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coursenotes tensor calculus and differential geometry 2wah0 lucflorack march10 2021 cover illustration papyrus fragment from euclid s elements of geometry book ii contents preface iii notation 1 1 prerequisites from ...

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                      CourseNotes
           Tensor Calculus and Differential Geometry
                        2WAH0
                       LucFlorack
                      March10,2021
        Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8].
                             Contents
                             Preface                                                                                                                                                                          iii
                             Notation                                                                                                                                                                           1
                             1     Prerequisites from Linear Algebra                                                                                                                                            3
                             2     Tensor Calculus                                                                                                                                                              7
                                   2.1       Vector Spaces and Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                7
                                   2.2       Dual Vector Spaces and Dual Bases                            .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . . .               8
                                   2.3       TheKroneckerTensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                 10
                                   2.4       Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                           11
                                   2.5       Reciprocal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                             14
                                   2.6       Bases, Dual Bases, Reciprocal Bases: Mutual Relations . . . . . . . . . . . . . . . . . . . . . .                                                                16
                                   2.7       Examples of Vectors and Covectors                            .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . . .             17
                                   2.8       Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                            18
                                             2.8.1       Tensors in all Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                        18
                                             2.8.2       Tensors Subject to Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                          22
                                             2.8.3       SymmetryandAntisymmetryPreserving Product Operators . . . . . . . . . . . . . . .                                                                    24
                                             2.8.4       Vector Spaces with an Oriented Volume . . . . . . . . . . . . . . . . . . . . . . . . . .                                                            31
                                             2.8.5       Tensors on an Inner Product Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                          34
                                             2.8.6       Tensor Transformations                     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . . . . .               36
                                                         2.8.6.1          “Absolute Tensors”                 .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . .            37
                             CONTENTS                                                                                                                                                                            i
                                                         2.8.6.2          “Relative Tensors” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                  38
                                                         2.8.6.3          “Pseudo Tensors” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                    41
                                             2.8.7       Contractions             .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . . . . . .                43
                                   2.9       TheHodgeStarOperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                 43
                             3     Differential Geometry                                                                                                                                                      47
                                   3.1       Euclidean Space: Cartesian and Curvilinear Coordinates                                        .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . .           47
                                   3.2       Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                             48
                                   3.3       Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                            49
                                   3.4       Tangent and Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                               50
                                   3.5       Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                            51
                                   3.6       AffineConnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                              52
                                   3.7       Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                           55
                                   3.8       Torsion        .  .  .  .  .  .  .   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . . . . . . .                 55
                                   3.9       Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                             56
                                   3.10 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                 57
                                   3.11 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                 58
                                   3.12 Push-Forward and Pull-Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                    59
                                   3.13 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                                  60
                                             3.13.1 Polar Coordinates in the Euclidean Plane                                   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . . . . . . . .             61
                                             3.13.2 A Helicoidal Extension of the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . .                                                                 63
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...Coursenotes tensor calculus and differential geometry wah lucflorack march cover illustration papyrus fragment from euclid s elements of book ii contents preface iii notation prerequisites linear algebra vector spaces bases dual thekroneckertensor inner products reciprocal mutual relations examples vectors covectors tensors in all generality subject to symmetries symmetryandantisymmetrypreserving product operators with an oriented volume on space transformations absolute i relative pseudo contractions thehodgestaroperator euclidean cartesian curvilinear coordinates differentiable manifolds tangent cotangent bundle exterior derivative afneconnection lie torsion levi civita connection geodesics curvature push forward pull back polar the plane a helicoidal extension...

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