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El. knyga: Differential Geometry: Basic Notions and Physical Examples

  • Formatas: PDF+DRM
  • Serija: Mathematical Engineering
  • Išleidimo metai: 02-Jul-2014
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319069203
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Differential Geometry: Basic Notions and Physical ExamplesDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Mathematical Engineering
  • Išleidimo metai: 02-Jul-2014
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319069203
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Differential Geometry offers a concise introduction to some basic notions of modern differential geometry and their applications to solid mechanics and physics.

Concepts such as manifolds, groups, fibre bundles and groupoids are first introduced within a purely topological framework. They are shown to be relevant to the description of space-time, configuration spaces of mechanical systems, symmetries in general, microstructure and local and distant symmetries of the constitutive response of continuous media.

Once these ideas have been grasped at the topological level, the differential structure needed for the description of physical fields is introduced in terms of differentiable manifolds and principal frame bundles. These mathematical concepts are then illustrated with examples from continuum kinematics, Lagrangian and Hamiltonian mechanics, Cauchy fluxes and dislocation theory.

This book will be useful for researchers and graduate students in science and engineering.

Recenzijos

The book under review has grown out of lecture notes for a mini-course given at a workshop on differential geometry and continuum mechanics at the International Centre for Mathematical Sciences in 2013. addressing researchers and engineers in particular, Epsteins book provides a quick way to appreciate modern differential geometry and topology and get to their essential ideas and usefulness. Surely, Epstein manages to give the reader a motivation to delve into the deep waters of these two fields. (Theophanes Grammenos, Mathematical Reviews, June, 2015)

This book is based on a short course on Differential Geometry and Continuum Mechanics given by Marcelo Epstein at the International Centre of Mathematical Sciences in Edinburgh in June 2013. The course provided a guided tour of differential geometry for researchers and graduate students in science and engineering many of whom had a particular interest in continuum mechanics. this book is a gold mine of aesthetically pleasing mathematical ideas, the presentation of which is highly inspirational. (P. N. Ruane, MAA Reviews, December, 2014)

1 Topological Constructs
1(24)
1.1 Topological Spaces
1(2)
1.1.1 Definition
1(1)
1.1.2 Nearness and Continuity
1(1)
1.1.3 Some Terminology
2(1)
1.2 Topological Manifolds
3(4)
1.2.1 Definition
3(1)
1.2.2 Maps and Their Representations
4(2)
1.2.3 Topological Manifolds with Boundary
6(1)
1.3 Topological Groups
7(5)
1.3.1 Definition
7(1)
1.3.2 Group Actions
8(4)
1.4 Topological Fibre Bundles
12(7)
1.4.1 Product Bundles
12(1)
1.4.2 Fibre Bundles
13(2)
1.4.3 Principal Bundles
15(2)
1.4.4 Cross Sections
17(2)
1.5 Topological Groupoids
19(6)
1.5.1 Definition
19(3)
1.5.2 From Groupoids to Principal Bundles
22(1)
References
23(2)
2 Physical Illustrations
25(12)
2.1 Manifolds
25(2)
2.1.1 The Configuration Space of a Mechanical System
25(1)
2.1.2 The Configuration Space of a Deformable Body
26(1)
2.2 Groups
27(1)
2.2.1 Local Symmetries of Constitutive Laws
27(1)
2.3 Fibre Bundles
28(3)
2.3.1 Space-Time
28(2)
2.3.2 Microstructure
30(1)
2.4 Groupoids
31(6)
2.4.1 Material Uniformity
31(4)
References
35(2)
3 Differential Constructs
37(76)
3.1 Differentiable Manifolds
37(2)
3.1.1 Definition
37(1)
3.1.2 Orientable Manifolds
38(1)
3.1.3 Differentiable Maps
38(1)
3.1.4 Smooth Constructs
39(1)
3.2 The Tangent Bundle of a Manifold
39(8)
3.2.1 Curves Through a Point
39(2)
3.2.2 Tangent Vectors
41(1)
3.2.3 The Tangent Space at a Point
42(1)
3.2.4 The Tangent Bundle
43(2)
3.2.5 The Differential of a Map
45(2)
3.3 Vector Fields and Flows
47(5)
3.3.1 Vector Fields
47(1)
3.3.2 The Lie Bracket
47(3)
3.3.3 The Flow of a Vector Field
50(1)
3.3.4 One-Parameter Groups of Transformations Generated by Flows
51(1)
3.4 The Principal Frame Bundle and Its Associated Bundles
52(19)
3.4.1 Definition
52(2)
3.4.2 Associated Bundles
54(2)
3.4.3 The Cotangent Bundle
56(2)
3.4.4 Exterior Algebra
58(8)
3.4.5 Interior Multiplication
66(1)
3.4.6 Non-canonical Isomorphisms
67(2)
3.4.7 Differential r-Forms
69(2)
3.5 Calculus of Differential Forms
71(10)
3.5.1 The Exterior Derivative of Forms
71(3)
3.5.2 Integration
74(6)
3.5.3 Currents of de Rham
80(1)
3.6 Lie Derivatives and Lie Groups
81(7)
3.6.1 Intuitive Considerations
81(1)
3.6.2 Relation to the Lie Bracket
82(2)
3.6.3 The Lie Derivative of Tensors
84(1)
3.6.4 One-Parameter Subgroups of a Lie Group
85(1)
3.6.5 The Lie Algebra of a Lie Group
86(2)
3.7 Distributions and Connections
88(25)
3.7.1 Distributions
88(1)
3.7.2 Integral Manifolds of a Distribution
89(1)
3.7.3 Involutivity and the Theorem of Frobenius
90(1)
3.7.4 The Idea of a Connection
91(3)
3.7.5 Ehresmann Connections
94(2)
3.7.6 Parallel Transport
96(2)
3.7.7 The Curvature of an Ehresmann Connection
98(2)
3.7.8 Principal-Bundle Connections
100(2)
3.7.9 Linear Connections
102(5)
3.7.10 Riemannian Connections
107(4)
References
111(2)
4 Physical Illustrations
113(24)
4.1 Mechanics in the Configuration Space
113(4)
4.1.1 Virtual Displacements and Velocity Vectors
113(1)
4.1.2 Force Fields
113(2)
4.1.3 The Lagrangian Function
115(1)
4.1.4 Lagrange's Postulate and the Equations of Motion
116(1)
4.2 Hamiltonian Mechanics
117(4)
4.2.1 Symplectic Vector Spaces
118(1)
4.2.2 Symplectic Manifolds
118(1)
4.2.3 Hamiltonian Systems
119(2)
4.3 Fluxes in Continuum Physics
121(4)
4.3.1 Extensive-Property Densities
121(1)
4.3.2 Balance Laws, Flux Densities and Sources
122(1)
4.3.3 Flux Forms and Cauchy's Formula
123(1)
4.3.4 Differential Expression of the Balance Law
124(1)
4.4 Microstructure
125(3)
4.4.1 Kinematics of a Cosserat Body
125(3)
4.5 Dislocations
128(9)
4.5.1 An Intuitive Picture
128(4)
4.5.2 Distant Parallelism
132(1)
4.5.3 Bravais Planes and Differential Forms
133(1)
4.5.4 Singular Dislocations and de Rham Currents
134(1)
References
135(2)
Index 137
Marcelo Epstein is Professor of Mechanical Engineering at the University of Calgary, where he has held the title of University Professor of Rational Mechanics. A Fellow of the American Academy of Mechanics and a recipient of the CANCAM award, he has published extensively in the field of the foundations and applications of continuum mechanics. He is the author or co-author of four books on various aspects of applied differential geometry, continuum mechanics and biomechanics.
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