balance law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  (It is in fact a linearity hypothesis, see theorem O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Thus, with (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41),  �  ωt  D(ρ⃗v)  Dt  + ρ⃗v div⃗v dΩt =  �  ωt  ⃗f dΩt +  �  ∂ωt  ⃗T∂ωt dΓt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  And with the conservation of mass hypothesis, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4), we get  �  ωt  ρD⃗v  Dt dΩt =  �  ωt  ⃗f dΩt +  �  ∂ωt  ⃗T∂ωt dΓt,  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  with D⃗v  Dt = ⃗γ = the Eulerian acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Cauchy theorem ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n (stress tensor σ)  Theorem O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (Cauchy first law: Cauchy stress tensor) If the master balance law (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) is satis-  fied, then ⃗T is linear in ⃗n, that is, there exists a Eulerian endomorphism σ, identified to a Eulerian tensor  σ ∈ T 1  1 (Ωt), called the Cauchy stress tensor, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' on all ∂ωt, in short  ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n,  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  where ⃗n is the unit outward normal to ∂ωt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗T(t, pt) = σ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n(t, pt) for all t and pt ∈ ∂ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The proof is based on:  Lemma O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Let ϕ :  �  Ω → R  p → ϕ(p)  �  ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and ψ :  �  Ω × ⃗R3 → R  (p, ⃗w) → ψ(p, ⃗w)  �  ∈ C1(Ω, ⃗R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If  ∀ω open in Ω,  �  p∈ω  ϕ(p) dΩ =  �  p∈∂ω  ψ(p,⃗n(p)) dΓ  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  (no dependence on the curvature or on higher derivatives since at any p ∈ ∂ω, ψ only depends on ⃗n(p)),  then  ∃⃗k ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗R3) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ψ = (⃗k,⃗n)g, and ϕ = div⃗k,  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ψ depends linearly on ⃗n, and ϕ is a divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  161  162  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cauchy theorem ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n (stress tensor σ)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Lemma O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') (This proof is standard: We recall it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Let p ∈ Ω ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the tetrahedral  defined by its vertices p, p + (h1, 0, 0), p + (0, h2, 0) and p + (0, 0, h3), with hi > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (On each  face of a tetrahedron, the unit normal vector is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Let Σ1 the side which outer unit normal is  − ⃗E1: It area is σ1 = 1  2h2h3 (square triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Idem for Σ2 and Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let Σ be the fourth side: its area  is σ = 1  2  �  h2  2h2  3 + h2  3h2  1 + h2  1h2  2 and its outer unit normal is ⃗n =  1  2σ(h2h3, h3h1, h1h2) (see exercise O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5),  that is ⃗n = (n1, n2, n3) with ni = σi  σ pour i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The volume of the tetrahedral is 1  6h1h2h3 =noted ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let M := supp∈Ω |ϕ(p)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have M < ∞, since ϕ is continuous in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6) give  Mℓ3 ≥ |  �  ∂ωt  ψ(p,⃗n(p)) dΓ|,  so  �  ∂ωt  ψ(p,⃗n(p)) dΓ = O(ℓ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  And ψ being continuous, the mean value theorem applied on Σi gives: There exists pi ∈ Σi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  Σi  ψ(p,⃗n(p)) dΓ = σiψ(pi,⃗ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  �  ∂ωt  ψ(p,⃗n(p)) dΓ =  �  σ1ψ(p1, − ⃗E1) + σ2ψ(t, p2, − ⃗E2) + σ3ψ(p3, − ⃗E3) + σψ(p4,⃗n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then, Ψ being continous, (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7) gives  σ1ψ(p1, − ⃗E1) + σ2ψ(p2, − ⃗E2) + σ3ψ(p3, − ⃗E3) + σψ(p4,⃗n) = O(ℓ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  We flatten the tetrahedron on the yz face by taking h2 = h3 =noted h and h1 = h2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus σ1 = 1  2h2,  σ2 = o(h2), σ3 = o(h2), σ ∼ σ1, ℓ3 = 1  6h4, with ⃗n ∼ −⃗n1 = ⃗E1 and pi ∼ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  ψ(p, − ⃗E1) + ψ(p, + ⃗E1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  Idem with xz and xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And for a fixed tetrahedron with h1, h2, h3 given, consider the smaller tetrahedron  with εh1, εh2, εh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then as ε → 0 (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8) with (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) give  ψ(p,⃗n) = − σi  σ ψ(p, − ⃗E1) − σ2  σ ψ(p, − ⃗E2) − σ3  σ ψ(p, − ⃗E3) =  3  �  i=1  niψ(p, ⃗Ei),  since ni = σi  σ pour i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The same steps can be done for any (inclined) tetrahedron (or apply a  change of variable to get back to the above tetrahedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ψp is a linear map in ⃗np, that is, there  exists a linear form αp s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ψp(⃗np) = αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗np for any p ∈ ∂ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the Riesz representation theorem gives:  ∃⃗kp s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗np = (⃗kp,⃗np)g =noted ⃗kp • ⃗np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Apply Lemma O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 component by component with ⃗ϕ = ρ D⃗v  Dt − ⃗f = �n  i=1ϕi⃗ei,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Corollary O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 With divσ := �n  i=1(�n  j=1  ∂σij  ∂xj )⃗ei (definition of “the matrix divergence” see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65)),  �  �  �  ⃗f + divσ = ρD⃗v  Dt  in Ωt,  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n = ⃗T  on Γt  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (With duality notations, divσ := �n  i=1(�n  j=1  ∂σi  j  ∂xj )⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Apply the divergence Formula to (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Consider a triangle T in R3 which vertices are A = (h1, 0, 0), B = (0, h2, 0), C = (0, 0, h3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Prove that ⃗n = (h2h3, h3h1, h1h2) is orthogonal to T and that σ = 1  2  �  h2  2h2  3 + h2  3h2  1 + h2  1h2  2 is its area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the parametric surface ⃗r(t, u) = A + t ⃗  AB + u ⃗  AC for t, u ∈ [0, 1] describing the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  ⃗n =  ∂⃗r  ∂t ∧ ∂⃗r  ∂u =  ⃗  AB ∧ ⃗  AC =  �  �  −h1  h2  0  �  � ∧  �  �  −h1  0  h3  �  � =  �  �  h2h3  h3h1  h1h2  �  � is orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And dσ = || ∂⃗r  ∂t ∧ ∂⃗r  ∂u||dudt =  �  h2  2h2  3 + h2  3h2  1 + h2  1h2  2dudt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus σ =  � 1  t=0  � 1  u=0 dσ =  �  h2  2h2  3 + h2  3h2  1 + h2  1h2  2 is twice the aera of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  162  163  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Tensorial product and multilinear forms  P  Balance of moment of momentum  Definition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 The balance of moment of momentum is satisfied by ρ, ⃗f and ⃗T iff for all regular sub-open  set ωt ⊂ Ωt  d  dt  �  ωt  ρ −−→  OM ∧ ⃗v dΩt =  �  ωt  ρ −−→  OM ∧ ⃗f dΩt +  �  ∂ωt  −−→  OM ∧ ⃗T dΓt,  (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  equality called the master balance of moment of momentum law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (This excludes e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Cosserat continua  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Theorem P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (Cauchy second law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') If the master balance law (so ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n) and the master balance of  moment of momentum law are satisfied then σ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Standard proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Let ⃗x = −−→  OM = �  i xi ⃗Ei, and ⃗T = �  i Ti ⃗Ei = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗n = �  ij σijnj ⃗Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  (first component) (⃗x ∧ ⃗T)1 = x2T3 − x3T2 = x2(σ31n1 + σ32n2 + σ33n3) − x3(σ21n1 + σ22n2 + σ23n3) =  (x2σ31 − x3σ21)n1 + (x2σ32 − x3σ22)n2 + (x2σ33 − x3σ23)n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  �  ∂ωt(⃗x ∧ ⃗T)1 dΓt =  �  ωt  ∂(x2σ31−x3σ21)  ∂x1  +  ∂(x2σ32−x3σ22)  ∂x2  + ∂(x2σ33−x3σ23)  ∂x3  dΩt =  �  ωt x2(divσ)3 + x3(divσ)2 + σ32 − σ23 dωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) gives ρ D⃗v  Dt − ⃗f = divσ, thus ⃗x ∧ (ρ⃗γ − ⃗f) = ⃗x ∧ divσ, so the first component of ⃗x ∧ (ρ⃗γ − ⃗f) is  x2(divσ)3−x3(divσ)2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) gives  �  ωt σ32−σ23 dωt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' True for all ωt, thus σ32−σ23 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Idem for the other components: σ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q  Uniform tensors in Lr  s(E)  Uniform tensors enable to define without ambiguity the “objective contraction rules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Uniform tensors  are scalar valued multilinear functions acting on both vectors and linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  NB: In classical mechanics courses, what is called a “tensor” generally not a tensor but a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' you may encounter the expression “Euclidean tensor” which means: The matrix representation of  “something” with respect to a Euclidean basis (based on the foot, metre,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') chosen by some observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (An “Euclidean tensor” is a non-sense, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' can you define a “Euclidean vector”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Tensorial product and multilinear forms  Let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', An be n finite dimension vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And A∗  i = L(Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) the set of linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Tensorial product of functions  Let f1 : A1 → R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', fn : An → R be n functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Their tensorial product is the function f1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⊗ fn :  A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × An → R defined by (separate variable function)  (f1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⊗ fn)(⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗xn) = f1(⃗x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='fn(⃗xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n = 2 and A1 = A2 = R and (cos ⊗ sin)(x, y) = cos(x) sin(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Tensorial product of linear forms: multilinear forms  Let L(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', An;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) be the set of R-multilinear forms on the Cartesian product A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × An, that is, the  set of the functions M : A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × An → R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n, all ⃗xi, ⃗yi ∈ Ai and all λ ∈ R,  M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗xi + λ⃗yi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') + λ M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗yi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='),  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  the other variables being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition: An elementary tensor is multilinear form M = ℓ1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. ⊗ ℓn, with ℓi ∈ A∗  i for all i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  ∀(⃗xi)i∈N∗ ∈  n  �  i=1  Ai,  (ℓ1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⊗ ℓn)(⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗xn) = (ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='(ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗xn) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  (The dot in ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗xi is not an inner dot product: It is the duality “outer product” ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗xi := ℓi(⃗xi), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  163  164  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Uniform tensors in L0  s(E)  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Uniform tensors in L0  s(E)  Let E be a real vector space, with dim(E) = n ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In this section we consider the first overlay on E  made of multilinear forms M on E, called the uniform tensors of type 0 s or of type  �0  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', M ∈ L0  1(E) a linear form, M ∈ L0  2(E) an inner dot product, M ∈ L0  n(E) a determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notations for quantification purposes: (⃗ei) is a basis in E, (πei) is its (covariant) dual basis (basis in  E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)), (∂i) is its bidual basis (basis in E∗∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of type  �0  s  �  uniform tensors  L0  0(E) := R, and if s ∈ N∗ then  L0  s(E) := L(E × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × E  �  ��  �  s times  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  is called the set of uniform tensors of type  �0  s  �  on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Example: Type  �0  1  �  uniform tensor = linear forms  A type  �0  1  �  uniform tensor is an element of L0  1(E) = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = E∗: It is a linear form ℓ ∈ L0  1(E) = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: With ℓi := ℓ(⃗ei) we have, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10),  ℓ =  n  �  i=1  ℓiπei,  and  [ℓ]|πe = ( ℓ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓn ) noted  =  [ℓ]|⃗e  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  (row matrix for a linear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: (ei) is the covariant dual basis and ℓ = �n  i=1ℓiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, if ⃗v ∈ E, ⃗v = �n  i=1vi⃗ei, then ⃗v is represented by [⃗v]|⃗e =  �  �  v1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  vn  �  � (column matrix for a vector),  and the matrix calculation rules give  ℓ(⃗v) = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗v]|⃗e = ( ℓ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ℓn ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  v1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  vn  �  � =  n  �  i=1  ℓivi  noted  =  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Duality notations: ⃗v = �n  i=1vi⃗ei and ℓ(⃗v) = �n  i=1ℓivi, and Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Example: Type  �0  2  �  uniform tensor  A type  �0  2  �  uniform tensor is an element of L0  2(E) = L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R): It is a bilinear form T ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: Let Tij := T(⃗ei,⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, with ⃗v = �n  i=1vi⃗ei and ⃗w = �n  i=1wi⃗ei,  T(⃗v, ⃗w) =  n  �  i,j=1  Tijviwj = [⃗v]T  |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗e,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  T =  n  �  i,j=1  Tijπei ⊗ πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Duality notations: T(⃗v, ⃗w) = �n  i,j=1Tijviwj, and Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  An elementary uniform tensor in L0  2(E) is a tensor T = ℓ ⊗ m, where ℓ, m ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And so, for all  ⃗v, ⃗w ∈ E,  (ℓ ⊗ m)(⃗v, ⃗w) = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v)(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Example: Determinant  The determinant is a alternating  �0  n  �  uniform tensor, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Uniform tensors in Lr  s(E)  In this section we consider an over-overlay on E: The multilinear forms acting on both vectors (∈ E) and  functions ∈ E∗ (linear forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  164  165  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Uniform tensors in Lr  s(E)  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definition of type  �r  s  �  uniform tensors  Let r, s ∈ N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' r + s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The set of multilinear forms  Lr  s(E) := L(E∗ × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × E∗  �  ��  �  r times  , E × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' × E  �  ��  �  s times  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  is called the set of uniform tensors of type  �r  s  �  on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The case r = 0 has been considered at § Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  When r ≥ 1, a tensor T ∈ Lr  s(E) is a functional: Its domain of definition contains a set of functions  (the set E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Example: Type  �1  0  �  uniform tensor: Identified with a vector  A uniform  �1  0  �  tensor is a element T ∈ L1  0(E) = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = L(L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = E∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With the natural  canonical isomorphism  J :  �  E → E∗∗ = L1  0(E)  ⃗w → J (⃗w) = w,  defined by  w(ℓ) := ℓ(⃗w),  ∀ℓ ∈ E∗,  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) and prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5,  w noted  =  ⃗w,  so  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ noted  =  ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ  (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  So a  �1  0  �  type uniform tensor w is identified (natural canonical) to the vector ⃗w = J −1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Interpretation:  E∗∗ is the set of directional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Indeed, if E is an affine space, if E is the  associated vector space, if p ∈ E, and if f is a differentiable function at p, then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='df(p) =(Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10) df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w is  the directional derivative along ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark: In differential geometry, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='df is written ⃗w(f), so ⃗w(f)(p) := df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w, the definition of a  vector being a directional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: For all i, j,  ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πej = δij = πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei,  thus  ∂i = J (⃗ei) noted  =  ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Duality notations: ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ej = δj  i = ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if f is a C1 function then df(p) = �n  i=1f|i(p) πei (=  �n  i=1f|i(p) ei) and  ∂i(df(p)) = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei = f|i(p) noted  =  ∂i(f)(p) noted  =  ⃗ei(f)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Example: Type  �1  1  �  uniform tensor  An elementary uniform tensor in L1  1(E) is a tensor T = u ⊗ β, where u ∈ E∗∗ and β ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And, with  ⃗u = J−1(u) ∈ E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10), we also write T = ⃗u ⊗ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, for all ℓ ∈ E∗ and ⃗w ∈ E  (u ⊗ β)(ℓ, ⃗w) = u(ℓ)β(⃗w) = ℓ(⃗u)β(⃗w) noted  =  ⃗u(ℓ)β(⃗w) noted  =  (⃗u ⊗ β)(ℓ, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  Quantification: Let T(πei,⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  T =  n  �  i,j=1  Tij ⃗ei ⊗ πej,  and  [T]|⃗e = [Tij],  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  [T]|⃗e = [Tij] being the matrix of T relative to the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: T(ei,⃗ej) = T ij, [T]|⃗e =  [T ij], T = �n  i,j=1T ij⃗ei ⊗ ej, and Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus with ℓ ∈ E∗, ℓ = �n  i=1ℓiei ∈ E∗, and ⃗w ∈ E, ⃗w = �n  i=1wi⃗ei ∈ E, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15) gives  T(ℓ, ⃗w) =  n  �  i,j=1  Tij⃗ei(ℓ)πej(⃗w) =  n  �  i,j=1  Tijℓiwj = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[T]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗e  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  ([ℓ]|⃗e is a row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: T(ℓ, ⃗w) = �n  i,j=1T ijℓiwj and Einstein convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  165  166  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exterior tensorial products  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Example: Type  �1  2  �  uniform tensor  The same steps are applied to any tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if T ∈ L1  2(E), then with duality notations, T ijk =  T(ei,⃗ej,⃗ek) and  T =  n  �  i,j,k=1  T i  jk⃗ei ⊗ ej ⊗ ek,  and  T(ℓ, ⃗u, ⃗w) =  n  �  i,j,k=1  T i  jkℓiujwk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Exterior tensorial products  Let T1 ∈ Lr1  s1(E) and T2 ∈ Lr2  s2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Their tensorial product is the tensor T1 ⊗ T2 ∈ Lr1+r2  s1+s2(E) defined by  (T1 ⊗ T2)(ℓ1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ℓ2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗u1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗u2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') := T1(ℓ1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗u1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')T2(ℓ2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗u2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  Particular case: with λ ∈ L0  0(E) = R and T ∈ Lr  s(E),  λ ⊗ T = T ⊗ λ := λT ∈ Lr  s(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 let T1, T2 ∈ L1  1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification:  Let T1 = �n  i,j=1(T1)i  j⃗ei ⊗ ej and let T2 =  �n  k,m=1(T2)k  m⃗ek ⊗ em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then T1 ⊗ T2 = �n  i,j,k,m=1(T1)i  k(T2)j  m⃗ei ⊗ ⃗ej ⊗ ek ⊗ em ∈ L2  2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Alternative  definition:  T1 �⊗T2  :=  �n  i,j,k,m=1(T1)i  j(T2)k  m⃗ei ⊗ ej ⊗ ⃗ek ⊗ em  ∈  L(E∗, E, E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And we get back to the previous definition thanks to the natural canonical  isomorphism �J : L(E∗, E, E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → L(E∗, E∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = L2  2(E) defined by �J( �T) = T where  T(ℓ, m,⃗v, ⃗w) = �T(ℓ,⃗v, m, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Contractions  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Contraction of a linear form with a vector  Let ℓ ∈ L0  1(E) = E∗ and ⃗w ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Their contraction is the value  ℓ(⃗w) linearity  =  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w noted  =  ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  And with a basis (⃗ei) and its dual basis (πei), ℓ = �n  i=1ℓiπei and ⃗w = �n  i=1wi⃗ei give  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w =  n  �  i=1  ℓiwi = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗e =  n  �  i=1  wiℓi = ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ = Tr(⃗w ⊗ ℓ),  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  where Tr is the objective trace operator Tr : L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) ≃ L1  1(E) → R (defined by Tr(⃗ei ⊗ πej) = δi  j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notations: ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = �n  i=1ℓiwi, and Einstein convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Use the change of coordinate formulas to prove that the computation ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w in (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21) gives  a result independent of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let P be the change of basis matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So [⃗w]new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]old and [ℓ]new = [ℓ]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28), thus  [ℓ]new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]new = ([ℓ]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]old) = [ℓ]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]old = [ℓ]old[⃗w]old (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Contraction of a  �1  1  �  tensor and a vector  Let ℓ ∈ E∗ and ⃗u ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The contraction of the elementary tensor ⃗w ⊗ ℓ ∈ L1  1(E) with ⃗u is defined by:  (⃗w ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u  ����  contraction  = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  Thus, if (⃗ei) is a basis in E and (πei) is the dual basis, and T = �n  i,j=1Tij⃗ei ⊗ πej ∈ L1  1(E) and  ⃗u = �n  j=1uj⃗ej ∈ E, then  T =  n  �  i,j=1  Tij⃗ei ⊗ ej  =⇒  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u =  n  �  i,j=1  Tijuj  j⃗ei  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  because πej(⃗u) = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = �n  i,j=1T i  juj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  166  167  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Contractions  Then, with the natural canonical isomorphism (L1  1(E) =) L(E, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E), see (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7), any  endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) defined by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1Lij⃗ei can be written, for calculation purpose,  �L =  n  �  i,j=1  Lij⃗ei ⊗ πej  noted  =  L,  which means  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  =  n  �  i=1  Lijuj⃗ei  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  when ⃗u = �  i uj⃗ej, since πej(⃗u) = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: L = �n  i,j=1Lij⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Contractions of uniform tensors  More generally, the contraction of two tensors, if meaningful, is defined thanks to (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20): Let T1 ∈ Lr1  s1(E),  T2 ∈ Lr2  s2(E), ℓ ∈ E∗ and ⃗u ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 The objective contraction of T1 ⊗ ℓ ∈ Lr2  s2+1(E) and ⃗u ⊗ T2 ∈ Lr2+1  s2  (E) is the tensor  (T1 ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u ⊗ T2) ∈ Lr1+r2  s1+s2 given by  (T1 ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u  ����  contraction  ⊗T2) := (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) T1 ⊗ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  In particular (T1 ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) T1 (as in (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)), and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u ⊗ T2) = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And the objective contraction of T1 ⊗ ⃗u ∈ Lr2+1  s2  (E) and ℓ ⊗ T2 ∈ Lr2  s2+1(E) is the tensor (T1 ⊗ ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (ℓ ⊗  T2) ∈ Lr1+r2  s1+s2 given by  (T1 ⊗ ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (ℓ ⊗ T2) = (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ) T1 ⊗ T2  (= (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) T1 ⊗ T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Quantification with a basis (⃗ei), examples to avoid cumbersome notations:  Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Let T ∈ L1  1(E) = L1  0+1(E), T = �n  i,j=1T i  j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  With ⃗w ∈ E ∼ E∗∗ = L1  0(E),  ⃗w = �n  j=1wj⃗ej, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25) gives T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w ∈ L1  0(E) ∼ E and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w =  n  �  i,j=1  T i  jwj⃗ei,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w]|⃗e = [T]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗e  (column matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Indeed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = �n  i,j,k=1T i  jwk(⃗ei ⊗ ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �n  i,j,k=1T i  jwk⃗ei(ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek) =  �n  i,j,k=1T i  jwk⃗ei(δj  k) = �n  i,j=1T i  jwj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With ℓ ∈ E∗ = L0  1(E), ℓ = �n  i=1ℓiei, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25) gives ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T ∈ L0  1(E) =  E∗ and  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T =  n  �  i,j=1  ℓiT i  jej,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T]|⃗e = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [T]|⃗e  (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Indeed ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T = (�n  i=1ℓiei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (�n  j,k=1T k  j ⃗ek⊗ej) = �n  i,j,k=1ℓiT k  j (ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek)ej =  �n  i,j=1ℓiT i  jej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 Let S, T ∈ L1  1(E), S = �n  i,k=1Si  k⃗ei ⊗ ek and T = �n  j,k=1T k  j ⃗ek ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T =  n  �  i,j,k=1  Si  kT k  j ⃗ei ⊗ ej,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T]|⃗e = [S]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [T]|⃗e  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Indeed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T  =  (�n  i,k=1Si  k⃗ei ⊗ ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (�n  j,m=1 T m  j ⃗em ⊗ ej)  =  �n  i,j,k,m=1Si  kT m  j ⃗ei(ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗em) ⊗ ej = �n  i,j,k=1Si  kT k  j ⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 Let T ∈ L1  2(E), T = �n  i,j,k=1T i  jk⃗ei ⊗ ej ⊗ ek, and ⃗u, ⃗w ∈ E ∼ L1  0(E), ⃗w = �n  i=1wi⃗ei and  ⃗u = �n  i=1ui⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w =  n  �  i,j,k=1  T i  jkwk⃗ei ⊗ ej ∈ L1  1(E),  and  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u =  n  �  i,j,k=1  T i  jkwkuj⃗ei  noted  =  T(⃗u, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') So [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w]|⃗e = [�n  k=1T i  jkwk] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=',n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And with ℓ ∈ E∗, ℓ = �n  i=1ℓiei,  ((T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ =  n  �  i,j,k=1  T i  jkwkujℓi = T(ℓ, ⃗u, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T(⃗u, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  167  168  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Contractions  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Objective double contractions of uniform tensors  Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 Let S, T ∈ L1  1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And let (⃗ei) be a basis in E, (ei) its dual basis, S = �n  i,j=1Si  j⃗ei ⊗ ej  and T = �n  i,j=1T i  j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The double objective contraction S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T of S and T is defined by  S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T =  n  �  i,j=1  Si  jT j  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  (Einstein convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T defined in (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32) is an invariant: It is the trace Tr(LS ◦ LT ) of the endo-  morphisms LS, LT ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) naturally canonically associated to S and T (given by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u := S(ℓ, ⃗u)  and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u := T(ℓ, ⃗u) for all (⃗u, ℓ) ∈ E × E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So the real value �n  i,j=1Si  jT j  i has the same real value  regardless of the chosen basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Which is not the case of the term to term matrix multiplication  S : T = �n  i,j=1Si  jT i  j, see next § Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 and example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ai) and (⃗bi) be two bases and P = [P i  j] be the transition matrix from (⃗ai) to (⃗bi),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ⃗bj  = �n  i=1P i  j⃗ai for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let Q = [Qi  j] := P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then bi  = �n  i=1Qi  jai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let S  =  �  ij(Sa)i  j⃗ai ⊗ aj = �  ij(Sb)i  j⃗bi ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So [(Sb)i  j] = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [(Sa)i  j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P (change of basis formula for  �1  1  �  tensors identified with endomorphisms), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Sb)i  j = �  km Qi  k(Sa)k  mP m  j  for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Idem with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus �  i,j(Sb)i  j(Tb)j  i = �  i,j,k,m,α,β Qi  k(Sa)k  mP m  j Qj  α(Ta)α  βP β  i  = �  i,j,k,m,α,β(Sa)k  m(Ta)α  βP β  i Qi  kP m  j Qj  α =  �  k,m,α,β(Sa)k  m(Ta)α  βδβ  k δm  α = �  k,m(Sa)k  m(Ta)m  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 More generally, the objective double contractions S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T of uniform tensors, is obtained  by applying the objective simple contraction twice consecutively, when applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', T1 ⊗ ℓ1,1 ⊗ ℓ1,2 and ⃗u2,1 ⊗ ⃗u2,2 ⊗ T2 give  (T1 ⊗ ℓ1,1 ⊗ ℓ1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗u2,1  �  ��  �  first  ⊗⃗u2,2 ⊗ T2) = (ℓ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2,1)(T1 ⊗ ℓ1,1) ⊗ (⃗u2,2  �  ��  �  second  ⊗T2)  = (ℓ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2,1)(ℓ1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2,2) T1 ⊗ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Let S ∈ L1  2(E), T ∈ L2  1(E), S = �n  i,j,k=1Si  jk⃗ei ⊗ej ⊗ej, T = �n  α,β,γ=1 T αβ  γ ⃗eα ⊗⃗eβ ⊗eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T =  n  �  i,j,k,β,γ=1  Si  jkT kβ  γ ⃗ei ⊗ ej ⊗ ⃗eβ ⊗ eγ,  and  S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T =  n  �  i,j,k,γ=1  Si  jkT kj  γ ⃗ei ⊗ eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 If S ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), T ∈ L(F, G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and U ∈ L(G, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) then prove  S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U) = (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. U = (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T  (circular permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  If S = � Si  j⃗ai ⊗ bj, T = � T i  j⃗bi ⊗ cj and U = � U i  j⃗ci ⊗ aj, then T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U = � T i  kU k  j ⃗bi ⊗ aj, thus  S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='U) = � Si  mT m  k U k  i , and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T = � Si  kT k  j ⃗ai ⊗ cj, so (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. U = � Si  kT k  mU m  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the second equality  thanks to the symmetry of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. U = U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T) = (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T with the previous calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  We define in the same way the triple objective contraction (apply the simple contraction three times  consecutively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', with (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34) we get  S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' T =  n  �  i,j,k=1  Si  jkT kj  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  168  169  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Kronecker (contraction) tensor, trace  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Non objective double contraction: Double matrix contraction  The engineers often use the double matrix contraction of second order tensors defined by (term to term  multiplication): If S = [Sij] = [Si  j] and T = [Tij] = [T i  j] then  S : T :=  n  �  i,j=1  SijTij =  n  �  i,j=1  Si  jT i  j  noted  =  Tr(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Einstein’s convention is not satisfied, and the result is observer dependent for associated endomorphism:  Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 Let (⃗ei) be a basis, let S ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) given by [S]⃗e =  �  0  4  2  0  �  (so S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e1 = 2⃗e2 and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e2 = 4⃗e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then the double matrix contraction (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37) gives  S : S = [S]⃗e : [S]⃗e = 4 ∗ 4 + 2 ∗ 2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  Change of basis: let ⃗b1 = ⃗e1 and ⃗b2 = 2⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The transition matrix from (⃗ei) to (⃗bi) is P =  �  1  0  0  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  [S]⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [S]⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P =  �  1  0  0  1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  0  8  2  0  �  =  �  0  8  1  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  S : S = [S]⃗b : [S]⃗b = 8 ∗ 8 + 1 ∗ 1 = 65 ̸= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  To be compared with the double objective contraction: [S]⃗e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. [S]⃗e = 4∗2+2∗4 = 16 = [S]⃗b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. [S]⃗b = S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. S  (observer independent result = objective result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So it is absurd to use S : S (double matrix contraction) if you need objectivity: Recall that the foot is  the international vertical unit in aviation, and thus the use of the double objective contraction is vital,  while the use of the double matrix contraction can be fatal (really).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Also see the Mars climate orbiter  probe crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 Let S ∈ L0  2(E) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' a metric), let (⃗ai) be a Euclidean basis in foot, and let (⃗bi) = (λ⃗ai)  be the related euclidean basis in metre (change of unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Give [S]|⃗a : [S]|⃗a and [S]|⃗b : [S]|⃗b and compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (The simple and double objective contractions are impossible here since S and T are not compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let S = �n  i,j=1Sa,ijai ⊗ aj = �n  i,j=1Sb,ijbi ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Since (⃗bi) = (λ⃗ai) we have bi =  1  λai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  �n  i,j=1Sa,ijai ⊗ aj = �n  i,j=1Sa,ijλ2bi ⊗ bj, thus λ2Sa,ij = Sb,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  [S]|⃗b : [S]|⃗b =  n  �  i,j=1  (Sb,ij)2 = λ4  n  �  i,j=1  (Sa,ij)2 = λ4[S]|⃗a : [S]|⃗a,  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  with λ4 ≥ 100: Quite a difference isn’t it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Kronecker (contraction) tensor, trace  Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 The Kronecker tensor is the  �1  1  �  uniform tensor δ ∈ L1  1(E) defined by  ∀(ℓ, ⃗u) ∈ E∗ × E,  δ(ℓ, ⃗u) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  And the Kronecker symbols relative to a basis (⃗ei) are the reals defined by, calling (πei) the dual basis,  ��ij := δ(πei,⃗ej) =  �  1 if i = j,  0 if i ̸= j,  �  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  δ :=  n  �  i=1  πei ⊗ ei,  [δ] = [δj] = [I]  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  (identity matrix whatever the basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Duality notations: δi  j := δ(ei,⃗ej), δ := �n  i=1 ⃗ei ⊗ ei and [δ] = [δi  j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 The trace of a  �1  1  �  uniform tensor T ∈ L1  1(E) is  �  Tr(T) = δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. T  (= Tr(LT ))  (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  (with the natural canonical isomorphism T ∈ L1  1(E) ≃ LT ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) given by T(ℓ,⃗v) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus �  Tr(T) = �n  i=1T ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In particular �  Tr(δ) = n, and �  Tr(⃗v ⊗ ℓ) = �  i viℓi = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v when ⃗v = �  i vi⃗ei and ℓ = �  j ℓjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  169  170  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Introduction, module, derivation  R  Tensors in T r  s (U)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Introduction, module, derivation  Let A and B be any sets, and let F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) be the set of functions A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The “plus” inner operation  and the “dot” outer operation are defined by, for all f, g ∈ F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B), all λ ∈ R and all p ∈ A,  � (f + g)(p) := f(p) + g(p),  and  (λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='f)(p) := λ f(p),  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='f noted  =  λf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  (F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', R) is thus a vector space on the field R (see any elementary course) called F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  But the field R is “too small” to define a tensor which can be seen as “a linear tool that satisfies the  change of coordinate system rules”:  Example R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 Fundamental counter-example: Derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let U be an open set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The  derivation d : ⃗w ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn) → d⃗w ∈ C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn)) is R-linear: In particular d(λ⃗w) = λ(d⃗w) for all  λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='but d doesn’t satisfy the change of coordinate system rules, see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So a derivation it not a tensor (it is a “spray”, see Abraham–Marsden [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In fact, one requirement for T to be a tensor is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with T = ⃗w a vector field: For all ϕ ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R),  and all ⃗w ∈ Γ(U) (C∞-vector field),  T(ϕ⃗w) = ϕ T(⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  While  d(ϕ⃗w) ̸= ϕ d(⃗w),  because  d(ϕ⃗w) = ϕ d⃗w + dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  Thus the elementary R-linearity requirement “T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (λ⃗w) = λ(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) for all λ ∈ R is not sufficient to charac-  terize a tensor: The R-linearity has to be replaced by the C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)-linearity, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus we will have to replace a real vector space (V, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', R) over the field R with the “module”  (V, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)) over the ring C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), which mainly amounts to consider (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) for all λ = ϕ ∈  C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Remark: The use of a module is very similar to the use of a vector space, but for the use of  the inverse: all real λ ̸= 0 has a multiplicative inverse in R (namely 1  λ), but a function f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  “f ̸= 0 and f vanishes at one point” doesn’t have a multiplicative inverse in C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Field of functions and vector fields  Framework of classical mechanics: U is an open set in an affine space E which associated vector  is E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the definition of tensors is done at a fixed time t (concerns the space variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' As before, the  approach is first qualitative, then quantitative with a basis (⃗ei(p)) and its dual basis (πei(p)) = (ei(p)),  at any p ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Field of functions  Let f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The associated function field is  �f :  �  U → U × R  p → �f(p) := (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' f(p)),  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  and p is called the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So Im �f = {(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' f(p)) : p ∈ U} is the graph of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Definition:  T 0  0 (U) := { �f : f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)} = {field of functions} = the set of  �0  0  �  type tensor on U,  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  or the set of tensors of order 0 on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Abusive short notations (to lighten the writings):  �f(p) noted  =  f(p),  and  T 0  0 (U) noted  =  C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R),  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  but keep the base point in mind (no ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In T 0  0 (U), the internal sum is defined by, for all �f, �g ∈ T 0  0 (U) with �f(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' f(p)) and �g(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' g(p)),  ( �f + �g)(p) := (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (f + g)(p))  (= (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' f(p) + g(p))),  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  and the external multiplication on the ring C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is defined by, for all ϕ ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R),  (ϕ �f)(p) := (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (ϕf)(p))  (= (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ϕ(p)f(p)))  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  (the base point p remains unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (T 0  0 (U), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') is a module over the ring C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  170  171  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Differential forms  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Vector fields  Let ⃗w ∈ C∞(U, E) be a vector valued function (at least Lipschitzian, to get integral curves, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Cauchy–  Lipschitz theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The associated vector field is  �⃗w :  �  U → U × E  p → �⃗w(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  So Im�⃗w = {(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗w(p)) : p ∈ U} is the graph of ⃗w, and the definition of �⃗w tells that the vector ⃗w(p) has to  be drawn at p (the base point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Abusive short notation:  �⃗w(p) noted  =  ⃗w(p)  instead of �⃗w(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  It lightens the notations, but keep the base point in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  Γ(U) := the set of vector fields on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  More precisely, we will use the following full definition of vector fields (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Abraham–Marsden [1]):  A vector field is built from tangent vectors to curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It makes sense on non planar surfaces, and more  generally on differential manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Differential forms  The basic concept is that of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A first over-layer is made of differential forms (which “measure  vector fields”):  Definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 Let α  �  U → E∗  p → α(p)  �  (so α(p) is a linear form at p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The associated differential form  (also called a 1-form) is “the field of linear forms” defined by  �α :  �  U → U × E∗  p → �α(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α(p))  ( = “a pointed linear form at p”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  And p is called the base point, and Im�α = {(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α(p)) : p ∈ U} is the graph of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, if �α ∈ Ω1(U) (differential form) and �⃗w ∈ Γ(U) (vector field), then �α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�⃗w ∈ T 0  0 (U) (field of scalar  valued functions) satisfies  �α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�⃗w :  �  U → U × R  p → (�α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='�⃗w)(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)(p)) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w(p)) ∈ U × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  Short notation:  �α(p) noted  =  α(p),  instead of �α(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' α(p)),  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  but keep the base point in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  Ω1(U) := the set of differential forms U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Tensors  A second over-layer is introduced with the tensors with are “functions defined on vector fields and on  differential forms” (which “measure vector fields and differential forms”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let r, s ∈ N, r+s ≥ 1, and let T :  �  U → Lr  s(E)  p → T(p)  �  (so T(p) is a uniform  �r  s  �  tensor for each p,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And consider the associated function  �T :  �  U → U × Lr  s(E)  p → �T(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' T(p))  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  Abusive short notation:  �T(p) noted  =  T(p)  instead of �T(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' T(p)),  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  but keep the base point in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  171  172  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  First Examples  Definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 (Abraham–Marsden [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') �T is a tensor of type  �r  s  �  iff T is C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)-multilinear (not only  R-multilinear), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), all z1, z2 vector field or differentiable form where applicable,  and all p ∈ U,  �  T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z1(p) + z2(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') + T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z2(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='),  and  T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', f(p)z1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = f(p) T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='),  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  written in short  �  T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z1 + z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') + T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='),  and  T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', fz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') = f T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  And  T r  s (U) := the set of  �r  s  �  type tensors on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  (Recall: T 0  0 (U) := C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) the set of function fields, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Remark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Definition in differential geometry lessons: A tensor is a section of a certain bundle over  a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' For classical mechanics, definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 gives an equivalent definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  First Examples  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Type  �0  1  �  tensor = differential forms  If T ∈ T 0  1 (U) then T(p) ∈ E∗, so T = α ∈ Ω1(U) is a differential form: T 0  1 (U) ⊂ Ω1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Converse: Does a differential form α ∈ Ω1(U) defines a  �0  1  �  type tensor on U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Yes: We have to  check (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18), which is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So α ∈ T 0  1 (U), so Ω1(U) ⊂ T 0  1 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  T 0  1 (U) = Ω1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Type  �1  0  �  tensor (identified to a vector field)  Let T ∈ T 0  1 (U), so T(p) ∈ L1  0(E) = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = E∗∗ for all p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, thanks to the natural canonical  isomorphism E∗∗ ≃ E, T(p) can be identified to a vector, thus T 0  1 (U) ⊂ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Converse: Does a vector field ⃗w ∈ Γ(U) defines a  �1  0  �  type tensor on U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Yes: We have to check (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18),  which is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So Γ(U) ⊂ T 1  0 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  T 1  0 (U) ≃ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  A metric is a  �0  2  �  tensor  Let T ∈ T 0  2 (U), so T(p) ∈ L0  2(E) for all p ∈ U, and T(⃗u, ⃗w) ∈ T 0  0 (U) for all ⃗u, ⃗w ∈ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 A metric g on U is a  �0  2  �  type tensor on U such that, for all p ∈ E, g(p) =noted gp is an  inner dot product on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  �1  1  �  tensor, identification with fields of endomorphisms  Let T ∈ T 1  1 (U), so T(p) ∈ L1  1(E) for all p ∈ U, and T(α, ⃗w) ∈ T 0  0 (U) for all α ∈ Ω1(U) and ⃗w ∈ Γ(U) (so  T(p)(α(p), ⃗w(p)) ∈ R for all p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The associated field of endomorphisms on U is �LT :  �  U → U × L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E)  p → �LT (p) = (p, LT (p))  �  where LT (p) is  identified with T(p) thanks to the natural canonical isomorphism L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) ≃ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = L1  1(E) given  by  ∀ℓ ∈ E∗, ∀⃗w ∈ E,  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (LT (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = T(p)(ℓ, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Unstationary tensor  Let t ∈ [t1, t2] ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (Tt)t∈[t1,t2] be a family of  �r  s  �  tensors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then T : t → T(t) := Tt is called  an unstationary tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the set of unstationary tensors is also noted T r  s (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', a Eulerian velocity  field is a  �1  0  �  unstationary vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  172  173  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Differential  S  Differential, its eventual gradients, divergences  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Differential  The definition of the differential of a function is observer independent: All observers have the same  definition (qualitative: no man made tool required, like a basis or an inner dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Framework  Classical Framework: E are F affine spaces associated with vector spaces E and F, and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||E and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||F  are norms in E and F such that (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||E) and (F, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||F ) are complete (we need “limit that stay in the  space as h → 0”, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' U is an open set in E, and Φ :  �  U → F  p → pF = Φ(p)  �  is a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If applicable, E  and/or F can be replaced by E and/or F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (The definitions can be generalized to manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Reminder:  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 Let p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The function Φ is said to be continuous at p iff Φ(q) −→  q→p Φ(p) relative to the  considered norms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', ||Φ(q) − Φ(p)||F −→||q−p||E→0 0, also written (Landau notation): Near p,  Φ(q) = Φ(p) + o(1),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  called “the zero-th order Taylor expansion of Φ near p”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In other words:  ∀ε > 0, ∃η > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∀q ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ||q − p||E < η we have ||Φ(q) − Φ(p)||F < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the set of functions that are continuous at all p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Directional derivative and differential (observer independent)  Let p ∈ U, ⃗u ∈ E, and let f : R → F defined by  f(h) := Φ(p + h⃗u)  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 The function Φ is differentiable at p in the direction ⃗u iff f is derivable at 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' iff the  limit f ′(0) = limh→0  Φ(p+h⃗u)−Φ(p)  h  =noted dΦ(p)(⃗u) exists in F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' iff, near p,  Φ(p + h⃗u) = Φ(p) + h dΦ(p)(⃗u) + o(h),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  equation called the first order Taylor expansion of Φ at p in the direction ⃗u (it is the first order Taylor  expansion of f near p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then dΦ(p)(⃗u) is called the directional derivative of Φ at p in the direction ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And if, for all ⃗u ∈ E, dΦ(p)(⃗u) exists (in F) then Φ is called Gâteaux differentiable at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Prove: If Φ is Gâteaux differentiable at p then dΦ(p) is homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dΦ(p)(λ⃗u) =  λ dΦ(p)(⃗u) for all ⃗u ∈ E and all λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' limh→0  Φ(p+h(λ⃗u))−Φ(p)  h  = λ limh→0  Φ(p+λh⃗u)−Φ(p)  λh  = λ limk→0  Φ(p+k⃗u)−Φ(p)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 If Φ is Gateaux differentiable and if moreover dΦ(p) is linear and continuous at p, then  Φ is said to be differentiable at p (or Fréchet differentiable at p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So  Φ(q) = Φ(p) + h dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−→  pq + o(||−→  pq||E),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  since then dΦ(p)(⃗u) =noted dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u for all ⃗u ∈ E (linearity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And the affine function affp : q → affp(q) := Φ(p) + dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='−→  pq is the affine approximation of Φ at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (So, the graph of affp is the tangent plane of Φ at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Φ : U → F is said to be differentiable in U iff Φ is differentiable at all p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then its  differential is the map  dΦ :  �  U → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  p → dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  And C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the set of differentiable functions ψ such that dΦ ∈ C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And C2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the set of differentiable functions ψ such that dΦ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And Ck(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the set of differentiable functions ψ such that dΦ ∈ Ck−1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='.  173  174  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A basis and the j-th partial derivative  Proposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 The differentiation (or derivation) operator d :  �  C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F))  Φ → dΦ  �  is  R-linear (“a derivation is linear”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d(Φ + λΨ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = limh→0  (Φ+λΨ)(p+h⃗u)−(Φ+λΨ)(p)  h  = limh→0  Φ(p+h⃗u)−Φ(p)+λΨ(p+h⃗u)−λΨ(p)  h  =  limh→0  Φ(p+h⃗u)−Φ(p)  h  + λ limh→0  Ψ(p+h⃗u)−Ψ(p)  h  = dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u + λdΨ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (dΦ(p) + λdΨ(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u for all p  and ⃗u, thus d(Φ + λΨ) = dΦ + λdΨ for all λ ∈ R and Φ, Ψ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 Prove: if f ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) (scalar values) and Φ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) then, for all ⃗u ∈ E,  d(fΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)Φ + f(dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  (and we also write d(fΦ) = Φ ⊗ df + f dΦ for a use with contraction rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  d(fΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = lim  h→0  f(p+h⃗u)Φ(p+h⃗u) − f(p)Φ(p)  h  = lim  h→0  f(p+h⃗u)Φ(p+h⃗u) − f(p)Φ(p+h⃗u)  h  + f(p)Φ(p+h⃗u) − f(p)Φ(p)  h  = lim  h→0  f(p+h⃗u) − f(p)  h  (Φ(p) + o(1)) + lim  h→0 f(p)Φ(p+h⃗u) − Φ(p)  h  = (df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)Φ(p) + f(p)(dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  Tensorial writing: d(fΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (Φ ⊗ df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u + (f dΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u, thanks to the contraction rule which gives (Φ ⊗ df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u +  (f dΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = Φ(df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u) + f(dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 In differential geometry, the definition of a tangent map is defined by, with definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4:  TΦ :  �  U × E → F × F  (p, ⃗u) → TΦ(p, ⃗u) = (Φ(p), dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  The two points p (input) and Φ(p) (output) are the base points, and the two vectors ⃗u (input) and  dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u (output) are the initial vector and its push-forward by Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Notation for the second order Differential  Let Φ ∈ C2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus dΦ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)), thus d(dΦ) ∈ C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So, for p ∈ U and ⃗u ∈  E, we have d(dΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = limh→0  dΦ(p+h⃗u)−dΦ(p)  h  ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), and, with ⃗v ∈ E we have (d(dΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 The bilinear map d2Φ(p) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is defined by  d2Φ(p)(⃗u,⃗v) = (d(dΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v,  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  thanks to the natural canonical isomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)) ↔ TL ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) given by  TL(⃗u1, ⃗u2) := (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2 for all ⃗u1, ⃗u2 ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus L =noted TL, thus d(dΦ) =noted d2Φ(p) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  This gives the usual second order Taylor expansion of Φ (supposed C2) near p in the direction ⃗u:  Φ(p + h⃗u) = Φ(p) + h dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u + h2  2 d2Φ(p)(⃗u, ⃗u) + o(h2)  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  (=the second order Taylor expansion of f : h → f(h) = Φ(p + h⃗u) near h = 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And Schwarz’s theorem tells that d2Φ(p) is symmetric when Φ is C2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d2Φ(p)(⃗u,⃗v) = d2Φ(p)(⃗v, ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  A basis and the j-th partial derivative  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 Let Φ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), ⃗u ∈ Γ(U) (a vector field), p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The derivative of Φ at p along ⃗u  is defined by  ∂⃗uΦ(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(p)  (= lim  h→0  Φ(p + h⃗u(p)) − Φ(p)  h  ∈ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  This defines the directional derivative operator along ⃗u:  ∂⃗u :  �  C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  Φ → ∂⃗u(Φ) := dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂⃗u(Φ)(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  (And ∂⃗u(Φ)(p) =noted ⃗u(Φ)(p) in differential geometry thanks to E ≃ E∗∗ which gives ∂⃗u ≃ ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  174  175  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Application 1: Scalar valued functions  In particular, if (⃗ei(p)) is a basis at p, then the j-th partial derivative of Φ at p is ∂⃗ejΦ(p) =noted ∂jΦ(p)  (the derivative along ⃗ej), and the j-th directional derivative operator is  ∂j :  � C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  Φ → ∂jΦ := dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂j(Φ)(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  (In differential geometry ∂jΦ =noted ⃗ej(Φ), so ⃗ej(Φ)(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Application 1: Scalar valued functions  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Differential of a scalar valued function (objective)  Here Φ noted  =  f :  �  U → R  p → f(p)  �  is a C1 scalar valued function, so df ∈ Ω1(U)∩C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) (a C0 differential  form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So df(p) ∈ E∗ for all p ∈ U, and df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = limh→0  f(p+h⃗u)−f(p)  h  ∈ R for all ⃗u ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 Prove: If f, g ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) then (derivative of a product)  d(fg) = (df)g + f(dg),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', d(fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)g + f(dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) for all ⃗w ∈ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' limh→0  f(p+h ⃗w)g(p+h ⃗w)−f(p)g(p)  h  = limh→0  f(p+h ⃗w)g(p+h ⃗w)−f(p)g(p+h ⃗w)  h  + limh→0  f(p)g(p+h ⃗w)−f(p)g(p)  h  =  limh→0  f(p+h ⃗w)−f(p)  h  (g(p) + o(1)) + limh→0 f(p) g(p+h ⃗w)−g(p)  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' calculation that only requires the first order (affine)  approximation of f and g: We get the same result as with the affine functions f(x) = a0+a1x and g(x) = b0+b1x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  which give (fg)(x) = a0b0 + (a0b1+a1b0)x + a1b1x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and then (fg)′(x) = a0b1+a1b0 + 2a1b1x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' which is indeed  equal to (f ′g + fg′)(x) = a1(b0+b1x) + (a0+a1x)b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Quantification  Let (⃗ei(p)) be a basis at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So ∂jf(p) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13) df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej(p) (= limh→0  f(p+h⃗ej(p))−f(p)  h  ), and we write  ∂jf(p) noted  =  f|j(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  So, with (πei(p)) the dual basis of the basis (⃗ei(p)), and with f|j(p) := πei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='df(p) (j-th component of  df(p) in the basis (πei(p))), we have  df =  n  �  j=1  f|jπej,  and  [df(p)]|⃗e = ( f|1(p)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  f|n(p) )  (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  So df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = �n  j=1f|juj = [df]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗u]|⃗e when ⃗u(p) = �  i ui(p)⃗ei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular with a Cartesian basis,  (πei(p)) =noted (dxj), and df = �n  j=1  ∂f  ∂xj dxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notations: πei = ei, ⃗u = �n  j=1uj⃗ej, df = �n  j=1f|j ej, df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = �n  j=1f|juj, and with a Cartesian  basis, πei = dxi and df = �n  j=1  ∂f  ∂xj dxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12 Prove: (fg)|j = f|j g + f g|j when f, g : U → R are C1 scalar valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Apply (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7): here d(fg) = g df + f dg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d(fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = (df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej) g + f (dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej) for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And df(p) ∈ E∗ satisfies the covariant change of basis formula for linear forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if (⃗ai(p)) and  (⃗bi(p)) are two bases at p and P(p) is the transition matrix from (⃗ai(p)) to (⃗bi(p)), then [df(p)]|⃗b =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  [df(p)]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P(p), or in short:  [df]|⃗b = [df]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  (covariance formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  175  176  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Application 2: Coordinate system basis and Christoffel symbols  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Gradients (subjective) associated with a differential through inner dot products  Let f ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) (a C1 scalar valued function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Choose (subjective) an inner dot product (·, ·)g in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13 The conjugate gradient  ⃗  gradgf(p) of f at p ∈ U relative to (·, ·)g, also called the  (·, ·)g-conjugate gradient of f at p, is the (·, ·)g-Riesz representation vector of the linear form df(p) ∈ E∗:  ⃗  gradgf(p) := ⃗Rg(df(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', the vector  ⃗  gradgf(p) ∈ E is characterized by, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2),  ∀⃗u ∈ E,  df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = ( ⃗  gradgf(p), ⃗u)g =  ⃗  gradgf(p) •g ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  Fundamental: An English observer with his Euclidean dot product (·, ·)a in foot and a French observer  with his Euclidean dot product (·, ·)b in metre have the same differential df (defined independently of  any unit of measurement);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' But do not have the same gradient:  ⃗  gradbf  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  =  λ2 ⃗  gradaf  with  λ2 > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  Quite different vectors isn’t it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The “gradient vector” strongly depends on the chosen inner dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And to forget this fact leads to accidents like the crash of the Mars Climate Orbiter probe, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Subjective first order Taylor expansion:  If an inner dot product (·, ·)g exists and is used, then the  first order Taylor expansion (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) gives  f(p + h⃗u) = f(p) + h ( ⃗  gradgf(p), ⃗u)g + o(h)  (= f(p) + h  ⃗  gradgf(p) •g ⃗u + o(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  Fundamental once again (we insist):  An inner dot product does not always exist (as a meaningful tool), see § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (thermodynamics),  thus, for a C1 function, a gradient does not always exists (contrary to a differential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  df(p) is a linear form (covariant) while  ⃗  gradgf(p) is a vector (contravariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular the  change of basis formulas differ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28):  [df]|new = [df]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P,  while  [ ⃗  gradg]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ ⃗  gradg]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22)  df cannot be identified  ⃗  gradf (with one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') (Recall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' there is no natural canonical isomorphims between  E and E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The differential df is also called the “covariant gradient”, and any of its associated gradient  vectors is also called the “contravariant gradient relative to an inner dot product”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Isometric Euclidean framework: If one Euclidean dot product can be imposed to all observers (foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') then  ⃗  gradgf =noted  ⃗  gradf = ⃗∇f and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) is written df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u =  ⃗  gradf • ⃗u = ⃗∇f • ⃗u (isometric  framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14 Cartesian basis (⃗ei) and (·, ·)g given by [g][⃗e =  �  1  0  0  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Give [df]|⃗e and [ ⃗  gradgf]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [df]|⃗e = ( ∂f  ∂x1  ∂f  ∂x2 ) (row matrix) and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) gives [ ⃗  gradgf]|⃗e =  �  ∂f  ∂x1  1  2  ∂f  ∂x2  �  (column matrix ̸= [df]T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Application 2: Coordinate system basis and Christoffel symbols  (Necessary when dealing with covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Coordinate system, and coordinate system basis  Consider a (open) set Upar = {⃗q ∈]a1, b1[×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='×]an, bn[}, called the set of parameters, in the Cartesian  space Rn, consider an open set U ⊂ Rn, called the set of geometric positions, and consider a C2-  diffeomorphism Ψ : ⃗q ∈ Upar → p ∈ U, called a coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (⃗ai) the canonical basis of the parameter space, let ⃗q = �  i qi⃗ai ∈ Upar (the qi are called the  parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', see the polar coordinate system at § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 where ⃗q = (q1, q2) = (r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  176  177  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Application 2: Coordinate system basis and Christoffel symbols  Ψ being a diffeomorphism, at any p = Ψ(⃗q) ∈ U the vectors  ⃗ai∗(p) := dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23)  make a basis in E at p, and (⃗ai∗(p)) is called the coordinate system basis at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its dual basis at p is made  of the linear forms dqi(p), so where, for all i, j,  dqi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(p) = δj  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24)  Duality notations: dqi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(p) = δi  j for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Parametric expression of the differential of a scalar valued function  With a coordinate system Ψ, a scalar valued function f :  �  U → R  p → f(p)  �  defined in U can be described  with the function g = f ◦ Ψ :  �  Upar → R  ⃗q → g(⃗q) := f(p) when p = Ψ(⃗q)  �  defined in Upar, and g is called the  parametric expression of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  dg(⃗q) = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(⃗q)  when  p = Ψ(⃗q),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25)  in particular,  ∂g  ∂qj  (⃗q) := dg(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj∗(p) noted  =  ∂f  ∂qj  (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  Warning, pay attention: f is a function of p, not a function of ⃗q, and the notations  ∂f  ∂qj (p) means  := ∂(f◦Ψ)  ∂qj  (⃗q) when p = Ψ(⃗q), and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus with (dqj(p)) the dual basis of the coordinate basis (⃗ai∗(p)) at p,  df(p)  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26)  =  n  �  j=1  ∂f  ∂qj  (p) dqj(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27)  Duality notations: df(p) = �  j  ∂f  ∂qj (p) dqj(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15 Pay attention to the notations that could contradict themselves:  1- In Upar the dual basis (πai) of the Cartesian basis (⃗ai) is a uniform basis (independent of ⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' and  is (almost) never written (dqi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Indeed, (dqi(p)) is the name reserved for the dual basis of (⃗ai∗(p)) in the geometric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Mind  the notations!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' for polar coordinates (dq1(p), dq2(p)) = (dr(p), dθ(p)) is the dual basis of the polar  coordinate system basis (⃗a1∗(p),⃗a2∗(p)) at p, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16 Bases (⃗ai) and (⃗bi) at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A vector ⃗x is expressed as ⃗x = �  i xa,i⃗ai = �  i xb,i⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Prove:  ⃗bi = λ⃗ai, ∀i  =⇒  ∂f  ∂xb,i  = λ ∂f  ∂xa,i  or  ∂f  ∂xa,i  =  ∂f  ∂(λxa,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28)  (Change of unit formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') Duality notations:  ∂f  ∂xj  b = λ ∂f  ∂xj  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj(p) = λdf(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj(p) (linearity of df(p)) reads (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Or [df]|⃗b = [df]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P with P = λI here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17 [df]|⃗b = [df]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∂f  ∂xj  b = �n  i=1  ∂f  ∂xia P i  j is also noted  ∂f  ∂xj  b  =  n  �  i=1  ∂f  ∂xia  ∂xi  a  ∂xj  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29)  Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  177  178  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Application 2: Coordinate system basis and Christoffel symbols  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Quick answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗x]|⃗b, which means [⃗x]|⃗a([⃗x]|⃗b) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗x]|⃗b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �  �  �  x1  a(x1  b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn  b )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  x1  a(x1  b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn  b )  �  �  � =  �  �  �  �n  j=1P 1  j xj  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �n  j=1P n  j xj  b  �  �  � ,  thus  ∂xi  a  ∂xj  b  (x1  b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn  b ) = P i  j , ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30)  Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29) means  ∂f  ∂xj  b  (p) =  n  �  i=1  ∂f  ∂xia  (p)∂xi  a  ∂xj  b  (x1  b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', xn  b )  thus  =  n  �  i=1  ∂f  ∂xia  (p) P i  j ,  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31)  as given in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Detailed answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let O be a point (origin) in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If p ∈ U, let ⃗x = −→  Op = �n  i=1xi  a⃗ai = �n  i=1xi  b⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  This define the function [⃗x]|⃗a : [⃗x]|⃗b → [⃗x]|⃗a([⃗x]|⃗b), and we have [⃗x]|⃗a([⃗x]|⃗b) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗x]|⃗b (change of basis formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then let fa, fb : Rn → R be defined by fa([⃗x]|⃗a) := f(p) and fb([⃗x]|⃗b) := f(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (NB: fa and fb don’t have the  same definition domain: They are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus fa([⃗x]|⃗a) = fb([⃗x]|⃗b) (= f(p)) when [⃗x([⃗x]|⃗b)]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='[⃗x]|⃗b, so (fa ◦ [⃗x]|⃗a)([⃗x]|⃗b) = fb([⃗x]|⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus  ∂(fa◦[⃗x]|⃗a)  ∂xi  b  ([⃗x]|⃗b) = ∂fb  ∂xi  b ([⃗x]|⃗b), thus the meaning of (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29) is  n  �  j=1  ∂fa  ∂xj  a  ([⃗x]|⃗a)∂xj  a  ∂xi  b  ([⃗x]|⃗b) = ∂fb  ∂xi  b  ([⃗x]|⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32)  Question: Why did we introduce fa and fb (and not just keep f)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: Because a vector is not just a collection of components (is not just a matrix), and −→  Op cannot be  reduced to a matrix of components (which one: [⃗x]|⃗a?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗x]|⃗b?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Here f is a function acting on a point p (independent  of a referential), while fa and fb are functions acting on matrices (dependent on the choice of a referential): The  domain of definitions are different, so the functions f, fa and fb are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Christoffel symbols  We use duality notations for readability and usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18 In a coordinate system basis (⃗ei(p)) in E (previously called (⃗ai∗(p)), the Christoffel  symbol γi  jk(p) ∈ R are the components of the vector d⃗ek(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d⃗ek(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej(p) = �n  k=1γi  jk(p)⃗ei(p), so  d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  i=1  γi  jk⃗ei ,  or  d⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei =  n  �  k=1  γk  ij⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  (So, with (ei(p)) the dual basis of (⃗ei(p)), γi  jk := ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej, and, for calculations with contractions,  d⃗ek = �  ij γi  jk⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (The Christoffel symbols vanish in a Cartesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  (Differential geometry in manifolds: ∇⃗ej⃗ek = �n  i=1γi  jk⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' the γi  jk = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∇⃗ej⃗ek are the component  of the connection ∇, the usual connection in a surface in Rn being the Riemannian connection, in which  case ∇⃗ej⃗ek is the orthogonal projection of d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej on the surface relative to a Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' for the polar coordinate system, see remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12, d⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗e2 = −r⃗e1, thus γ1  22 = −r and γ2  22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19 Prove: If (⃗ei(p)) is the coordinate system basis of a C2 coordinate system, then:  ∀i, j, d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = d⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei  (=  ∂2Ψ  ∂qi∂qj ),  and  ∀i, j, k, γk  ji = γk  ij  (symmetry for lower indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='34)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗ei(p) = (⃗ei ◦ Ψ)(⃗q) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23) dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai gives d(⃗ei ◦ Ψ)(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = d(dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj, thus d⃗ei(Ψ(⃗q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj =  ∂ ∂Ψ  ∂qi  ∂qj  =  ∂ ∂Ψ  ∂qj  ∂qi  (Schwarz theorem since Ψ is C2) = d⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei =noted  ∂2Ψ  ∂qj∂qi , thus �n  k=1γk  ij⃗ek = �n  k=1γk  ji⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20 Consider two coordinate system bases (⃗ai(p)) and (⃗bi(p)) at p, P(p) = [P i  j(p)] the tran-  sition matrix from (⃗ai(p)) to (⃗bi(p)), and Q = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Using the generic notation d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1γi  jk,e⃗ei,  prove the change of basis formula for the Christoffel symbols:  γi  jk,b =  n  �  λ,µ,ν=1  Qi  λP µ  j P ν  k γλ  µν,a+  n  �  λ,µ=1  Qi  λP µ  j (dP λ  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ)  (=  n  �  λ,µ,ν=1  Qi  λP µ  j P ν  k γλ  µν,a+  n  �  λ=1  Qi  λ(dP λ  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35)  (Because of the term �  µν Qi  λP µ  j (dP λ  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ), a derivation is not a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  178  179  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Application 3: Differential of a vector field  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⃗bk(p) = �  ν P ν  k (p)⃗aν(p) gives d⃗bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = �  ν(dP ν  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='���bj)⃗aν + �  ν P ν  k (d⃗aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj) = �  µν P µ  j (dP ν  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ)⃗aν +  �  µν P ν  k P µ  j (d⃗aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And bi = �  λ Qi  λaλ, thus  γi  jk,b = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='d⃗bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj =  �  λµν  Qi  λP µ  j (dP ν  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ)aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν +  �  λµν  Qi  λP µ  j P ν  k aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) =  �  λµ  Qi  λP µ  j (dP λ  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) +  �  λµν  Qi  λP µ  j P ν  k γλ  µν,a,  thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Application 3: Differential of a vector field  Here F = E = ⃗Rn, Φ =noted ⃗w ∈ Γ(U) is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus d⃗w(p) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) and d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u is a vector field  in E for all ⃗u ∈ Γ(U), given by (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)(p) = d⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u(p) = limh→0  ⃗w(p+h⃗u(p))− ⃗w(p)  h  ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: (⃗ei(p)) is a basis at p in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Call wi(p) ∈ R the components of ⃗w(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗w(p) =  �n  i=1wi(p)⃗ei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And call wi|j(p) the components of d⃗w(p) (endomorphism in E):  ⃗w =  n  �  i=1  wi⃗ei,  d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  i=1  wi|j⃗ei,  [d⃗w]|⃗e = [wi|j]  (Jacobian matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36)  And tensorial notations for calculations with contractions: (πei(p)) being the dual basis,  d⃗w =  n  �  i,j=1  wi|j⃗ei ⊗ πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='37)  Duality notations: ⃗w = ���n  i=1wi⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i,j=1wi  |j⃗ei, [d⃗w]|⃗e = [wi  |j], and d⃗w = �n  i,j=1wi  |j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a Cartesian basis: Here (⃗ei) is uniform, so ⃗w(p) = �n  i=1wi(p)⃗ei gives d⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1(dwi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)⃗ei,  thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='36) gives  wi|j = ∂wi  ∂xj  (p) noted  =  wi,j,  so  [d⃗w]|⃗e = [∂wi  ∂xj  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='38)  Duality notations: wi  |j = ∂wi  ∂xj and [d⃗w]|⃗e = [ ∂wi  ∂xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a coordinate system basis: With the coordinate system described in § S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 and the duality notations  for readability (and usage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗w(p) = �n  i=1wi(p)⃗ei(p) gives, for all j,  d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  i=1  (dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)⃗ei +  n  �  i=1  wi(d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)  (=  n  �  i=1  wi  |j⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='39)  (Tensorial notations to be used with contractions: d⃗w = �  i ⃗ei ⊗ dwi + �  i wi d⃗ei = �  ij wi  |j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  And �  i wi(d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33) �  ik wiγk  ji⃗ek = �  ik wkγi  jk⃗ei, thus, for all i, j,  wi  |j = ∂wi  ∂qj +  n  �  k=1  wkγi  jk  where  ∂wi  ∂qj := dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40)  ( ∂wi  ∂qj := dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej is the derivation along the j-th coordinate line of the scalar valued function wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (In particular, if ⃗w = ⃗eℓ = �  i δi  ℓ⃗ei, we recover d⃗eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �  i 0⃗ei + �  ik δk  ℓ γi  jk⃗ei = �  i γi  jℓ⃗ei, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21 With exercise S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20, and ⃗w = �n  i=1ui⃗ai = �  n vi⃗bi, check with calculations (d⃗w is an  endomorphism defined independently of any basis):  [d⃗w]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [d⃗w]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  vi  |j =  n  �  k,ℓ=1  Qi  kuk  |ℓP ℓ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗bj = �  ℓ P ℓ  j⃗aℓ for all i, Q = P |1, and [⃗w]|⃗b = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [⃗w]|⃗a reads vi = �  k Qi  kuk for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Cartesian basis: dvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = d(�  k Qi  kuk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (�  ℓ P ℓ  j⃗aℓ) = �  kℓ Qi  k(duk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aℓ)P ℓ  j , qed (here the Qi  k are uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  independent of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  179  180  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Application 4: Differential of a differential form  Coordinate system basis: vi = �  λ Qi  λuλ gives dvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = �  λ(dQi  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj)uλ + �  λ Qi  λ(duλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  vi  |j  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33)  =  dvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj +  �  k  vkγi  jk,b  =  �  λµ  uλP µ  j (dQi  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) +  �  λµ  Qi  λP µ  j (duλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) +  �  kλµν  (Qk  λuλ)Qi  νP µ  j (dP ν  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) +  �  kωλµν  (Qk  ωuω)Qi  λP µ  j P ν  k γλ  µν,a  And Qk  ωP λ  k = δλ  ω gives (dQk  ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ)P λ  k + Qk  ω(dP λ  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) = 0, thus the third term reads  �  kλµν  uλQi  νP µ  j Qk  λ(dP ν  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) = −  �  kλµν  uλQi  νP µ  j P ν  k (dQk  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) = −  �  λµ  uλP µ  j (dQi  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ),  which cancels the first term: Thus vi  |j = �  λµ Qi  λP µ  j (duλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aµ) + �  λµν uνQi  λP µ  j γλ  µν = �  λµ Qi  λui  |jP µ  j , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6  Application 4: Differential of a differential form  Here F = R, Φ =noted ℓ ∈ Ω1(U) (differential form) supposed C1, p ∈ U, so ℓ(p) ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its differential at p  in a direction ⃗u is dℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = limh→0  ℓ(p+h⃗u)−ℓ(p)  h  ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (dℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = limh→0  ℓ(p+h⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v−ℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v  h  ∈ R  for all ⃗u,⃗v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: (πei(p) its the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Call ℓi(p) ∈ R the components of ℓ(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ℓ(p) = �n  i=1ℓi(p)πei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And call ℓi|j(p) the components  of dℓ(p) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗):  ℓ =  n  �  i=1  ℓiπei,  dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =  n  �  i=1  ℓi|jπei,  [dℓ]|⃗e = [ℓi|j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='42)  Tensorial notations, to be used with contractions: dℓ = �n  i,j=1ℓi|jπei ⊗ πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Duality notations: ℓ = �  i ℓiei, dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1ℓi|jei, [dℓ]|⃗e = [ℓi|j], and dℓ = �n  i,j=1ℓi|jei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a Cartesian basis: Here (⃗ei) is uniform, so  ℓi|j = ∂ℓi  ∂xj  (p) noted  =  ℓi,j,  so  [dℓ]|⃗e = [ ∂ℓi  ∂xj  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='43)  Duality notations: ℓi|j = dℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = ∂ℓi  ∂xj and [dℓ]|⃗e = [ ∂ℓi  ∂xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  In a coordinate system basis: With duality notations and Christoffel symbols:  dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = −  n  �  k=1  γi  jkek .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='44)  Indeed, ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = δi  k gives (dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek + ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej) = 0, thus (dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = −ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  ℓ γℓ  jk⃗eℓ = −γi  jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  ℓi|j = ∂ℓi  ∂qj −  n  �  k=1  ℓkγk  ji  where  ∂ℓi  ∂qj (p) := dℓi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45)  Indeed, ℓ = �  i ℓiei gives dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �  i(dℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)ei + �  i ℓi(dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej) = �  i(dℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej)ei − �  ik ℓiγi  jkek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7  Application 5: Differential of a 1 1 tensor  Consider a C1 �1  1  �  tensor τ :  �  U → L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  p → τ(p)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Its differential dτ :  �  U → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R))  p → dτ(p)  �  is  defined by dτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = limh→0  τ(p+h⃗u)−τ(p)  h  ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), so (dτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u)(ℓ,⃗v) = limh→0  τ(p+h⃗u)(ℓ,⃗v)−τ(p)(ℓ,⃗v)  h  (∈ R), for all ⃗u,⃗v ∈ E and ℓ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification (duality notations): Basis (⃗ei(p)) in E at p, dual basis (ei(p)), call τ i  j(p) the components  of τ(p), call τ i  j|k(p) the components of dτ(p):  τ =  �  ij  τij⃗ei ⊗ ej,  dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek =  n  �  i,j=1  τ i  j|k⃗ei ⊗ ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46)  Tensorial notations, to be used with contractions: dτ = �n  i,j,k=1τ i  j|k⃗ei ⊗ ej ⊗ ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Classical notations: τ = �  ij τij⃗ei ⊗ πej, dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  ij τij|k⃗ei ⊗ πej, and dτ = �  ijk τij|k⃗ei ⊗ πej ⊗ πek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  180  181  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Divergence of a vector field: Invariant  Cartesian basis: dτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  ij(dτ i  j(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek)⃗ei ⊗ ej, so  τ i  j|k = ∂τ i  j  ∂xk  noted  =  τ i  j,k  (:= dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='47)  Coordinate system basis: τ(p) = �n  i,j=1τ i  j(p)⃗ei(p) ⊗ ej(p) gives, for all k,  dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  ij(dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek)⃗ei ⊗ ej + �  ij τ i  j(d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek) ⊗ ej + �  ij τ i  j⃗ei ⊗ (dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek)  = �  ij(dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek)⃗ei ⊗ ej + �  ijℓ τ i  jγℓ  ki⃗eℓ ⊗ ej − �  ijℓ τ i  jγj  kℓ⃗ei ⊗ eℓ  = �  ij(dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek)⃗ei ⊗ ej + �  ijℓ τ ℓ  j γi  kℓ⃗ei ⊗ ej − �  ijℓ τ i  ℓγℓ  kj⃗ei ⊗ ej  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='48)  thus  τ i  j|k = ∂τ i  j  ∂qk +  n  �  ℓ=1  τ ℓ  j γi  kℓ −  n  �  ℓ=1  τ i  ℓγℓ  kj  where  ∂τ i  j  ∂qk := dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='49)  (We have the + sign from vector fields, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40), and the − sign from differential forms, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='22 If ⃗u ∈ E, ℓ ∈ E∗ then for the elementary  �1  1  �  tensor τ = ⃗u ⊗ ℓ prove:  d(⃗u ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = (d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek) ⊗ ℓ + ⃗u ⊗ (dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek),  and  (⃗u ⊗ ℓ)i  j|k = ui  |kℓj + uiℓj|k,  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)  when ⃗u = �  i ui⃗ei, ℓ = �  j ℓjej, d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  i ui  |k⃗ei, dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  j ℓj|kej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' τ = ⃗u ⊗ ℓ = �  ij τ i  j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' where τ i  j = uiℓj, and dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �n  i,j=1τ i  j|k⃗ei ⊗ ej where τ i  j|k = (uiℓj)|k =  ui  |kℓj + uiℓj|k = (⃗u ⊗ ℓ)i  j|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (similar to the derivation of a product):  d(⃗u ⊗ ℓ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek(p) = lim  h→0  (⃗u ⊗ ℓ)(p+h⃗ek(p)) − (⃗u ⊗ ℓ)(p)  h  = lim  h→0  ⃗u(p+h⃗ek(p)) ⊗ ℓ(p+h⃗ek(p)) − ⃗u(p) ⊗ ℓ(p)  h  = lim  h→0  ⃗u(p+h⃗ek(p)) ⊗ ℓ(p+h⃗ek(p)) − ⃗u(p+h⃗ek(p)) ⊗ ℓ(p)  h  + lim  h→0  ⃗u(p+h⃗ek(p)) ⊗ ℓ(p) − ⃗u(p) ⊗ ℓ(p)  h  = lim  h→0(⃗u(p+h⃗ek(p)) ⊗ (ℓ(p+h⃗ek(p)) − ℓ(p)  h  ) + lim  h→0(⃗u(p+h⃗ek(p)) − ⃗u(p)  h  ) ⊗ ℓ(p)  = ⃗u(p) ⊗ (dℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek(p)) + (d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek(p)) ⊗ ℓ(p),  thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Which gives d(⃗u ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = (�  i ui⃗ei) ⊗ (�  j ℓj|kej) + (�  i ui  |k⃗ei) ⊗ (�  j ℓjej), thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8  Divergence of a vector field: Invariant  Γ(U) is the set of C1 vector fields in U, and Tr : L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) → R is the trace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='23 The divergence operator is  div := Tr ◦ d :  �  Γ(U) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  ⃗w → div⃗w := Tr(d⃗w),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  div⃗w(p) := Tr(d⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='51)  (So div⃗w(p) = trace of the endomorphism d⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Tr and d are linear, hence div = Tr ◦ d is R-linear (composed of two R-linear maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='24 The divergence of a vector field is objective (is an invariant): Same value for all  observers (objective quantity) intrinsic to ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The differential and the trace are objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Or computation: ⃗w = �  i ui⃗ai = �  i vi⃗bi gives  vi  |j = �  kℓ Qi  kuk  |ℓP ℓ  j , see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='41), thus �  i vi  |i = �  ikℓ P ℓ  i Qi  kuk  |ℓ = �  kℓ δℓ  kuk  |ℓ = �  k uk  |k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Quantification: ⃗w ∈ Γ(U), (⃗ei) is a basis, ⃗w = �n  i=1wi⃗ei with classical notations, and wi|j(p) are the  components of the vector d⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej(p) in the basis (⃗ei(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus  div⃗w =  n  �  i=1  wi|i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='52)  Duality notations: ⃗w = �n  i=1wi⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �n  i=1wi  |j⃗ei, [d⃗w]|⃗e = [wi  |j], div⃗w = �n  i=1wi  |i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  181  182  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Objective divergence for 1 1 tensors  Cartesian basis (⃗ei) (classical notations): dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej =noted ∂wi  ∂xj and  wi|j = ∂wi  ∂xi  ,  thus  div⃗w =  n  �  i=1  ∂wi  ∂xi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='53)  Coordinate system basis (⃗ei) (duality notations): With the Christoffel symbols, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='33), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='40) gives  wi  |i = ∂wi  ∂qi +  n  �  i=1  wkγi  ik,  thus  div⃗w =  n  �  i=1  ∂wi  ∂qi +  n  �  i,k=1  wkγi  ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='25 Prove:  div(f ⃗w) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + f div⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='55)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' d(f ⃗w) = ⃗w ⊗ df + f d⃗w, thus Tr(d(f ⃗w)) = Tr(⃗w ⊗ df) + Tr(f d⃗w) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + f Tr(d⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Use a coordinate  system if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='26 If α is a differential form, if (⃗ei) is a basis and (ei) its dual basis, and if α = �n  i=1αiei,  then dα = �n  i=1αi|jei ⊗ ej, with αi|j := ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Here it is impossible to define an objective trace of dα  like �n  i=1αi|i: The result depends on the choice of the basis (the Einstein convention is not satisfied, and  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' with a Euclidean basis the result depends on the choice of unit of length: Foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Meter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus the  objective (or intrinsic) divergence of a differential form is a nonsense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9  Objective divergence for 1 1 tensors  To create an objective divergence for a second order  �1  1  �  tensor τ ∈ T 1  1 (U), in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46) we have to contract an  admissible index with the “differential index k”, So, no choice: Contract i and k to get �  divτ := �n  i,j=1τ i  j|iej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let us start with:  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='27 Let ⃗u ∈ Γ(U) and ℓ ∈ Ω1(U) be C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The objective divergence of the elementary  �1  1  �  tensor ⃗u ⊗ ℓ ∈ T 1  1 (U) is the differential form �  div(⃗u ⊗ ℓ) ∈ Ω1(U) defined by  �  div(⃗u ⊗ ℓ) = (div⃗u)ℓ + dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' defined by �  div(⃗u ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = (div⃗u)(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) + (dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w)⃗u for all ⃗w ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the objective divergence operator  �  div :  �  T 1  1 (U) → Ω1(U)  τ → �  divτ  �  is the linear map defined on elementary tensors with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Quantification: (⃗ei) is a basis, (ei) its dual basis, ⃗u = �  i ui⃗ei, ℓ = �  j ℓjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus ⃗u⊗ℓ = �  ij uiℓj⃗ei⊗ej,  and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='50)-(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56) give  �  div(⃗u ⊗ ℓ) =  n  �  i,j=1  (ui  |iℓj + ℓj|iui)ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='57)  So for an elementary tensor τ = ⃗u ⊗ ℓ, τ = �  ij τ i  j⃗ei ⊗ ej, and dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  ijk τ i  j|k⃗ei ⊗ ej and �  div(τ) =  �  ij τ i  j|iei, with τ i  j|k = ui  |kℓj + uiℓj|k, here with τ i  j = uiℓj, and τ i  j|k = ui  |kℓj + uiℓj|k, so τ i  j|i = ui  |iℓj + uiℓj|i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus, by linearity of �  div, for all tensors τ ∈ T 1  1 (U), we have with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46):  �  divτ =  n  �  i,j=1  τ i  j|iej ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [ �  divτ]|⃗e =  � �  i τ i  1|i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  i τ i  n|i  �  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='58)  (row matrix since �  divτ is a differential form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', we have contracted i and k in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (Classical notations: �  divτ := �n  i,j=1τij|iπej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [ �  divτ]|⃗e = ( �  i τi1|i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' �  i τin|i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  So  Cartesian bases: �  divτ =  n  �  i,j=1  ∂τ i  j  ∂xi ej,  Coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' bases: �  divτ =  n  �  i,j=1  �∂τ i  j  ∂qi +  n  �  k=1  τ k  j γi  ik −  n  �  k=1  τ k  i γi  kj  �  ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)  Indeed: With τ = �  j(�  i τ i  j⃗ei) ⊗ ej = �  j ⃗wj ⊗ ej where ⃗wj = �  i τ i  j⃗ei, the linearity of �  div gives  �  divτ = �  j �  div(⃗wj ⊗ ej);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus, with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='56): 1- Cartesian basis: div⃗wj = �  i  ∂τ i  j  ∂xi and dej = 0 give �  divτ =  182  183  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Objective divergence for 1 1 tensors  �  j  �  i  ∂τ i  j  ∂xi ej = �  ij τ i  j,iej, thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And 2- Coordinate system basis: div⃗wj=(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='54)�  i  ∂τ i  j  ∂qi +�  ik τ k  j γi  ik  and dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wj = �  k τ k  j dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ek = �  k τ k  j (− �  i γj  kiei), thus �  j dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗wj = − �  ijk τ k  j γj  kiei = − �  ijk τ k  i γi  kjej,  thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='28 Prove: If f ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and τ = �n  i,j=1τ i  j⃗ei ⊗ ej ∈ T 1  1 (U) ∩ C1 then  �  div(fτ) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='τ + f �  divτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='60)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' fτ = �  ij fτ i  j⃗ei ⊗ ej gives d(fτ) = �  ijk(fτ i  j)|k⃗ei ⊗ ej ⊗ ek = �  ijk(f|kτ i  j + fτ i  j|k)⃗ei ⊗ ej ⊗ ek, thus  �  div(fτ) = �  ij(f|iτ i  j + fτ i  j|i)ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='τ + f �  divτ = �  ij f|iτ i  jej + f �  ij τ i  j|iej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='29 Prove: If τ ∈ T 1  1 (U) and ⃗w ∈ Γ(U) then  div(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = �  div(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w + τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='. d⃗w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='61)  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' τ = �  ij τ i  j⃗ei ⊗ ej and ⃗w = �  i wi⃗ei give τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w = �  ij τ i  jwj⃗ei, thus div(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗w) = �  ij τ i  j|iwj + τ i  jwj  |i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='30 If τ ∈ T 1  1 (U) check with component calculations (since �  div(τ) is objective):  [ �  div(τ)]|b = [ �  div(τ)]|a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  (covariance formula),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='62)  where P is the transition matrix from a basis (⃗ai) to a basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let τ = �  ij σi  j⃗ai ⊗ aj = �  ij τ i  j⃗bi ⊗ bj, so τ i  j = �  λµ Qi  λσλ  µP µ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1- Cartesian bases: �  i τ i  j|i = �  i dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bi = �  i d(�  λµ Qi  λσλ  µP µ  j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (�  ν P ν  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) = �  iλµν Qi  λP µ  j P ν  i (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) =  �  λµν δν  λP µ  j (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) = �  λµ P µ  j (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aλ) = �  µ(�  λ σλ  µ|λ)P µ  j as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Coordinate system bases: �  i τ i  j|i =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='59) �  i dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ei + �  iℓ τ ℓ  j γi  iℓ,b − �  iℓ τ i  ℓγℓ  ij,b (with j fixed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' With  �  i  (dτ i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bi) =  �  iλµ  Qi  λ (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bi) P µ  j +  �  iλµ  (dQi  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bi) σλ  µ P µ  j +  �  iλµ  Qi  λ σλ  µ (dP µ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bi)  =  �  iλµν  Qi  λP µ  j P ν  i (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) +  �  iλµν  σλ  µ P µ  j P ν  i (dQi  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) +  �  iλµν  σλ  µ Qi  λP ν  i (dP µ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν)  =  �  λµ  P µ  j (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aλ) −  �  iλµν  σλ  µ P µ  j Qi  λ(dP ν  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) +  �  λµ  σλ  µ (dP µ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aλ)  since P ν  i Qi  λ = δν  λ gives P ν  i (dQi  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν) − Qi  λ(dP ν  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And, with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='35),  �  iℓ  τ ℓ  j γi  iℓ,b =  �  iℓ  (  �  λµ  Qℓ  λσλ  µP µ  j )(  �  αβω  Qi  αP β  i P ω  ℓ γα  βω,a +  �  αβ  Qi  αP β  i (dP α  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aβ))  =  �  λµα  σλ  µP µ  j γα  αλ,a +  �  ℓλµα  σλ  µQℓ  λP µ  j (dP α  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aα),  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='63)  and  −  �  iℓ  τ i  ℓγℓ  ij,b = −  �  iℓ  (  �  λµ  Qi  λσλ  µP µ  ℓ )(  �  αβω  P α  i P β  j Qℓ  ωγω  αβ,a +  �  αω  P α  i Qℓ  ω(dP ω  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aα))  = −  �  λµβ  σλ  µP β  j γµ  λβ,a −  �  λµ  σλ  µ(dP µ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='64)  Thus �  i τ i  j|i = �  λµ P µ  j (dσλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aλ) + �  λµα σλ  µP µ  j γα  αλ,a − �  λµβ σλ  µP β  j γµ  λβ,a = �  λµ P µ  j σλ  µ|λ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Divergence of a 2 0 tensor  Let τ ∈ T 2  0 (U) and τ = �n  i,j=1τ ij⃗ei⊗⃗ej, thus dτ = �n  i,j,k=1τ ij  |k⃗ei⊗⃗ej⊗ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then two objective divergences  may be defined: by contracting k with i, or k with j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (The Einstein convention is then satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Divergence of a 0 2 tensor  Let τ = ��n  i,j=1τijei ⊗ ej ∈ T 0  2 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus dτ = �n  i,j,k=1τij|kei ⊗ ej ⊗ ek, and there are no indices to  contract to satisfy Einstein convention: There is no objective divergence of 0 2 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  183  184  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Euclidean framework and “classic divergence” of a tensor (subjective)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10  Euclidean framework and “classic divergence” of a tensor (subjective)  Let σ be a C1 tensor of order 2 of any kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' An observer chooses a (Cartesian) Euclidean basis (⃗ei)  and call (·, ·)g the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And he calls σij the components of σ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' writes  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗ej = �  i σij⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='31 (Usual divergence in classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') The divergence divσ of σ relative to the  basis (⃗ei), is the column matrix (it is not a vector)  diveσ =  �  �  �  �n  j=1  ∂σ1j  ∂xj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  �n  j=1  ∂σnj  ∂xj  �  �  �  noted  =  divσ  (a matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65)  (Take the divergences of the “row vectors” of [σ]|e = [σij] to make the “column vector” [diveσ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='32 The “so called vector” divσ, in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67), is not a vector: It does not satisfy the change  of basis formula: If (⃗ai) and (⃗bi) are bases, if P is the transition matrix from (⃗ai) to (⃗bi), if [σ]|⃗a = [Aij]  and [σ]|⃗b = [Bij], with the divergence of σ relative to (⃗ai) and (⃗bi) called divaσ and divbσ, then neither a  contravariant nor a covariant change of basis formula applies in general:  neither  [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [divaσ]|⃗a  nor  [divbσ]T  |⃗b = [divaσ]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='66)  (compare with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='62)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So divσ as given in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67) is neither a contravariant vector nor a covariant vector  (it is just a matrix which depends on an observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the simple case ⃗bi = λ⃗ai, for all i, λ > 1: Transition matrix P = λI, and P −1 = 1  λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For a  �1  1  �  tensor: σ = �  ij(σb)i  j⃗bi ⊗ bj = �  ij(σa)i  j⃗ai ⊗ aj, [σ]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = 1  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='λ = [σ]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (σa)i  j = (σb)i  j for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67) gives divbσ = �  ij(d(σb)i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj)⃗bi = �  ij(d(σa)i  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (λ⃗aj))(λ⃗ai) = λ2divaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [divbσ]|⃗a and [divbσ]T  |⃗b ̸= [divaσ]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For a  �0  2  �  tensor: σ = �  ij σb,ijbi ⊗ bj = �  ij σa,ijai ⊗ aj, and [σ]|⃗b = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P = λ2[σ]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' σb,ij =  λ2σa,ij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67) gives divbσ = �  ij(dσb,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj)⃗bi = λ2 �  ij(dσa,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (λ⃗aj))(λ⃗ai) = λ4divaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [divbσ]|⃗a and [divbσ]T  |⃗b ̸= [divaσ]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For a  �2  0  �  tensor: σ = �  ij σij  b ⃗bi ⊗ ⃗bj = �  ij σij  a ⃗ai ⊗ ⃗aj, and [σ]|⃗b = P −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P −1 =  1  λ2 [σ]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  σij  b =  1  λ2 σij  a for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67) gives divbσ = �  ij(dσij  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj)⃗bi =  1  λ2  �  ij(dσij  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (λ⃗aj))(λ⃗ai) = divaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' [divbσ]|⃗a and [divbσ]T  |⃗b ̸= [divaσ]T  |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Remark: (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='65) can be written  divσ =  �  ij  ∂σij  ∂xj  ⃗Ei  (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='67)  where ( ⃗Ei) is the canonical basis in Mn1 the space of n ∗ 1 column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  T  Natural canonical isomorphisms  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  The adjoint of a linear map  Setting of § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12: E and F are vector spaces, E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) and F ∗ = L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) are their dual spaces,  and the adjoint of a linear map P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) is the linear map P∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) canonically defined by  ∀ℓ ∈ F ∗,  P∗(ℓ) := ℓ ◦ P,  written  P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  (dot notations P∗(ℓ) =noted P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ and ℓ◦P =noted ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P since ℓ and P∗ are linear), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for all (ℓ, ⃗u) ∈ F ∗×E,  P∗(ℓ)(⃗u) = ℓ(P(⃗u)),  written  (P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  Interpretation: If P is the push-forward of vector fields, then P∗ is the pull-back of differential forms,  see remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In particular, it will be interpreted with P ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) (linear and invertible = a change  of observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  184  185  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  An isomorphism E ≃ E∗ is never natural (never objective)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  An isomorphism E ≃ E∗ is never natural (never objective)  Two observers A and B consider a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) be the change of observer  endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Willing to work together, A and B (“naturally”) consider the diagram  E  L  −→ E∗  ← considered by observer A  P ↓  ↑ P∗  E −→  L  E∗  ← considered by observer B  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  Definition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 (Spivak [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') A linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) is natural iff the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) commutes for  all P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E):  L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) is natural  ⇐⇒  ∀P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E), P∗ ◦ L ◦ P = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  (In that case, if A computes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u with the top line of the diagram, if B computes with the bottom line of  the diagram, then they can easily check their results since here L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u = (P∗ ◦ L ◦ P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Question: Does there exist an endomorphism L such that the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) commutes for all change  of observers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' That is, do we have  ∃?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E), ∀P ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E),  P∗ ◦ L ◦ P = L ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  Answer: Always no (if L ̸= 0):  Theorem T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 A (non-zero) linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) is not natural: If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) − {0}, then  ∃P ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  L ̸= P∗ ◦ L ◦ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Spivak [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') It suffices to prove this proposition for E = ⃗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let L ∈ L(⃗R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (⃗R)∗), L ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (⃗a1) be a basis in ⃗R (chosen by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗b1) be a basis in ⃗R (chosen by B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Consider P ∈ Li(⃗R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗R) defined by P(⃗a1) = ⃗b1 (change of observer), and let λ ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗b1 = λ⃗a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1) gives P∗(ℓ)(⃗a1) := ℓ(P(⃗a1)) = ℓ(⃗b1) = ℓ(λ⃗a1) = λℓ(⃗a1), thus P∗(ℓ) = λℓ for all ℓ ∈ (⃗R)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus P∗(L(P(⃗a1))) = P∗(L(λ⃗a1)) = λP∗(L(⃗a1)) = λ2L(⃗a1) ̸= L(⃗a1) when λ2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', P = 2I gives  L ̸= P∗ ◦ L ◦ P (= 4L), thus (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6): A (non-zero) linear map E → E∗ cannot be natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 Consider E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' dim E = 1, and consider the linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) which sends a basis  (⃗a1) onto its dual basis (πa1), so L is defined by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1 := πa1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Question: If (⃗b1) is another basis, λ ̸= ±1 and ⃗b1 = λ⃗a1 (change of unit of measurement), does  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b1 = πb1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' does L also sends (⃗b1) onto its dual basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Indeed, ⃗b1 = λ⃗a1 gives πb1 =  1  λπa1, thus L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b1 = λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1 = λπa1 = λ2πb1 ̸= πb1 since  λ2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In words: L is not natural, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  A different presentation: Let LA and LB be defined by LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = πaj and LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = πbj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And  suppose that ⃗bj = λ⃗aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then, LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = λLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = λπaj = λ2πbj = λ2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj ̸= LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj when λ2 ̸= 1,  that is, LA ̸= LB when λ2 ̸= 1: An operator that sends a basis onto its dual basis is not natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Let (·, ·)g be an inner dot product in E = ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗Rg ∈ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) be the Riesz rep-  resentation map, that is, defined by ⃗Rg(ℓ) = ⃗ℓg where ⃗ℓg is defined by (⃗ℓg,⃗v)g = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v for all ⃗v ∈ ⃗Rn,  cf (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Question: Is ⃗Rg natural?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: No: Consider the diagram  �  E∗  ⃗Rg  −→ E  P∗ ↓  ↑ P  E∗ −→  ⃗Rg  E  �  with P = λI, λ ̸= ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then P∗ = λI, and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ = λ2 ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ ̸= ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ gives P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P∗ ̸= ⃗Rg: So ⃗Rg is not natural, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (You may prefer to  consider the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3) with L = ⃗R−1  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  A different presentation: Consider two distinct Euclidean dot products (·, ·)g and (·, ·)h (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', built with  a foot and built with a metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So (·, ·)h = λ2(·, ·)g with λ2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let ⃗Rg, ⃗Rh ∈ L(Rn∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Rn) be the Riesz  operators relative to (·, ·)g and (·, ·)h, that is ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ = ⃗ℓg and ⃗Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ = ⃗ℓh are given by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v = (⃗ℓg,⃗v)g = (⃗ℓh,⃗v)h  for all ⃗v ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have ⃗ℓh = λ2⃗ℓg, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11), thus ⃗Rh = λ2 ⃗Rg ̸= ⃗Rg since λ2 ̸= 1: A Riesz representation  operator is not natural (it is observer dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  185  186  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Natural canonical isomorphism E ≃ E∗∗  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Natural canonical isomorphism E ≃ E∗∗  Two observers A and B consider the same linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗∗) (where E∗∗ = (E∗)∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Willing to work together, they (“naturally”) consider the diagram  E  L  −→ E∗∗  ← considered by observer A  P ↓  ↓ P∗∗  E −→  L  E∗∗  ← considered by observer B  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  where P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E) is a linear diffeomorphism, P∗ ∈ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗) its adjoint, given by P∗(ℓ) = ℓ ◦ P  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), and P∗∗ ∈ Li(E∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗∗) the adjoint of P∗, thus given by P∗∗(u) = u ◦ P∗ for all u ∈ E∗∗  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' P∗∗ is given by, for all (ℓ, u) ∈ E∗ × E∗∗,  (P∗∗(u))(ℓ) = u(ℓ ◦ P),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (P∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  Question: Does there exist a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E∗∗) that is natural?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Answer: Yes (particular case of the next proposition):  Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 The canonical isomorphism  JE :  �  E → E∗∗  ⃗u → u = JE(⃗u)  defined by  JE(⃗u)(ℓ) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  ∀ℓ ∈ E∗,  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  is natural, that is, F being another finite dimensional vector space, the diagram  E JE  −→ E∗∗  P ↓  ↓ P∗∗  F −→  JF  F ∗∗  written  E  J  −→ E∗∗  P ↓  ↓ P∗∗  F −→  J  F ∗∗  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  commutes for all P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ∀P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F),  P∗∗ ◦ JE = JF ◦ P,  and we write  E ≃ E∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  Thus we can use the unambiguous notation (observer independent)  J (⃗u) noted  =  ⃗u,  and  J (⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ noted  =  ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ℓ  (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  (And u = J (⃗u) is the derivation operator in the direction ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Spivak [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  It is trivial that JE is linear and bijective (E is finite dimensional): It is an  isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (P∗∗ ◦ JE(⃗u))(ℓ)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  = JE(⃗u)(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  = (ℓ ◦ P)(⃗u) = ℓ(P(⃗u))  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  = JF (P(⃗u))(ℓ), for all  ℓ ∈ F ∗ and all ⃗u ∈ E, thus P∗∗ ◦ JE(⃗u) = JF (P(⃗u)), for all ⃗u ∈ E, thus P∗∗ ◦ JE = JF ◦ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 (Characterization of JE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') JE sends any basis (⃗ai) onto its bidual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (Expected,  since JE(⃗u) is the directional derivative in the direction ⃗u, whatever ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let (⃗ai) be a basis and (πai) be its dual basis (defined by πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = δij for all i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  gives JE(⃗aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πai = πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗aj = δij for all i, j, thus (JE(⃗aj)) is the dual basis of (πai), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', is the bidual basis  of (⃗ai);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' True for all basis: JE(⃗bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='πbi = πbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗bj = δij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4  Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) ≃ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) ≃ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F ∗)  E, F, A, B are finite dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider the canonical isomorphism  JEF :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  L → �L = JEF (L)  where  �L(ℓ, ⃗u) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u,  ∀(ℓ, ⃗u) ∈ F ∗ × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  186  187  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) ≃ L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  Let P1 ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A) and P2 ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B), and consider the diagram  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) JEF  −→ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  IP ↓  ↓ �  IP  L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) −→  JAB  L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  where  IP(L) = P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P−1  1  and  �  IP(�L)(b,⃗a) = �L(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a)  ∀(b,⃗a) ∈ B∗ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  (IP and �  IP are the push-forwards for linear maps L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) and for bilinear forms �L ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 The canonical isomorphism JEF is natural, that is, the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14) commutes for  all P1 ∈ Li(E, A) and all P2 ∈ L(F, B):  �  IP ◦ JEF = JAB ◦ IP,  and  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  natural  ≃  L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  Thus L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F ∗)  natural  ≃  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' JAB(IP(L))(b,⃗a)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  =  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='IP(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  =  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P−1  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a = (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  =  JEF (L)(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  =  �  IP(JEF (L))(b,⃗a), true for all L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F), b ∈ B∗, ⃗a ∈ A, thus (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Thus L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F ∗)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  ≃ L((F ∗)∗, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  ≃ L(F, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  ≃ L(E∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Consider the canonical isomorphism (defines the transposed of a bilinear map)  KEF :  �  L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  T → KEF (T)  �  ,  KEF (T)(⃗u,⃗v) := T(⃗v, ⃗u),  ∀(⃗u,⃗v) ∈ E × F,  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  and ZAB ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) → L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) defined by ZAB(T)(⃗a,⃗b) := T(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a, P−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b) for all (⃗a,⃗b) ∈ A × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 The canonical isomorphism KEF is natural: For all (P1, P2) ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' A)×L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B), the  diagram  L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) KEF  −→ L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  ZAB ↓  ↓ ZBA  L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) −→  KAB  L(B, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  commutes: L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  natural  ≃  L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' KEF (ZAB(T))(⃗b,⃗a) = ZAB(T)(⃗a,⃗b) = T(P−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a) and ZBA(KEF (T))(⃗a,⃗b) = KEF (T)(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a, P−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b) =  T(P−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗b, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a), thus KAB ◦ ZAB = ZBA ◦ KEF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5  Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) ≃ L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  For application to the second order derivative d(d⃗u) ≃ d2⃗u and, with ⃗u ∈ T 1  0 (U), the notation d⃗u ∈ T 1  1 (U),  then d2⃗u ∈ T 1  2 (U), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', dk⃗u ∈ T 1  k (U), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Consider the canonical isomorphism  J12E :  �  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)) → L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  T1 → T2 = J12E(T1)  �  ,  J12E(T1)(⃗u1, ⃗u2) := T1(⃗u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗u2 ∈ F,  ∀⃗u1, ⃗u2 ∈ E,  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  and the canonical isomorphism  J23E :  �  L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) → L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  T2 → J23E(T2) = T3  �  ,  T3(ℓ, ⃗u,⃗v) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T2(⃗u1, ⃗u2),  ∀⃗u1, ⃗u2 ∈ E, ∀ℓ ∈ F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 J12 and J23 are natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Thus J23 ◦ J12 is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  187  188  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definitions  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 1- We have to prove that the following diagram commutes:  L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)) J12E  −→  L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)  ZAB ↓  ↓ YAB  L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B)) J12A  −→  L(A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B)  where  ZAB(T1)(⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2 := T1(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2),  YAB(T2)(⃗a1,⃗a2) = T2(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2),  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  (the “push-forwards) for all ⃗a1,⃗a2 ∈ A and LAB ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let T1 ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have  J12A(ZAB(T1))(⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2 = ZAB(T1)(⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2 = T1(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2), and  YAB(J12E(T1))(⃗a1,⃗a2) = J12E(T1)(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2) = T1(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2),  thus J12A ◦ ZAB = YAB ◦ J12E, thus J12 is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- We have to prove that the following diagram commutes:  L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F) J23E  −→  L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  ZAB ↓  ↓ YAB  L(A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' B) J23A  −→  L(B∗, A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  where  ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ZAB(T2)(⃗a1,⃗a2) := (ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T2(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2),  YAB(T3)(ℓB,⃗a1,⃗a2) = T3(ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2),  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='21)  (the “push-forwards) for all ⃗a1,⃗a2 ∈ A and ℓB ∈ B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let T2 ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We have  J23A(ℓB, ZAB(T2)(⃗a1,⃗a2)) = ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ZAB(T2)(⃗a1,⃗a2) = (ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T2(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2), and  YAB(J23A(T2))(ℓB,⃗a1,⃗a2) = J23A(T2)(ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2) = ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='T2(P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a1, P−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗a2)  thus J23A ◦ ZAB = YAB ◦ J23E, thus J23 is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  U  Distribution in brief: A covariant concept  We refer to the books of Laurent Schwartz for a full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In continuum mechanics, with Ω an  open set in Rn and for the space of the finite energy functions L2(Ω) and its sub-spaces, a distribution  gives a covariant formulation for the virtual power, as used by Germain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Definitions  Usual notations: Let p ∈ [1, ∞[ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' p = 2 for finite energy functions), and let  Lp(Ω) := {f : Ω → R :  �  Ω  |f(x)|p dΩ < ∞}  and  ||f||p = (  �  Ω  |f(x)|p dΩ)  1  p ,  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1)  the space of functions such that |f|p is Lebesgue integrable with ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||p its usual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (Lp(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||Lp)  is a Banach space (a complete normed space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And let  L∞(Ω) := {f : Ω → R : sup  x∈Ω  (|f(x)|) < ∞},  and  ||f||∞ = sup  x∈Ω  (|f(x)|),  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2)  the space of Lebesgue measurable bounded functions with ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||∞ its usual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then (L∞(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||L∞)  is a Banach space (a complete normed space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1 If f ∈ F(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R), then its support is the set  supp(f) := {x ∈ Ω : f(x) ̸= 0} = the closure of {x ∈ Ω : f(x) ̸= 0}  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3)  = the set where it is interesting to study f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The closure is required: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', if Ω =]0, 2π[ and f(x) = sin x for all x ∈ ω, then {x ∈ Ω : f(x) ̸= 0} =  ]0, π[∪]π, 2π[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And the point π is a point of interest since sin varies in its vicinity: f ′(π) = cos(π) = −1 ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So it is the closure supp(f) := ]0, π[∪]π, 2π[ = [0, 2π] that is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2 (Schwartz notation, D being the letter after C:) Let  D(Ω) := C∞  c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) = {ϕ ∈ C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' supp(ϕ) is compact in Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4)  188  189  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Derivation of a distribution  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Ω = R, ϕ(x) := e−  1  1−x2 if x ∈]−1, 1[ and ϕ(x) := 0 elsewhere: ϕ ∈ D(R) with supp(ϕ) = [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  And D(Ω) is a vector space which is dense in (Lp(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||Lp) for any p ∈ [1, ∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3 A distribution in Ω is a linear D(Ω)-continuous3 function  T :  � D(Ω) → R  ϕ → T(ϕ) noted  =  ⟨T, ϕ⟩  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5)  The space of distribution in Ω is named D′(Ω) (the dual of D(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  The notation ⟨T, ϕ⟩D′(Ω),D(Ω) = ⟨T, ϕ⟩ is the “duality bracket” = the “covariance–contravariance  bracket” between a linear function T ∈ D′(Ω) and a vector ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='4 Let f ∈ Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The regular distribution Tf ∈ D′(Ω) associated to f is defined by  Tf(ϕ) :=  �  Ω  f(x)ϕ(x) dΩ,  ∀ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6)  So Tf is a measuring instrument with density dmf(x) = f(x) dΩ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Tf(ϕ) :=  �  Ω ϕ(x) dmf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5 Let x0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' The Dirac measure at x0 is the distribution T noted  =  δx0 ∈ D′(R) defined  by, for all ϕ ∈ D(R),  δx0(ϕ) = ϕ(x0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⟨δx0, ϕ⟩ = ϕ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7)  And δx0 is not a regular distribution (δx0 is not a density measure): There is no integrable function f  such that Tf = δx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Interpretation: δx0 corresponds to an ideal measuring device: The precision is perfect  at x0 (gives the exact value ϕ(x0) at x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' In real life δx0 is the ideal approximation of Tfn where fn is  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' given by fn(x) = n1[x0,x0+ 1  n ] (drawing): For all ϕ ∈ D(Ω), Tfn(ϕ) −→n→∞ δx0(ϕ) = ϕ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Generalization of the definition: In (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='5) D(Ω) = C∞  c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is replaced by C∞  c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' So if you  consider a basis (⃗ei) then ⃗ϕ ∈ C∞  c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ⃗Rn) reads ⃗ϕ = �n  i=1ϕi⃗ei with ϕi ∈ D(Ω) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6 Power:  Let α : Ω → T 0  1 (Ω) be a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Then P  = Tα defined by  P(⃗v) =  �  Ω α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗v dΩ gives the virtual power associated to α relative to the vector field ⃗v (mechanics and  thermodynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Derivation of a distribution  Let O be a point in Rn (an origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If p ∈ Rn and if (⃗ei) is a basis in ⃗Rn, let ⃗x = −→  Op = �n  i=1xi⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='7 The derivative  ∂T  ∂xi of a distribution T ∈ D′(Ω) is the distribution ∈ D′(Ω) defined by,  for all ϕ ∈ D(Ω),  ∂T  ∂xi  (ϕ) := −T( ∂ϕ  ∂xi  ),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  ⟨ ∂T  ∂xi  , ϕ⟩ := −⟨T, ∂ϕ  ∂xi  ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  ( ∂T  ∂xi is indeed a distribution: Easy check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Example U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8 If T = Tf is a regular distribution with f ∈ C1(Ω), then ∂(Tf )  ∂xi  = T( ∂f  ∂xi ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Indeed, for all  ϕ ∈ D(Ω), ∂(Tf )  ∂xi (ϕ) = −Tf( ∂ϕ  ∂xi ) = −  �  Ω f(x) ∂ϕ  ∂xi dΩ = +  �  Ω  ∂f  ∂xi ϕ(x) dΩ +  �  Γ 0 dΓ, since ϕ vanishes on  Γ = ∂Ω (the support of ϕ is compact in Ω), thus ∂(Tf )  ∂xi (ϕ) = T( ∂f  ∂xi )(ϕ) for all ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Example U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9 Consider the Heaviside function (the unit step function) H0 := 1R+ and the associated  distribution T = TH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then ⟨(TH0)′, ϕ⟩ := −⟨TH0, ϕ′⟩ = −  �  Ω H0(x)ϕ′(x) dx = −  � ∞  0  ϕ′(x) dx = ϕ(0) =  ⟨δ0, ϕ⟩ for any ϕ ∈ D(R), thus (TH0)′ = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Written H0  ′ = δ0 in D′(Ω), which is not in a equality between  functions, because H0 is not derivable at 0 as a function, and δ0 is not a function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' It is equality between  distributions: The notation H0  ′ can only be used to compute H0  ′(ϕ) (= ⟨H0  ′, ϕ⟩ := −⟨H0, ϕ′⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3The D(Ω)-continuity of T is defined by: 1- A sequence (ϕn)N∗ in D(Ω) converges in D(Ω) towards a function ϕ ∈ D(Ω)  iff there exists a compact K ⊂ Ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' supp(ϕn) ⊂ K for all n, and ||  ∂kϕ  ∂xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂xik −  ∂kϕn  ∂xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='∂xik ||∞ −→n→∞ 0 for all k ∈ N  and all ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 2- T is continuous at ϕ ∈ D(Ω) iff T(ϕn) −→  n→∞ T(ϕ) for any sequence (ϕn)N ∈ D(Ω)N −→  n→∞ ϕ in D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  189  190  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Hilbert space H1(Ω)  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Hilbert space H1(Ω)  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='1  Motivation  Consider the hat function Λ(x)  �  �  �  �  �  = x + 1 if x ∈ [−1, 0],  = 1 − x if x ∈ [0, 1],  = 0 otherwise  �  �  �  �  �  (drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' When applying the finite element  method, it is well-known that, if you use integrals (if you use the virtual power principle which makes  you compute average values), then you can consider the derivative of the hat function Λ as if it was the  usual derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' at the points where the usual computation of Λ′ is meaningful, that is,  Λ′(x)  �  �  �  �  �  = 1 if x ∈] − 1, 0[,  = −1 if x ∈]0, 1[,  = 0 if x ∈ R − {−1, 0, 1}  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9)  (drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Problem: Λ′ is not defined at −1, 0, 1 (the function Λ is not derivable at −1, 0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Question: So does the “usual” computation I =  �  R Λ′(x)ϕ(x) dx with (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='9) gives the good result?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (This  is not a trivial question: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', with H0 = 1R+ instead of Λ, we would get the absurd result H′  0 = 0,  absurd since H′  0 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Answer: Yes:  1- Consider TΛ the regular distribution associated to Λ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2- Then consider (TΛ)′, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8):  We get ⟨(TΛ)′, ϕ⟩  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='8)  = −⟨TΛ, ϕ′⟩  =  −  �  R  Λ(x)ϕ′(x) dx  =  −  � 0  −1  Λ(x)ϕ′(x) dx −  � 1  0  Λ(x)ϕ′(x) dx = +  � 0  −1  1]−1,0[(x)ϕ(x) dx +  � 1  0  1]0,1[ϕ(x) dx, for any ϕ ∈ D(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  3- Thus (TΛ)′ = Tf where f = 1]−1,0[ + 1]0,1[, that is (TΛ)′ is a regular distribution, And its is named  f = Λ′ within the distribution framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for computations ⟨Λ′, ϕ⟩ := ⟨(TΛ)′, ϕ⟩ with ϕ ∈ D(R)  (value =  �  R f(x)ϕ(x) dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='2  Definition of H1(Ω)  The space C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R) is too small in many applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', for the Λ function above);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' We need a larger  space where the functions are “derivable is a weaker sense” which is the distribution sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Consider a  basis in Rn:  Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10 The Sobolev space H1(Ω) is the subspace of L2(Ω) restricted to functions whose gen-  eralized derivatives are in L2(Ω):  H1(Ω) = {v ∈ L2(Ω) : ∂v  ∂xi  ∈ L2(Ω), ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='10)  Usual shortened notation: H1(Ω) = {v ∈ L2(Ω) :  ⃗  gradv ∈ L2(Ω)  n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  So to check that v ∈ H1(Ω), even if ∂v  ∂xi does not exists in the classic way (see the above hat function Λ),  you have to:  1- Consider its associated regular distribution Tv,  2- Compute ∂Tv  ∂xi in D′(Ω),  3- And if, for all i, there exists fi ∈ L2(Ω) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' ∂Tv  ∂xi = Tfi, then v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  4- And then fi is noted  ∂v  ∂xi when used within the Lebesgue integrals  �  Ω  ∂v  ∂xi (x)ϕ(x) dx with ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Λ ∈ H1(R) since (TΛ)′ = Tf with f = 1]−1,0[ + 1]0,1[ ∈ L2(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (TΛ)′ =noted Λ′ (= f) in the  distribution context (integral computations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Let (·, ·)L2 and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||L2 be the usual inner dot product and norm in L2(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (u, v)L2 =  �  Ω  u(x)v(x) dΩ,  and  ||v||L2 =  �  (v, v)L2 = (  �  Ω  v(x)2 dΩ)  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11)  (L2(Ω), (·, ·)L2) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Then define, for all u, v ∈ H1(Ω),  (u, v)H1 = (u, v)L2 +  n  �  i=1  ( ∂u  ∂xi  , ∂v  ∂xi  )L2,  and  ||v||H1 = (v, v)  1  2  H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='12)  Then (H1(Ω), (·, ·)H1) is a Hilbert space (Riesz–Fisher theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  190  191  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Hilbert space H1(Ω)  With a Euclidean dot product (·, ·)g in ⃗Rn and a (·, ·)g-orthonormal basis,  (u, v)H1 = (u, v)L2 + ( ⃗  gradu,  ⃗  gradv)L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='13)  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='3  Subspace H1  0(Ω) and its dual space H−1(Ω)  The boundary Γ = ∂Ω of Ω is supposed to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Let  H1  0(Ω) := {v ∈ H1(Ω) : v|Γ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='14)  Then (H1  0(Ω), (·, ·)H1) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  More generally (without any regularity assumption on Γ), H1  0(Ω) := D(Ω)  H1  = the closure of D(Ω)  in (H1(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='||H1): This closure of D(Ω) in H1(Ω) enables the use of the distribution framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Notation : The dual space of H1  0(Ω) is the space  H−1(Ω) = (H1  0(Ω))′ = L(H1  0(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' R)  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='15)  equipped with the (usual) norm ||T||H−1 :=  sup  ||v||H1  0 =1  |T(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' And (duality bracket), if v ∈ H1  0(Ω) and  T ∈ H−1(Ω) then  T(v) noted  =  ⟨T, v⟩H−1,H1  0  noted  =  ⟨T, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='16)  Theorem U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='11 (Characterization of H−1(Ω) = (H1  0(Ω))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=') A distribution T is in H−1(Ω) iff  ∃(f,⃗g) ∈ L2(Ω) × L2(Ω)  n  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  T = f − div⃗g (∈ D′(Ω)),  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  that is, for all v ∈ H1  0(Ω),  ⟨T, v⟩H−1,H1  0 =  �  Ω  fv dΩ +  �  Ω  dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='⃗g dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18)  And if Ω is bounded then we can choose f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' If moreover ⃗g ∈ H1(Ω)  n then  ⟨T, v⟩H−1,H1  0 =  �  Ω  f(x)v(x) dx −  �  Ω  div⃗g(x)v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19)  (In fact we only need ⃗g ∈ Hdiv(Ω) = {⃗g ∈ L2(Ω)  n : div⃗g ∈ L2(Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', see Brezis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  For boundary value problems with Neumann boundary conditions, we then need (H1(Ω))′ the dual  space of H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Characterization of (H1(Ω))′: We still have (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='18), but we have to replace (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='17)  or (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='19) by, with a Euclidean dot product in ⃗Rn, see Brezis [4],  ⟨T, v⟩(H1)′,H1 =  �  Ω  f(x)v(x) dx −  �  Ω  div⃗g(x)v(x) dx +  �  Γ  ⃗g(x) • ⃗n(x) v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='20)  191  192  REFERENCES  References  [1] Abraham R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Marsden J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : Foundation of mechanics, 2nd edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Addison-Wesley, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [2] Arnold V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : Equations différentielles ordinaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Editions Mir, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [3] Arnold V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : Mathematical Methods of Classical Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Second Edition, Springer 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [4] Brézis H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': Functional Analysis, Sobolev spaces and Partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [5] Cartan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': Formes différentielles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hermann 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [6] Cartan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': Cours de calcul différentiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Hermann 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [7] Forest  et  al:  Mécanique  de  milieux  continus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  Ecole  des  Mines  de  Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  http://mms2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='ensmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='fr/mmc_paris/poly/MMC2020web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='pdf  [8] Germain P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': Cours de Mécanique des Milieux Continus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Tome 1: théorie générale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Masson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [9] Germain P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': The Method of Virtual Power in Continuum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Part 2: Microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' SIAM  Journal on Applied Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' 3 (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', 1973), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : A short Course in Fluid Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' Publish or Perish, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' Université  Pierre et Marie Curie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=' Dover publications, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [13] Maxwell: A treatise on electricity and magnetism, Clarendon press series, 1873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='  [14] Misner C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content=', Thorne K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' : Gravitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  [17] Spivak M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': A comprehensive introduction to differential geometry, Volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  [18] Strang G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  [19] Truesdell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=', Noll W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
+page_content=': The Non-Linear Field Theories of Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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+page_content='  [20] http://planet-terre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'}
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