Tensors and manifolds: with applications to physics
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
Oxford
Oxford University Press
2004
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 462 S. |
| ISBN: | 0198510594 |
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| adam_text | Titel: Tensors and manifolds
Autor: Wasserman, Robert H
Jahr: 2004
CONTENTS
1 VECTOR SPACES 1
1.1 Definitions, properties, and examples 1
1.2 Representation of vector spaces 5
1.3 Linear mappings 6
1.4 Representation of linear mappings 8
2 MULTILINEAR MAPPINGS AND DUAL SPACES 11
2.1 Vector spaces of linear mappings 11
2.2 Vector spaces of multilinear mappings 15
2.3 Nondegenerate bilinear functions 19
2.4 Orthogonal complements and the transpose of a linear mapping 20
3 TENSOR PRODUCT SPACES 25
3.1 The tensor product of two finite-dimensional vector spaces 25
3.2 Generalizations, isomorphisms, and a characterization 28
3.3 Tensor products of infinite-dimensional vector spaces 31
4 TENSORS 34
4.1 Definitions and alternative interpretations 34
4.2 The components of tensors 36
4.3 Mappings of the spaces VJ 39
5 SYMMETRIC AND SKEW-SYMMETRIC TENSORS 46
5.1 Symmetry and skew-symmetry 46
5.2 The symmetric subspace of V5° 48
5.3 The skew-symmetric (alternating) subspace of Va° 54
5.4 Some special properties of S2(V*) and A2(T*) 59
6 EXTERIOR (GRASSMANN) ALGEBRA 71
6.1 Tensor algebras 71
6.2 Definition and properties of the exterior product 72
6.3 Some more properties of the exterior product 77
7 THE TANGENT MAP OF REAL CARTESIAN SPACES 84
7.1 Maps of real cartesian spaces 84
7.2 The tangent and cotangent spaces at a point of Rn 88
7.3 The tangent map 96
8 TOPOLOGICAL SPACES 102
8.1 Definitions, properties, and examples 102
8.2 Continuous mappings 106
xii
CONTENTS
108
108
116
119
125
9 differentiable manifolds
9.1 Definitions and examples
9.2 Mappings of differentiable manifolds
9.3 The tangent and cotangent spaces at a point of M
9.4 Some properties of mappings
10 SUBMANIFOLDS
10.1 Parametrized submanifolds
10.2 Differentiable varieties as submanifolds 133
11 VECTOR FIELDS, 1-FORMS, AND OTHER
TENSOR FIELDS 136
11.1 Vector fields *3^
11.2 1-Form fields 143
11.3 Tensor fields and differential forms 146
11.4 Mappings of tensor fields and differential forms 149
12 DIFFERENTIATION AND INTEGRATION OF
DIFFERENTIAL FORMS 153
12.1 Exterior differentiation of differential forms 153
12.2 Integration of differential forms 158
13 THE FLOW AND THE LIE DERIVATIVE OF A
VECTOR FIELD 168
13.1 Integral curves and the flow of a vector field 168
13.2 Flow boxes (local flows) and complete vector fields 172
13.3 Coordinate vector fields 176
13.4 The Lie derivative 178
14 INTEGRABILITY CONDITIONS FOR
DISTRIBUTIONS AND FOR PFAFFIAN SYSTEMS 186
14.1 Completely integrable distributions 186
14.2 Completely integrable Pfaffian systems 191
14.3 The characteristic distribution of a differential system 192
15 PSEUDO-RIEMANNIAN GEOMETRY 198
15.1 Pseudo-Riemannian manifolds 198
15.2 Length and distance 204
15.3 Flat spaces 208
16 CONNECTION 1-FORMS 212
16.1 The Levi-Civita connection and its covariant derivative 212
16.2 Geodesies of the Levi-Civita connection 216
16.3 The torsion and curvature of a linear, or affine connection 219
16.4 The exponential map and normal coordinates 225
16.5 Connections on pseudo-Riemannian manifolds 226
CONTENTS xiii
17 CONNECTIONS ON MANIFOLDS 230
17.1 Connections between tangent spaces 230
17.2 Coordinate-free description of a connection 231
17.3 The torsion and curvature of a connection 235
17.4 Some geometry of submanifolds 242
18 MECHANICS 248
18.1 Symplectic forms, symplectic mappings, Hamiltonian
vector fields, and Poisson brackets 248
18.2 The Darboux theorem, and the natural symplectic
structure of T*M 253
18.3 Hamilton s equations. Examples of mechanical systems 258
18.4 The Legendre transformation and Lagrangian vector fields 263
19 ADDITIONAL TOPICS IN MECHANICS 268
19.1 The configuration space as a pseudo-Riemannian manifold 268
19.2 The momentum mapping and Noether s theorem 271
19.3 Hamilton-Jacobi theory 275
20 A SPACETIME 282
20.1 Newton s mechanics and Maxwell s electromagnetic theory 282
20.2 Frames of reference generalized 287
20.3 The Lorentz transformations 289
20.4 Some properties and forms of the Lorentz transformations 294
20.5 Minkowski spacetime 298
21 SOME PHYSICS ON MINKOWSKI SPACETIME 306
21.1 Time dilation and the Lorentz-Fitzgerald contraction 306
21.2 Particle dynamics on Minkowski spacetime 313
21.3 Electromagnetism on Minkowski spacetime 317
21.4 Perfect fluids on Minkowski spacetime 322
22 EINSTEIN SPACETIMES 326
22.1 Gravity, acceleration, and geodesies 326
22.2 Gravity is a manifestation of curvature 328
22.3 The field equation in empty space 331
22.4 Einstein s field equation (Sitz, der Preuss Acad.
Wissen., 1917) 334
23 SPACETIMES NEAR AN ISOLATED STAR 339
23.1 Schwarzschild s exterior solution 339
23.2 Two applications of Schwarzschild s solution 344
23.3 The Kruskal extension of Schwarzschild spacetime 348
23.4 The field of a rotating star 352
xiv
CONTENTS
24 NONEMPTY SPACETIMES 356
24.1 Schwarzschild s interior solution 356
24.2 The form of the Friedmann-Robertson-Walker metric
tensor and its properties 361
24.3 Friedmann-Robertson-Walker spacetimes 365
25 LIE GROUPS 369
25.1 Definition and examples 369
25.2 Vector fields on a Lie group 371
25.3 Differential forms on a Lie group 377
25.4 The action of a Lie group on a manifold 380
26 FIBER BUNDLES 384
26.1 Principal fiber bundles 384
26.2 Examples i - ; 388
26.3 Associated bundles 390
26.4 Examples of associated bundles 392
27 CONNECTIONS ON FIBER BUNDLES 394
27.1 Connections on principal fiber bundles . ; • 394
27.2 Curvature 398
27.3 Linear Connections , 401
27.4 Connections on vector bundles 404
28 GAUGE THEORY 409
28.1 Gauge transformation of a principal bundle 409
28.2 Gauge transformations of a vector bundle ; 413
28.3 How fiber bundles with connections form the basic
framework of the Standard Model of elementary particle
physics 417
References 423
Notation 427
Index 435
|
| adam_txt |
Titel: Tensors and manifolds
Autor: Wasserman, Robert H
Jahr: 2004
CONTENTS
1 VECTOR SPACES 1
1.1 Definitions, properties, and examples 1
1.2 Representation of vector spaces 5
1.3 Linear mappings 6
1.4 Representation of linear mappings 8
2 MULTILINEAR MAPPINGS AND DUAL SPACES 11
2.1 Vector spaces of linear mappings 11
2.2 Vector spaces of multilinear mappings 15
2.3 Nondegenerate bilinear functions 19
2.4 Orthogonal complements and the transpose of a linear mapping 20
3 TENSOR PRODUCT SPACES 25
3.1 The tensor product of two finite-dimensional vector spaces 25
3.2 Generalizations, isomorphisms, and a characterization 28
3.3 Tensor products of infinite-dimensional vector spaces 31
4 TENSORS 34
4.1 Definitions and alternative interpretations 34
4.2 The components of tensors 36
4.3 Mappings of the spaces VJ 39
5 SYMMETRIC AND SKEW-SYMMETRIC TENSORS 46
5.1 Symmetry and skew-symmetry 46
5.2 The symmetric subspace of V5° 48
5.3 The skew-symmetric (alternating) subspace of Va° 54
5.4 Some special properties of S2(V*) and A2(T*) 59
6 EXTERIOR (GRASSMANN) ALGEBRA 71
6.1 Tensor algebras 71
6.2 Definition and properties of the exterior product 72
6.3 Some more properties of the exterior product 77
7 THE TANGENT MAP OF REAL CARTESIAN SPACES 84
7.1 Maps of real cartesian spaces 84
7.2 The tangent and cotangent spaces at a point of Rn 88
7.3 The tangent map 96
8 TOPOLOGICAL SPACES 102
8.1 Definitions, properties, and examples 102
8.2 Continuous mappings 106
xii
CONTENTS
108
108
116
119
125
9 differentiable manifolds
9.1 Definitions and examples
9.2 Mappings of differentiable manifolds
9.3 The tangent and cotangent spaces at a point of M
9.4 Some properties of mappings
10 SUBMANIFOLDS
10.1 Parametrized submanifolds
10.2 Differentiable varieties as submanifolds 133
11 VECTOR FIELDS, 1-FORMS, AND OTHER
TENSOR FIELDS 136
11.1 Vector fields *3^
11.2 1-Form fields 143
11.3 Tensor fields and differential forms 146
11.4 Mappings of tensor fields and differential forms 149
12 DIFFERENTIATION AND INTEGRATION OF
DIFFERENTIAL FORMS 153
12.1 Exterior differentiation of differential forms 153
12.2 Integration of differential forms 158
13 THE FLOW AND THE LIE DERIVATIVE OF A
VECTOR FIELD 168
13.1 Integral curves and the flow of a vector field 168
13.2 Flow boxes (local flows) and complete vector fields 172
13.3 Coordinate vector fields 176
13.4 The Lie derivative 178
14 INTEGRABILITY CONDITIONS FOR
DISTRIBUTIONS AND FOR PFAFFIAN SYSTEMS 186
14.1 Completely integrable distributions 186
14.2 Completely integrable Pfaffian systems 191
14.3 The characteristic distribution of a differential system 192
15 PSEUDO-RIEMANNIAN GEOMETRY 198
15.1 Pseudo-Riemannian manifolds 198
15.2 Length and distance 204
15.3 Flat spaces 208
16 CONNECTION 1-FORMS 212
16.1 The Levi-Civita connection and its covariant derivative 212
16.2 Geodesies of the Levi-Civita connection 216
16.3 The torsion and curvature of a linear, or affine connection 219
16.4 The exponential map and normal coordinates 225
16.5 Connections on pseudo-Riemannian manifolds 226
CONTENTS xiii
17 CONNECTIONS ON MANIFOLDS 230
17.1 Connections between tangent spaces 230
17.2 Coordinate-free description of a connection 231
17.3 The torsion and curvature of a connection 235
17.4 Some geometry of submanifolds 242
18 MECHANICS 248
18.1 Symplectic forms, symplectic mappings, Hamiltonian
vector fields, and Poisson brackets 248
18.2 The Darboux theorem, and the natural symplectic
structure of T*M 253
18.3 Hamilton's equations. Examples of mechanical systems 258
18.4 The Legendre transformation and Lagrangian vector fields 263
19 ADDITIONAL TOPICS IN MECHANICS 268
19.1 The configuration space as a pseudo-Riemannian manifold 268
19.2 The momentum mapping and Noether's theorem 271
19.3 Hamilton-Jacobi theory 275
20 A SPACETIME 282
20.1 Newton's mechanics and Maxwell's electromagnetic theory 282
20.2 Frames of reference generalized 287
20.3 The Lorentz transformations 289
20.4 Some properties and forms of the Lorentz transformations 294
20.5 Minkowski spacetime 298
21 SOME PHYSICS ON MINKOWSKI SPACETIME 306
21.1 Time dilation and the Lorentz-Fitzgerald contraction 306
21.2 Particle dynamics on Minkowski spacetime 313
21.3 Electromagnetism on Minkowski spacetime 317
21.4 Perfect fluids on Minkowski spacetime 322
22 EINSTEIN SPACETIMES 326
22.1 Gravity, acceleration, and geodesies 326
22.2 Gravity is a manifestation of curvature 328
22.3 The field equation in empty space 331
22.4 Einstein's field equation (Sitz, der Preuss Acad.
Wissen., 1917) 334
23 SPACETIMES NEAR AN ISOLATED STAR 339
23.1 Schwarzschild's exterior solution 339
23.2 Two applications of Schwarzschild's solution 344
23.3 The Kruskal extension of Schwarzschild spacetime 348
23.4 The field of a rotating star 352
xiv
CONTENTS
24 NONEMPTY SPACETIMES 356
24.1 Schwarzschild's interior solution 356
24.2 The form of the Friedmann-Robertson-Walker metric
tensor and its properties 361
24.3 Friedmann-Robertson-Walker spacetimes ' 365
25 LIE GROUPS 369
25.1 Definition and examples 369
25.2 Vector fields on a Lie group 371
25.3 Differential forms on a Lie group 377
25.4 The action of a Lie group on a manifold 380
26 FIBER BUNDLES 384
26.1 Principal fiber bundles 384
26.2 Examples i - ; 388
26.3 Associated bundles 390
26.4 Examples of associated bundles 392
27 CONNECTIONS ON FIBER BUNDLES 394
27.1 Connections on principal fiber bundles . ; • 394
27.2 Curvature 398
27.3 Linear Connections , 401
27.4 Connections on vector bundles 404
28 GAUGE THEORY 409
28.1 Gauge transformation of a principal bundle 409
28.2 Gauge transformations of a vector bundle ' ; 413
28.3 How fiber bundles with connections form the basic
framework of the Standard Model of elementary particle
physics 417
References 423
Notation 427
Index 435 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:52:26Z |
| indexdate | 2024-07-09T20:35:06Z |
| institution | BVB |
| isbn | 0198510594 |
| language | Undetermined |
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| physical | XIV, 462 S. |
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| spelling | Wasserman, Robert H. Verfasser aut Tensors and manifolds with applications to physics Robert H. Wasserman 2. ed. Oxford Oxford University Press 2004 XIV, 462 S. txt rdacontent n rdamedia nc rdacarrier Tensorrechnung (DE-588)4192487-3 gnd rswk-swf Tensoranalysis (DE-588)4204323-2 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Tensoranalysis (DE-588)4204323-2 s DE-604 Mannigfaltigkeit (DE-588)4037379-4 s Tensorrechnung (DE-588)4192487-3 s Mathematische Physik (DE-588)4037952-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014621220&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wasserman, Robert H. Tensors and manifolds with applications to physics Tensorrechnung (DE-588)4192487-3 gnd Tensoranalysis (DE-588)4204323-2 gnd Mathematische Physik (DE-588)4037952-8 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd |
| subject_GND | (DE-588)4192487-3 (DE-588)4204323-2 (DE-588)4037952-8 (DE-588)4037379-4 |
| title | Tensors and manifolds with applications to physics |
| title_auth | Tensors and manifolds with applications to physics |
| title_exact_search | Tensors and manifolds with applications to physics |
| title_exact_search_txtP | Tensors and manifolds with applications to physics |
| title_full | Tensors and manifolds with applications to physics Robert H. Wasserman |
| title_fullStr | Tensors and manifolds with applications to physics Robert H. Wasserman |
| title_full_unstemmed | Tensors and manifolds with applications to physics Robert H. Wasserman |
| title_short | Tensors and manifolds |
| title_sort | tensors and manifolds with applications to physics |
| title_sub | with applications to physics |
| topic | Tensorrechnung (DE-588)4192487-3 gnd Tensoranalysis (DE-588)4204323-2 gnd Mathematische Physik (DE-588)4037952-8 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd |
| topic_facet | Tensorrechnung Tensoranalysis Mathematische Physik Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014621220&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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