Tensor geometry: the geometric viewpoint and its uses
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u.a.
Springer
1991
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
130 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 432 S. graph. Darst. |
| ISBN: | 354052018X 038752018X |
Staff View
MARC
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| 245 | 1 | 0 | |a Tensor geometry |b the geometric viewpoint and its uses |c C. T. J. Dodson ; T. Poston |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin u.a. |b Springer |c 1991 | |
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Record in the Search Index
| _version_ | 1804118820846043136 |
|---|---|
| adam_text | Contents
Introduction XI
0. Fundamental Not(at)ions 1
1. Sets 1
2. Functions 6
3. Physical Background 13
1. Real Vector Spaces 18
1. Spaces 18
Subspace geometry, components
2. Maps 24
Linearity, singularity, matrices
3. Operators 31
Projections, eigenvalues, determinant, trace
II. Affine Spaces 43
1. Spaces 43
Tangent vectors, parallelism, coordinates
2. Combinations of Points 49
Midpoints, convexity
3. Maps 53
Linear parts, translations, components
III. Dual Spaces 57
1. Contours, Co- and Contravariance, Dual Basis 57
IV. Metric Vector Spaces 64
1. Metrics 64
Basic geometry and examples, Lorentz geometry
2. Maps 76
Isometries, orthogonal projections and complements, adjoints
3. Coordinates 83
Orthonormal bases
VIII Contents
4. Diagonalising Symmetric Operators 92
Principal directions, isotropy
V. Tensors and Multilinear Forms 98
1. Multilinear Forms 98
Tensor Products, Degree, Contraction, Raising Indices
VI. Topological Vector Spaces 114
1. Continuity 114
Metrics, topologies, homeomorphisms
2. Limits 125
Convergence and continuity
3. The Usual Topology 128
Continuity in finite dimensions
4. Compactness and Completeness 136
Intermediate Value Theorem, convergence, extrema
VII. Differentiation and Manifolds 149
1. Differentiation 149
Derivative as local linear approxiamation
2. Manifolds 160
Charts, maps, diffeomorphisms
3. Bundles and Fields 170
Tangent and tensor bundles, metric tensors
4. Components 182
Hairy Ball Theorem, transformation formulae, raising indices
5. Curves 189
Parametrisation, length, integration
6. Vector Fields and Flows 195
First order ordinary differential equations
7. Lie Brackets 200
Commuting vector fields and flows
VIII. Connections and Covariant Differentiation 205
1. Curves and Tangent Vectors 205
Representing a vector by a curve
2. Rolling Without Turning 207
Differentiation along curves in embedded manifolds
3. Differentiating Sections 212
Connections, horizontal vectors, Christoffel symbols
4. Parallel Transport 222
Integrating a connection
Contents IX
5. Torsion and Symmetry 228
Torsion tensor of a connection
6. Metric Tensors and Connections 232
Levi-Civita connection
7. Covariant Differentiation of Tensors 240
Parallel transport, Ricci s Lemma, components, constancy
IX. Geodesies 246
1. Local Characterisation 246
Undeviating curves
2. Geodesies from a Point 249
Completeness, exponential map, normal coordinates
3. Global Characterisation 256
Criticality of length and energy, First Variation Formula
4. Maxima, Minima, Uniqueness 264
Saddle points, mirages, Twins Paradox
5. Geodesies in Embedded Manifolds 275
Characterisation, examples
6. An Example of Lie Group Geometry 281
2x2 matrices as a pseudo-Riemannian manifold
X. Curvature 298
1. Flat Spaces 298
Intrinsic description of local flatness
2. The Curvature Tensor 304
Properties and Components
3. Curved Surfaces 319
Gaussian curvature, Gauss-Bonnet Theorem
4. Geodesic Deviation 324
Tidal effects in spacetime
5. Sectional Curvature 326
Schur s Theorem, constant curvature
6. Ricci and Einstein Tensors 329
Signs, geometry, Einstein manifolds, conservation equation
7. The Weyl Tensor 337
XL Special Relativity 340
1. Orienting Spacetimes 340
Causality, particle histories
2. Motion in Flat Spacetime 342
Inertial frames, momentum, rest mass, mass-energy
3. Fields 355
Matter tensor, conservation
X Contents
4. Forces 367
No scalar potentials
5. Gravitational Red Shift and Curvature 369
Measurement gives a curved metric tensor
XII. General Relativity 372
1. How Geometry Governs Matter 372
Equivalence principle, free fall
2. What Matter does to Geometry 377
Einstein s equation, shape of spacetime
3. The Stars in Their Courses 384
Geometry of the solar system, Schwarzschild solution
4. Farewell Particle 398
Appendix. Existence and Smoothness of Flows 400
1. Completeness 400
2. Two Fixed Point Theorems 401
3. Sequences of Functions 404
4. Integrating Vector Quantities 408
5. The Main Proof 408
6. Inverse Function Theorem 415
Bibliography 418
Index of Notations 421
Index 424
|
| any_adam_object | 1 |
| author | Dodson, Christopher T. J. 1941- Poston, Timothy |
| author_GND | (DE-588)12958472X |
| author_facet | Dodson, Christopher T. J. 1941- Poston, Timothy |
| author_role | aut aut |
| author_sort | Dodson, Christopher T. J. 1941- |
| author_variant | c t j d ctj ctjd t p tp |
| building | Verbundindex |
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| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| institution | BVB |
| isbn | 354052018X 038752018X |
| language | English |
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| spelling | Dodson, Christopher T. J. 1941- Verfasser (DE-588)12958472X aut Tensor geometry the geometric viewpoint and its uses C. T. J. Dodson ; T. Poston 2. ed. Berlin u.a. Springer 1991 XIV, 432 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 130 Calcul tensoriel Calcul tensoriel ram Differentiaalmeetkunde gtt Géométrie différentielle Géométrie différentielle ram Relativité (Physique) - Mathématiques Tensoren gtt Calculus of tensors Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Tensor (DE-588)4184723-4 gnd rswk-swf Tensorrechnung (DE-588)4192487-3 gnd rswk-swf Tensorrechnung (DE-588)4192487-3 s Differentialgeometrie (DE-588)4012248-7 s DE-604 Tensor (DE-588)4184723-4 s 1\p DE-604 Poston, Timothy Verfasser aut Graduate texts in mathematics 130 (DE-604)BV000000067 130 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002889950&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Dodson, Christopher T. J. 1941- Poston, Timothy Tensor geometry the geometric viewpoint and its uses Graduate texts in mathematics Calcul tensoriel Calcul tensoriel ram Differentiaalmeetkunde gtt Géométrie différentielle Géométrie différentielle ram Relativité (Physique) - Mathématiques Tensoren gtt Calculus of tensors Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd Tensor (DE-588)4184723-4 gnd Tensorrechnung (DE-588)4192487-3 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4184723-4 (DE-588)4192487-3 |
| title | Tensor geometry the geometric viewpoint and its uses |
| title_auth | Tensor geometry the geometric viewpoint and its uses |
| title_exact_search | Tensor geometry the geometric viewpoint and its uses |
| title_full | Tensor geometry the geometric viewpoint and its uses C. T. J. Dodson ; T. Poston |
| title_fullStr | Tensor geometry the geometric viewpoint and its uses C. T. J. Dodson ; T. Poston |
| title_full_unstemmed | Tensor geometry the geometric viewpoint and its uses C. T. J. Dodson ; T. Poston |
| title_short | Tensor geometry |
| title_sort | tensor geometry the geometric viewpoint and its uses |
| title_sub | the geometric viewpoint and its uses |
| topic | Calcul tensoriel Calcul tensoriel ram Differentiaalmeetkunde gtt Géométrie différentielle Géométrie différentielle ram Relativité (Physique) - Mathématiques Tensoren gtt Calculus of tensors Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd Tensor (DE-588)4184723-4 gnd Tensorrechnung (DE-588)4192487-3 gnd |
| topic_facet | Calcul tensoriel Differentiaalmeetkunde Géométrie différentielle Relativité (Physique) - Mathématiques Tensoren Calculus of tensors Geometry, Differential Differentialgeometrie Tensor Tensorrechnung |
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