Cover 1
Half-title 3
Title 5
Copyright 6
Dedication 7
Contents 9
Preface 13
Part One: Motivation and background 15
1 The Case for Differential Geometry 17
1.1 Classical Space-Time and Fibre Bundles 18
1.1.1 Aristotelian Space-Time 18
1.1.2 Galilean Space-Time 19
1.1.3 Observer Transformations 21
1.1.4 Cross Sections 22
1.1.5 Relativistic Space-time 23
1.2 Configuration Manifolds and Their Tangent and Cotangent Spaces 24
1.2.1 The Configuration Space 24
1.2.2 Virtual Displacements and Tangent Vectors 24
1.2.3 The Tangent Bundle 25
1.2.4 The Cotangent Bundle 26
1.3 The Infinite-dimensional Case 27
1.3.1 How Many Degrees of Freedom Does a Bar Have? 28
1.3.2 What Is a Configuration of a Deformable Bar? 29
1.3.3 Is Continuity Sufficient? 31
1.3.4 The Configuration Space 31
1.3.5 The Tangent Bundle and Its Physical Meaning 32
1.3.6 The Cotangent Bundle 35
1.4 Elasticity 36
1.5 Material or Configurational Forces 37
2 Vector and Affine Spaces 38
2.1 Vector Spaces: Definition and Examples 38
2.2 Linear Independence and Dimension 40
2.3 Change of Basis and the Summation Convention 44
2.4 The Dual Space 45
2.5 Linear Operators and the Tensor Product 48
2.6 Isomorphisms and Iterated Dual 50
2.7 Inner-product Spaces 55
2.7.1 Generalities and Definition 55
2.7.2 The Isomorphism between V and V 57
2.7.3 The Reciprocal Basis 57
2.7.4 Consequences 58
2.8 Affine Spaces 60
2.8.1 Introduction 60
2.8.2 Definition 61
2.8.3 Affine Simplexes 64
2.8.4 Euclidean or Inner-product Affine Structures 65
2.9 Banach Spaces 66
2.9.1 Basic Definitions 66
2.9.2 The Dual Space of a Normed Space 69
2.9.3 Completion of a Normed Space 69
3 Tensor Algebras and Multivectors 71
3.1 The Algebra of Tensors on a Vector Space 71
3.1.1 The Direct Sum of Vector Spaces 71
3.1.2 Tensors on a Vector Space 72
3.1.3 The Tensor Algebra 72
3.1.4 The Operation of Contraction 73
3.2 The Contravariant and Covariant Subalgebras 74
3.3 Exterior Algebra 76
3.3.1 Introduction 76
3.3.2 The Exterior Product 77
3.4 Multivectors and Oriented Affine Simplexes 83
3.5 The Faces of an Oriented Affine Simplex 85
3.6 Multicovectors or r-Forms 86
3.7 The Physical Meaning of r-Forms 89
3.8 Some Useful Isomorphisms 90
Part Two: Differential Geometry 93
4 Differentiable Manifolds 95
4.1 Introduction 95
4.2 Some Topological Notions 97
4.3 Topological Manifolds 99
4.4 Differentiable Manifolds 100
4.5 Differentiability 101
4.6 Tangent Vectors 103
4.7 The Tangent Bundle 108
4.8 The Lie Bracket 110
4.9 The Differential of a Map 115
4.9.1 Push-forwards 118
4.10 Immersions, Embeddings, Submanifolds 119
4.10.1 Linear Maps of Vector Spaces 119
4.10.2 The Inverse Function Theorem of Calculus 120
4.10.3 Implications for Differentiable Manifolds 120
4.11 The Cotangent Bundle 123
4.12 Tensor Bundles 124
4.13 Pull-backs 126
4.14 Exterior Differentiation of Differential Forms 128
4.15 Some Properties of the Exterior Derivative 131
4.16 Riemannian Manifolds 132
4.17 Manifolds with Boundary 133
4.18 Differential Spaces and Generalized Bodies 134
4.18.1 Differential Spaces 135
4.18.2. Mechanics of Differential Spaces 137
5 Lie Derivatives, Lie Groups, Lie Algebras 140
5.1 Introduction 140
5.2 The Fundamental Theorem of the Theory of ODEs 141
5.3 The Flow of a Vector Field 142
5.4 One-parameter Groups of Transformations Generated by Flows 143
5.5 Time-Dependent Vector Fields 144
5.6 The Lie Derivative 145
5.6.1 The Lie Derivative of a Scalar 146
5.6.2 The Lie Derivative of a Vector Field 146
5.6.3 The Lie Derivative of a One-Form 147
5.6.4 The Lie Derivative of Arbitrary Tensor Fields 147
5.6.5 The Lie Derivative in Components 148
5.6.6 The Nonautonomous Lie Derivative 149
5.7 Invariant Tensor Fields 149
5.8 Lie Groups 152
5.9 Group Actions 154
5.10 One-Parameter Subgroups 156
5.11 Left-and Right-Invariant Vector Fields on a Lie Group 157
5.12 The Lie Algebra of a Lie Group 159
5.12.1 The Structure Constants of a Lie Group 162
5.13 Down-to-Earth Considerations 163
5.14 The Adjoint Representation 167
6 Integration and Fluxes 169
6.1 Integration of Forms in Affine Spaces 169
6.1.1 Simplicial Complexes 169
6.1.2 The Riemann Integral of an r-Form 171
6.1.3 Simplicial Chains and the Boundary Operator 171
6.1.4 Integration of n-Forms in Rn 173
6.2 Integration of Forms on Chains in Manifolds 174
6.2.1 Singular Chains in a Manifold 174
6.2.2 Integration of Forms over Chains in a Manifold 176
6.2.3 Stokes\u2019 Theorem for Chains 177
6.3 Integration of Forms on Oriented Manifolds 180
6.3.1 Partitions of Unity 180
6.3.2 Definition of the Integral 182
6.3.3 Stokes\u2019 Theorem 183
6.4 Fluxes in Continuum Physics 183
6.4.1 Extensive-Property Densities 184
6.4.2 Balance Laws, Flux Densities and Sources 185
6.4.3 Flux Forms and Cauchy\u2019s Formula 186
6.4.4 Differential version of the Balance Law 187
6.5 General Bodies and Whitney's Geometric Integration Theory 188
6.5.1 Polyhedral Chains 189
6.5.2 The Flat Norm 190
6.5.3 Flat Cochains 193
6.5.4 Significance for Continuum Mechanics 194
6.5.5 Cochains and Differential Forms 195
6.5.6 Continuous Chains 195
6.5.7 Balance Laws and Virtual Work in Terms of Flat Chains 197
6.5.8 The Sharp Norm 198
6.5.9 Fields on Chains as Chains 199
Part Three: Further Topics 203
7 Fibre Bundles 205
7.1 Product Bundles 205
7.2 Trivial Bundles 207
7.3 General Fibre Bundles 210
7.3.1 Adapted Coordinate Systems 211
7.4 The Fundamental Existence Theorem 212
7.5 The Tangent and Cotangent Bundles 213
7.6 The Bundle of Linear Frames 215
7.7 Principal Bundles 217
7.8 Associated Bundles 220
7.9 Fibre-Bundle Morphisms 223
7.10 Cross Sections 226
7.11 Iterated Fibre Bundles 228
7.11.1 The Tangent Bundle of a Fibre Bundle 229
7.11.2 The Iterated Tangent Bundle 231
8 Inhomogeneity Theory 234
8.1 Material Uniformity 234
8.1.1 Material Response 234
8.1.2 Germ Locality 235
8.1.3 Jet Locality 236
8.1.4 First and Second-Grade Materials 239
8.1.5 Material Isomorphism 240
8.1.6 Material Symmetries and the Nonuniqueness of Material Isomorphisms 245
8.2 The material Lie groupoid 247
8.2.1 Introduction 247
8.2.2 Groupoids 247
8.3 The Material Principal Bundle 251
8.3.1 Introduction 251
8.3.2 Fromthe Groupoid to the Principal Bundle 251
8.4 Flatness and Homogeneity 253
8.5 Distributions and the Theorem of Frobenius 254
8.6 Jet Bundles and Differential Equations 256
8.6.1 Jets of Sections 256
8.6.2 Jet Bundles 257
8.6.3 Differential Equations 258
9 Connection, Curvature, Torsion 259
9.1 Ehresmann Connection 259
9.1.1 Definition 259
9.1.2 Parallel Transport Along a Curve 261
9.2 Connections in Principal Bundles 262
9.2.1 Definition 262
9.2.2 Existence 263
9.2.3 Curvature 264
9.2.4 Cartan\u2019s Structural Equation 265
9.2.5 Bianchi Identities 265
9.3 Linear Connections 266
9.3.1 The Canonical 1-Form 266
9.3.2 The Christoffel Symbols 267
9.3.3 Parallel Transport and the Covariant Derivative 268
9.3.4 Curvature and Torsion 269
9.4 G-Connections 272
9.4.1 Reduction of Principal Bundles 273
9.4.2 G-structures 275
9.4.3 Local Flatness 276
9.5 Riemannian Connections 278
9.6 Material Homogeneity 279
9.6.1 Uniformity and Homogeneity 279
9.6.2 Homogeneity in Terms of a Material Connection 280
9.6.3 Homogeneity in Terms of a Material G-structure 282
9.6.4 Homogeneity in Terms of the Material Groupoid 283
9.7 Homogeneity Criteria 284
9.7.1 Solids 284
9.7.2 Fluids 286
9.7.3 Fluid Crystals 287
Appendix A: A Primer in Continuum Mechanics 288
A.1 Bodies and Configurations 288
A.2 Observers and Frames 289
A.3 Strain 290
A.4 Volume and Area 294
A.5 The Material Time Derivative 295
A.6 Change of Reference 296
A.7 Transport Theorems 298
A.8 The General Balance Equation 299
A.9 The Fundamental Balance Equations of Continuum Mechanics 303
A.10 A Modicum of Constitutive Theory 309
Index 320