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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010297673 | QA808.2 E67 2010 | Open Access Book | Book | Searching... |
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Summary
Summary
Epstein presents the fundamental concepts of modern differential geometry within the framework of continuum mechanics. 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 materialisation 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.
Reviews 1
Choice Review
The Geometrical Language of Continuum Mechanics brings a fresh quality to this subject by blending differential topology with tensor analysis. Although initially the book appears to be nonmethodical, it is organized into three parts. Epstein (mechanical and manufacturing engineering, Univ. of Calgary, Canada) starts with the basic concepts of vector and affine spaces in part 1. The book migrates smoothly in part 2 to a discussion of differential manifolds and Lie derivatives, groups, and algebras. Part 3 includes chapters titled "Fibre Bundles," "Inhomogeneity Theory," and "Connection, Curvature, Torsion." In this section, Epstein introduces innovatively the concept of fiber bundles coupled with an elucidative mapping analysis, which are useful research topics in bodies with microstructure. The book is mathematically rigorous. However, the author is commended for making the materials accessible to the reader by including in the first chapter excellent introductory background topics, which are essential for unraveling interconnected abstractions in subsequent chapters. The book contains well over 100 exercises scattered within various sections; however, several exercises are worded nebulously. Furthermore, including Mathematica in order to explore parametrically topology and differential geometry cases relevant to continuum mechanics could have enhanced the book. Summing Up: Recommended. Graduate students and researchers/faculty. R. N. Laoulache University of Massachusetts Dartmouth
Table of Contents
| Preface | p. xi |
| Part 1 Motivation and Background | p. 1 |
| 1 The Case for Differential Geometry | p. 3 |
| 1.1 Classical Space-Time and Fibre Bundles | p. 4 |
| 1.2 Configuration Manifolds and Their Tangent and Cotangent Spaces | p. 10 |
| 1.3 The Infinite-dimensional Case | p. 13 |
| 1.4 Elasticity | p. 22 |
| 1.5 Material or Configurational Forces | p. 23 |
| 2 Vector and Affine Spaces | p. 24 |
| 2.1 Vector Spaces: Definition and Examples | p. 24 |
| 2.2 Linear Independence and Dimension | p. 26 |
| 2.3 Change of Basis and the Summation Convention | p. 30 |
| 2.4 The Dual Space | p. 31 |
| 2.5 Linear Operators and the Tensor Product | p. 34 |
| 2.6 Isomorphisms and Iterated Dual | p. 36 |
| 2.7 Inner-product Spaces | p. 41 |
| 2.8 Affine Spaces | p. 46 |
| 2.9 Banach Spaces | p. 52 |
| 3 Tensor Algebras and Multivectors | p. 57 |
| 3.1 The Algebra of Tensors on a Vector Space | p. 57 |
| 3.2 The Contravariant and Covariant Subalgebras | p. 60 |
| 3.3 Exterior Algebra | p. 62 |
| 3.4 Multivectors and Oriented Affine Simplexes | p. 69 |
| 3.5 The Faces of an Oriented Affine Simplex | p. 71 |
| 3.6 Multicovectors or r-Forms | p. 72 |
| 3.7 The Physical Meaning of r-Forms | p. 75 |
| 3.8 Some Useful Isomorphisms | p. 76 |
| Part 2 Differential Geometry | p. 79 |
| 4 Differentiable Manifolds | p. 81 |
| 4.1 Introduction | p. 81 |
| 4.2 Some Topological Notions | p. 83 |
| 4.3 Topological Manifolds | p. 85 |
| 4.4 Differentiable Manifolds | p. 86 |
| 4.5 Differentiability | p. 87 |
| 4.6 Tangent Vectors | p. 89 |
| 4.7 The Tangent Bundle | p. 94 |
| 4.8 The Lie Bracket | p. 96 |
| 4.9 The Differential of a Map | p. 101 |
| 4.10 Immersions, Embeddings, Submanifolds | p. 105 |
| 4.11 The Cotangent Bundle | p. 109 |
| 4.12 Tensor Bundles | p. 110 |
| 4.13 Pull-backs | p. 112 |
| 4.14 Exterior Differentiation of Differential Forms | p. 114 |
| 4.15 Some Properties of the Exterior Derivative | p. 117 |
| 4.16 Riemannian Manifolds | p. 118 |
| 4.17 Manifolds with Boundary | p. 119 |
| 4.18 Differential Spaces and Generalized Bodies | p. 120 |
| 5 Lie Derivatives, Lie Groups, Lie Algebras | p. 126 |
| 5.1 Introduction | p. 126 |
| 5.2 The Fundamental Theorem of the Theory of ODEs | p. 127 |
| 5.3 The Flow of a Vector Field | p. 128 |
| 5.4 One-parameter Groups of Transformations Generated by Flows | p. 129 |
| 5.5 Time-Dependent Vector Fields | p. 130 |
| 5.6 The Lie Derivative | p. 131 |
| 5.7 Invariant Tensor Fields | p. 135 |
| 5.8 Lie Groups | p. 138 |
| 5.9 Group Actions | p. 140 |
| 5.10 "One-Parameter Subgroups | p. 142 |
| 5.11 Left-and Right-Invariant Vector Fields on a Lie Group | p. 143 |
| 5.12 The Lie Algebra of a Lie Group | p. 145 |
| 5.13 Down-to-Earth Considerations | p. 149 |
| 5.14 The Adjoint Representation | p. 153 |
| 6 Integration and Fluxes | p. 155 |
| 6.1 Integration of Forms in Affine Spaces | p. 155 |
| 6.2 Integration of Forms on Chains in Manifolds | p. 160 |
| 6.3 Integration of Forms on Oriented Manifolds | p. 166 |
| 6.4 Fluxes in Continuum Physics | p. 169 |
| 6.5 General Bodies and Whitney's Geometric Integration Theory | p. 174 |
| Part 3 Further Topics | p. 189 |
| 7 Fibre Bundles | p. 191 |
| 7.1 Product Bundles | p. 191 |
| 7.2 Trivial Bundles | p. 193 |
| 7.3 General Fibre Bundles | p. 196 |
| 7.4 The Fundamental Existence Theorem | p. 198 |
| 7.5 The Tangent and Cotangent Bundles | p. 199 |
| 7.6 The Bundle of Linear Frames | p. 201 |
| 7.7 Principal Bundles | p. 203 |
| 7.8 Associated Bundles | p. 206 |
| 7.9 Fibre-Bundle Morphisms | p. 209 |
| 7.10 Cross Sections | p. 212 |
| 7.11 Iterated Fibre Bundles | p. 214 |
| 8 Inhomogeneity Theory | p. 220 |
| 8.1 Material Uniformity | p. 220 |
| 8.2 The Material Lie groupoid | p. 233 |
| 8.3 The Material Principal Bundle | p. 237 |
| 8.4 Flatness and Homogeneity | p. 239 |
| 8.5 Distributions and the Theorem of Frobenius | p. 240 |
| 8.6 JetBundles-and -Differential Equations | p. 242 |
| 9 Connection, Curvature, Torsion | p. 245 |
| 9.1 Ehresmann Connection | p. 245 |
| 9.2 Connections in Principal Bundles | p. 248 |
| 9.3 Linear Connections | p. 252 |
| 9.4 G-Connections | p. 258 |
| 9.5 Riemannian Connections | p. 264 |
| 9.6 Material Homogeneity | p. 265 |
| 9.7 Homogeneity Criteria | p. 270 |
| Appendix A A Primer in Continuum Mechanics | p. 274 |
| A.1 Bodies and Configurations | p. 274 |
| A.2 Observers and Frames | p. 275 |
| A.3 Strain | p. 276 |
| A.4 Volume and Area | p. 280 |
| A.5 The Material Time Derivative | p. 281 |
| A.6 Change of Reference | p. 282 |
| A.7 Transport Theorems | p. 284 |
| A.8 The General Balance Equation | p. 285 |
| A.9 The Fundamental Balance Equations of Continuum Mechanics | p. 289 |
| A.10 A Modicum of Constitutive Theory | p. 295 |
| Index | p. 306 |
