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Geometrical Language of Continuum Mechanics [Hardback]

(University of Calgary)
  • Formāts: Hardback, 324 pages, height x width x depth: 260x185x24 mm, weight: 820 g, Worked examples or Exercises; 39 Line drawings, unspecified
  • Izdošanas datums: 26-Jul-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521198550
  • ISBN-13: 9780521198554
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  • Cena: 114,54 €
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Geometrical Language of Continuum MechanicsPalielināt
  • Formāts: Hardback, 324 pages, height x width x depth: 260x185x24 mm, weight: 820 g, Worked examples or Exercises; 39 Line drawings, unspecified
  • Izdošanas datums: 26-Jul-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521198550
  • ISBN-13: 9780521198554
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521198554.html
Keywords:
"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"--

Provided by publisher.

Recenzijas

'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

Papildus informācija

Presents the fundamental concepts of modern differential geometry within the framework of continuum mechanics.
Preface xi
PART 1 MOTIVATION AND BACKGROUND
1(78)
1 The Case for Differential Geometry
3(21)
1.1 Classical Space-Time and Fibre Bundles
4(6)
1.2 Configuration Manifolds and Their Tangent and Cotangent Spaces
10(3)
1.3 The Infinite-dimensional Case
13(9)
1.4 Elasticity
22(1)
1.5 Material or Configurational Forces
23(1)
2 Vector and Affine Spaces
24(33)
2.1 Vector Spaces: Definition and Examples
24(2)
2.2 Linear Independence and Dimension
26(4)
2.3 Change of Basis and the Summation Convention
30(1)
2.4 The Dual Space
31(3)
2.5 Linear Operators and the Tensor Product
34(2)
2.6 Isomorphisms and Iterated Dual
36(5)
2.7 Inner-product Spaces
41(5)
2.8 Affine Spaces
46(6)
2.9 Banach Spaces
52(5)
3 Tensor Algebras and Multivectors
57(22)
3.1 The Algebra of Tensors on a Vector Space
57(3)
3.2 The Contravariant and Covariant Subalgebras
60(2)
3.3 Exterior Algebra
62(7)
3.4 Multivectors and Oriented Affine Simplexes
69(2)
3.5 The Faces of an Oriented Affine Simplex
71(1)
3.6 Multicovectors or r-Forms
72(3)
3.7 The Physical Meaning of r-Forms
75(1)
3.8 Some Useful Isomorphisms
76(3)
PART 2 DIFFERENTIAL GEOMETRY
79(110)
4 Differentiable Manifolds
81(45)
4.1 Introduction
81(2)
4.2 Some Topological Notions
83(2)
4.3 Topological Manifolds
85(1)
4.4 Differentiable Manifolds
86(1)
4.5 Differentiability
87(2)
4.6 Tangent Vectors
89(5)
4.7 The Tangent Bundle
94(2)
4.8 The Lie Bracket
96(5)
4.9 The Differential of a Map
101(4)
4.10 Immersions, Embeddings, Submanifolds
105(4)
4.11 The Cotangent Bundle
109(1)
4.12 Tensor Bundles
110(2)
4.13 Pull-backs
112(2)
4.14 Exterior Differentiation of Differential Forms
114(3)
4.15 Some Properties of the Exterior Derivative
117(1)
4.16 Riemannian Manifolds
118(1)
4.17 Manifolds with Boundary
119(1)
4.18 Differential Spaces and Generalized Bodies
120(6)
5 Lie Derivatives, Lie Groups, Lie Algebras
126(29)
5.1 Introduction
126(1)
5.2 The Fundamental Theorem of the Theory of ODEs
127(1)
5.3 The Flow of a Vector Field
128(1)
5.4 One-parameter Groups of Transformations Generated by Flows
129(1)
5.5 Time-Dependent Vector Fields
130(1)
5.6 The Lie Derivative
131(4)
5.7 Invariant Tensor Fields
135(3)
5.8 Lie Groups
138(2)
5.9 Group Actions
140(2)
5.10 One-Parameter Subgroups
142(1)
5.11 Left- and Right-Invariant Vector Fields on a Lie Group
143(2)
5.12 The Lie Algebra of a Lie Group
145(4)
5.13 Down-to-Earth Considerations
149(4)
5.14 The Adjoint Representation
153(2)
6 Integration and Fluxes
155(34)
6.1 Integration of Forms in Affine Spaces
155(5)
6.2 Integration of Forms on Chains in Manifolds
160(6)
6.3 Integration of Forms on Oriented Manifolds
166(3)
6.4 Fluxes in Continuum Physics
169(5)
6.5 General Bodies and Whitney's Geometric Integration Theory
174(15)
PART 3 FURTHER TOPICS
189(85)
7 Fibre Bundles
191(29)
7.1 Product Bundles
191(2)
7.2 Trivial Bundles
193(3)
7.3 General Fibre Bundles
196(2)
7.4 The Fundamental Existence Theorem
198(1)
7.5 The Tangent and Cotangent Bundles
199(2)
7.6 The Bundle of Linear Frames
201(2)
7.7 Principal Bundles
203(3)
7.8 Associated Bundles
206(3)
7.9 Fibre-Bundle Morphisms
209(3)
7.10 Cross Sections
212(2)
7.11 Iterated Fibre Bundles
214(6)
8 Inhomogeneity Theory
220(25)
8.1 Material Uniformity
220(13)
8.2 The Material Lie groupoid
233(4)
8.3 The Material Principal Bundle
237(2)
8.4 Flatness and Homogeneity
239(1)
8.5 Distributions and the Theorem of Frobenius
240(2)
8.6 Jet Bundles and Differential Equations
242(3)
9 Connection, Curvature, Torsion
245(29)
9.1 Ehresmann Connection
245(3)
9.2 Connections in Principal Bundles
248(4)
9.3 Linear Connections
252(6)
9.4 G-Connections
258(6)
9.5 Riemannian Connections
264(1)
9.6 Material Homogeneity
265(5)
9.7 Homogeneity Criteria
270(4)
APPENDIX A A PRIMER IN CONTINUUM MECHANICS
274(32)
A.1 Bodies and Configurations
274(1)
A.2 Observers and Frames
275(1)
A.3 Strain
276(4)
A.4 Volume and Area
280(1)
A.5 The Material Time Derivative
281(1)
A.6 Change of Reference
282(2)
A.7 Transport Theorems
284(1)
A.8 The General Balance Equation
285(4)
A.9 The Fundamental Balance Equations of Continuum Mechanics
289(6)
A.10 A Modicum of Constitutive Theory
295(11)
Index 306
Marcelo Epstein is currently a Professor of Mechanical Engineering at the University of Calgary, Canada. His main research has centered around the various aspects of modern continuum mechanics and its applications. A secondary related area of interest is biomechanics. He is a Fellow of the American Academy of Mechanics, recipient of the Cancam prize and University Professor of Rational Mechanics. He is also adjunct Professor in the Faculties of Humanities and Kinesiology at the University of Calgary.