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The Geometrical Language of Continuum Mechanics - Marcelo Epstein

The Geometrical Language of Continuum Mechanics


Published: 26th July 2010
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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.

'The book is suitable for graduate students in the field of continuum mechanics who seek an introduction to the fundamentals of modern differential geometry and its applications in theoretical continuum mechanics. It will also be useful to researchers in the field of mechanics who look for overviews of the more specialized topics. The book is written in a very enjoyable and literary style in which the rich and picturesque language sheds light on the mathematics.' Mathematical Reviews
'I warmly recommend this book to all interested in differential geometry and mechanics.' Zentralblatt MATH

Prefacep. xi
Motivation and Backgroundp. 1
The Case for Differential Geometryp. 3
Classical Space-Time and Fibre Bundlesp. 4
Configuration Manifolds and Their Tangent and Cotangent Spacesp. 10
The Infinite-dimensional Casep. 13
Elasticityp. 22
Material or Configurational Forcesp. 23
Vector and Affine Spacesp. 24
Vector Spaces: Definition and Examplesp. 24
Linear Independence and Dimensionp. 26
Change of Basis and the Summation Conventionp. 30
The Dual Spacep. 31
Linear Operators and the Tensor Productp. 34
Isomorphisms and Iterated Dualp. 36
Inner-product Spacesp. 41
Affine Spacesp. 46
Banach Spacesp. 52
Tensor Algebras and Multivectorsp. 57
The Algebra of Tensors on a Vector Spacep. 57
The Contravariant and Covariant Subalgebrasp. 60
Exterior Algebrap. 62
Multivectors and Oriented Affine Simplexesp. 69
The Faces of an Oriented Affine Simplexp. 71
Multicovectors or r-Formsp. 72
The Physical Meaning of r-Formsp. 75
Some Useful Isomorphismsp. 76
Differential Geometryp. 79
Differentiable Manifoldsp. 81
Introductionp. 81
Some Topological Notionsp. 83
Topological Manifoldsp. 85
Differentiable Manifoldsp. 86
Differentiabilityp. 87
Tangent Vectorsp. 89
The Tangent Bundlep. 94
The Lie Bracketp. 96
The Differential of a Mapp. 101
Immersions, Embeddings, Submanifoldsp. 105
The Cotangent Bundlep. 109
Tensor Bundlesp. 110
Pull-backsp. 112
Exterior Differentiation of Differential Formsp. 114
Some Properties of the Exterior Derivativep. 117
Riemannian Manifoldsp. 118
Manifolds with Boundaryp. 119
Differential Spaces and Generalized Bodiesp. 120
Lie Derivatives, Lie Groups, Lie Algebrasp. 126
Introductionp. 126
The Fundamental Theorem of the Theory of ODEsp. 127
The Flow of a Vector Fieldp. 128
One-parameter Groups of Transformations Generated by Flowsp. 129
Time-Dependent Vector Fieldsp. 130
The Lie Derivativep. 131
Invariant Tensor Fieldsp. 135
Lie Groupsp. 138
Group Actionsp. 140
"One-Parameter Subgroupsp. 142
Left-and Right-Invariant Vector Fields on a Lie Groupp. 143
The Lie Algebra of a Lie Groupp. 145
Down-to-Earth Considerationsp. 149
The Adjoint Representationp. 153
Integration and Fluxesp. 155
Integration of Forms in Affine Spacesp. 155
Integration of Forms on Chains in Manifoldsp. 160
Integration of Forms on Oriented Manifoldsp. 166
Fluxes in Continuum Physicsp. 169
General Bodies and Whitney's Geometric Integration Theoryp. 174
Further Topicsp. 189
Fibre Bundlesp. 191
Product Bundlesp. 191
Trivial Bundlesp. 193
General Fibre Bundlesp. 196
The Fundamental Existence Theoremp. 198
The Tangent and Cotangent Bundlesp. 199
The Bundle of Linear Framesp. 201
Principal Bundlesp. 203
Associated Bundlesp. 206
Fibre-Bundle Morphismsp. 209
Cross Sectionsp. 212
Iterated Fibre Bundlesp. 214
Inhomogeneity Theoryp. 220
Material Uniformityp. 220
The Material Lie groupoidp. 233
The Material Principal Bundlep. 237
Flatness and Homogeneityp. 239
Distributions and the Theorem of Frobeniusp. 240
JetBundles-and -Differential Equationsp. 242
Connection, Curvature, Torsionp. 245
Ehresmann Connectionp. 245
Connections in Principal Bundlesp. 248
Linear Connectionsp. 252
G-Connectionsp. 258
Riemannian Connectionsp. 264
Material Homogeneityp. 265
Homogeneity Criteriap. 270
A Primer in Continuum Mechanicsp. 274
Bodies and Configurationsp. 274
Observers and Framesp. 275
Strainp. 276
Volume and Areap. 280
The Material Time Derivativep. 281
Change of Referencep. 282
Transport Theoremsp. 284
The General Balance Equationp. 285
The Fundamental Balance Equations of Continuum Mechanicsp. 289
A Modicum of Constitutive Theoryp. 295
Indexp. 306
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521198554
ISBN-10: 0521198550
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 324
Published: 26th July 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 26.0 x 18.5  x 2.4
Weight (kg): 0.82