The Geometrical Language of Continuum

This book presents the fundamental concepts of modern differential geometry within the framework of continuum mechanics. It is divided into three parts of roughly equal length. The book opens with a motivational chapter to impress upon the reader that differential geometry is indeed the natural language of continuum mechanics or, better still, that the latter is a prime example of the application and materialization of the former. In the second part, the fundamental notions of differential geometry are presented with rigor using a writing style that is as informal as possible. Differentiable manifolds, tangent bundles, exterior derivatives, Lie derivatives, and Lie groups are illustrated in terms of their mechanical interpretations. The third part includes the theory of fiber bundles, G-structures, and groupoids, which are applicable to bodies with internal structure and to the description of material inhomogeneity. The abstract notions of differential geometry are thus illuminated by practical and intuitively meaningful engineering applications.

Preface	p. xi
Motivation and Background	p. 1
The Case for Differential Geometry	p. 3
Classical Space-Time and Fibre Bundles	p. 4
Configuration Manifolds and Their Tangent and Cotangent Spaces	p. 10
The Infinite-dimensional Case	p. 13
Elasticity	p. 22
Material or Configurational Forces	p. 23
Vector and Affine Spaces	p. 24
Vector Spaces: Definition and Examples	p. 24
Linear Independence and Dimension	p. 26
Change of Basis and the Summation Convention	p. 30
The Dual Space	p. 31
Linear Operators and the Tensor Product	p. 34
Isomorphisms and Iterated Dual	p. 36
Inner-product Spaces	p. 41
Affine Spaces	p. 46
Banach Spaces	p. 52
Tensor Algebras and Multivectors	p. 57
The Algebra of Tensors on a Vector Space	p. 57
The Contravariant and Covariant Subalgebras	p. 60
Exterior Algebra	p. 62
Multivectors and Oriented Affine Simplexes	p. 69
The Faces of an Oriented Affine Simplex	p. 71
Multicovectors or r-Forms	p. 72
The Physical Meaning of r-Forms	p. 75
Some Useful Isomorphisms	p. 76
Differential Geometry	p. 79
Differentiable Manifolds	p. 81
Introduction	p. 81
Some Topological Notions	p. 83
Topological Manifolds	p. 85
Differentiable Manifolds	p. 86
Differentiability	p. 87
Tangent Vectors	p. 89
The Tangent Bundle	p. 94
The Lie Bracket	p. 96
The Differential of a Map	p. 101
Immersions, Embeddings, Submanifolds	p. 105
The Cotangent Bundle	p. 109
Tensor Bundles	p. 110
Pull-backs	p. 112
Exterior Differentiation of Differential Forms	p. 114
Some Properties of the Exterior Derivative	p. 117
Riemannian Manifolds	p. 118
Manifolds with Boundary	p. 119
Differential Spaces and Generalized Bodies	p. 120
Lie Derivatives, Lie Groups, Lie Algebras	p. 126
Introduction	p. 126
The Fundamental Theorem of the Theory of ODEs	p. 127
The Flow of a Vector Field	p. 128
One-parameter Groups of Transformations Generated by Flows	p. 129
Time-Dependent Vector Fields	p. 130
The Lie Derivative	p. 131
Invariant Tensor Fields	p. 135
Lie Groups	p. 138
Group Actions	p. 140
"One-Parameter Subgroups	p. 142
Left-and Right-Invariant Vector Fields on a Lie Group	p. 143
The Lie Algebra of a Lie Group	p. 145
Down-to-Earth Considerations	p. 149
The Adjoint Representation	p. 153
Integration and Fluxes	p. 155
Integration of Forms in Affine Spaces	p. 155
Integration of Forms on Chains in Manifolds	p. 160
Integration of Forms on Oriented Manifolds	p. 166
Fluxes in Continuum Physics	p. 169
General Bodies and Whitney's Geometric Integration Theory	p. 174
Further Topics	p. 189
Fibre Bundles	p. 191
Product Bundles	p. 191
Trivial Bundles	p. 193
General Fibre Bundles	p. 196
The Fundamental Existence Theorem	p. 198
The Tangent and Cotangent Bundles	p. 199
The Bundle of Linear Frames	p. 201
Principal Bundles	p. 203
Associated Bundles	p. 206
Fibre-Bundle Morphisms	p. 209
Cross Sections	p. 212
Iterated Fibre Bundles	p. 214
Inhomogeneity Theory	p. 220
Material Uniformity	p. 220
The Material Lie groupoid	p. 233
The Material Principal Bundle	p. 237
Flatness and Homogeneity	p. 239
Distributions and the Theorem of Frobenius	p. 240
JetBundles-and -Differential Equations	p. 242
Connection, Curvature, Torsion	p. 245
Ehresmann Connection	p. 245
Connections in Principal Bundles	p. 248
Linear Connections	p. 252
G-Connections	p. 258
Riemannian Connections	p. 264
Material Homogeneity	p. 265
Homogeneity Criteria	p. 270
A Primer in Continuum Mechanics	p. 274
Bodies and Configurations	p. 274
Observers and Frames	p. 275
Strain	p. 276
Volume and Area	p. 280
The Material Time Derivative	p. 281
Change of Reference	p. 282
Transport Theorems	p. 284
The General Balance Equation	p. 285
The Fundamental Balance Equations of Continuum Mechanics	p. 289
A Modicum of Constitutive Theory	p. 295
Index	p. 306
Table of Contents provided by Ingram. All Rights Reserved.

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The Geometrical Language of Continuum Mechanics

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