Tensors and Manifolds: With Applications to Physics / Edition 2

Tensors and Manifolds: With Applications to Physics / Edition 2

by Robert H. Wasserman
     
 

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ISBN-10: 0198510594

ISBN-13: 9780198510598

Pub. Date: 07/08/2004

Publisher: Oxford University Press, USA

An advance undergraduate or beginning graduate textbook introducing the expanse of modern mathematics and its application in modern physics to students who are intending to go into pure mathematics, and may not otherwise be exposed to applications for a long time. It seeks to harmonize the two approaches that are variously termed classical, index or local on the one

Overview

An advance undergraduate or beginning graduate textbook introducing the expanse of modern mathematics and its application in modern physics to students who are intending to go into pure mathematics, and may not otherwise be exposed to applications for a long time. It seeks to harmonize the two approaches that are variously termed classical, index or local on the one hand and modern, intrinsic, or global on the other. The 1992 edition has been corrected, expanded, and clarified and a few problems have been added. Annotation ©2004 Book News, Inc., Portland, OR

Product Details

ISBN-13:
9780198510598
Publisher:
Oxford University Press, USA
Publication date:
07/08/2004
Edition description:
REV
Pages:
464
Product dimensions:
6.20(w) x 9.30(h) x 1.30(d)

Table of Contents

1Vector Spaces1
1.1Definitions, properties, and examples1
1.2Representation of vector spaces5
1.3Linear mappings6
1.4Representation of linear mappings8
2Multilinear Mappings and Dual Spaces11
2.1Vector spaces of linear mappings11
2.2Vector spaces of multilinear mappings15
2.3Nondegenerate bilinear functions19
2.4Orthogonal complements and the transpose of a linear mapping20
3Tensor Product Spaces25
3.1The tensor product of two finite-dimensional vector spaces25
3.2Generalizations, isomorphisms, and a characterization28
3.3Tensor products of infinite-dimensional vector spaces31
4Tensors34
4.1Definitions and alternative interpretations34
4.2The components of tensors36
4.3Mappings of the spaces V[superscript r subscript s]39
5Symmetric and Skew-Symmetric Tensors46
5.1Symmetry and skew-symmetry46
5.2The symmetric subspace of V[superscript 0 subscript s]48
5.3The skew-symmetric (alternating) subspace of V[superscript 0 subscript s]54
5.4Some special properties of S[superscript 2](V*) and [Lambda superscript 2](V*)59
6Exterior (Grassmann) Algebra71
6.1Tensor algebras71
6.2Definition and properties of the exterior product72
6.3Some more properties of the exterior product77
7The Tangent Map of Real Cartesian Spaces84
7.1Maps of real cartesian spaces84
7.2The tangent and cotangent spaces at a point of R[superscript n]88
7.3The tangent map96
8Topological Spaces102
8.1Definitions, properties, and examples102
8.2Continuous mappings106
9Differentiable Manifolds108
9.1Definitions and examples108
9.2Mappings of differentiable manifolds116
9.3The tangent and cotangent spaces at a point of M119
9.4Some properties of mappings125
10Submanifolds131
10.1Parametrized submanifolds131
10.2Differentiable varieties as submanifolds133
11Vector Fields, 1-Forms, and Other Tensor Fields136
11.1Vector fields136
11.21-Form fields143
11.3Tensor fields and differential forms146
11.4Mappings of tensor fields and differential forms149
12Differentiation and Intergration of Differential Forms153
12.1Exterior differentiation of differential forms153
12.2Integration of differential forms158
13The Flow and the Lie Derivative of A Vector Field168
13.1Integral curves and the flow of a vector field168
13.2Flow boxes (local flows) and complete vector fields172
13.3Coordinate vector fields176
13.4The Lie derivative178
14Integrability Conditions for Distributions and for Pfaffian Systems186
14.1Completely integrable distributions186
14.2Completely integrable Pfaffian systems191
14.3The characteristic distribution of a differential system192
15Pseudo-Riemannian Geometry198
15.1Pseudo-Riemannian manifolds198
15.2Length and distance204
15.3Flat spaces208
16Connection 1-Forms212
16.1The Levi-Civita connection and its covariant derivative212
16.2Geodesics of the Levi-Civita connection216
16.3The torsion and curvature of a linear, or affine connection219
16.4The exponential map and normal coordinates225
16.5Connections on pseudo-Riemannian manifolds226
17Connections on Manifolds230
17.1Connections between tangent spaces230
17.2Coordinate-free description of a connection231
17.3The torsion and curvature of a connection235
17.4Some geometry of submanifolds242
18Mechanics248
18.1Symplectic forms, symplectic mappings, Hamiltonian vector fields, and Poisson brackets248
18.2The Darboux theorem, and the natural symplectic structure of T* M253
18.3Hamilton's equations. Examples of mechanical systems258
18.4The Legendre transformation and Lagrangian vector fields263
19Additional Topics in Mechanics268
19.1The configuration space as a pseudo-Riemannian manifold268
19.2The momentum mapping and Noether's theorem271
19.3Hamilton-Jacobi theory275
20A Spacetime282
20.1Newton's mechanics and Maxwell's electromagnetic theory282
20.2Frames of reference generalized287
20.3The Lorentz transformations289
20.4Some properties and forms of the Lorentz transformations294
20.5Minkowski spacetime298
21Some Physics on Minkowski Spacetime306
21.1Time dilation and the Lorentz-Fitzgerald contraction306
21.2Particle dynamics on Minkowski spacetime313
21.3Electromagnetism on Minkowski spacetime317
21.4Perfect fluids on Minkowski spacetime322
22Einstein Spacetimes326
22.1Gravity, acceleration, and geodesics326
22.2Gravity is a manifestation of curvature328
22.3The field equation in empty space331
22.4Einstein's field equation (Sitz, der Preuss Acad. Wissen., 1917)334
23Spacetimes Near an Isolated Star339
23.1Schwarzschild's exterior solution339
23.2Two applications of Schwarzschild's solution344
23.3The Kruskal extension of Schwarzschild spacetime348
23.4The field of a rotating star352
24Nonempty Spacetimes356
24.1Schwarzschild's interior solution356
24.2The form of the Friedmann-Robertson-Walker metric tensor and its properties361
24.3Friedmann-Robertson-Walker spacetimes365
25Lie Groups369
25.1Definition and examples369
25.2Vector fields on a Lie group371
25.3Differential forms on a Lie group377
25.4The action of a Lie group on a manifold380
26Fiber Bundles384
26.1Principal fiber bundles384
26.2Examples388
26.3Associated bundles390
26.4Examples of associated bundles392
27Connections on Fiber Bundles394
27.1Connections on principal fiber bundles394
27.2Curvature398
27.3Linear Connections401
27.4Connections on vector bundles404
28Gauge Theory409
28.1Gauge transformation of a principal bundle409
28.2Gauge transformations of a vector bundle413
28.3How fiber bundles with connections form the basic framework of the Standard Model of elementary particle physics417
References423
Notation427
Index435

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