Tensor Geometry: The Geometric Viewpoint and its UsesSpringer Science & Business Media, 23 de nov. de 2009 - 434 páginas We have been very encouraged by the reactions of students and teachers using our book over the past ten years and so this is a complete retype in TEX, with corrections of known errors and the addition of a supplementary bibliography. Thanks are due to the Springer staff in Heidelberg for their enthusiastic sup port and to the typist, Armin Kollner for the excellence of the final result. Once again, it has been achieved with the authors in yet two other countries. November 1990 Kit Dodson Toronto, Canada Tim Poston Pohang, Korea Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI O. Fundamental Not(at)ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I. 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 Contents VIII 4. Diagonalising Symmetric Operators 92 Principal directions, isotropy V. Tensors and Multilinear Forms 98 1. Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Tensor Products, Degree, Contraction, Raising Indices VE Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 1. Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Metrics, topologies, homeomorphisms 2. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Convergence and continuity 3. The Usual Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
Conteúdo
Real Vector Spaces | 18 |
2 Maps | 24 |
3 Operators | 31 |
Affine Spaces | 43 |
2 Combinations of Points | 49 |
3 Maps | 53 |
Dual Spaces | 57 |
Metric Vector Spaces | 64 |
2 Geodesies from a Point | 249 |
3 Global Characterisation | 256 |
4 Maxima Minima Uniqueness | 264 |
5 Geodesies in Embedded Manifolds | 275 |
6 An Example of Lie Group Geometry | 281 |
Curvature | 298 |
2 The Curvature Tensor | 304 |
3 Curved Surfaces | 319 |
2 Maps | 76 |
3 Coordinates | 83 |
4 Diagonalising Symmetric Operators | 92 |
Tensors and Multilinear Forms | 98 |
Topological Vector Spaces | 114 |
2 Limits | 125 |
3 The Usual Topology | 128 |
4 Compactness and Completeness | 136 |
Differentiation and Manifolds | 149 |
2 Manifolds | 160 |
3 Bundles and Fields | 170 |
4 Components | 182 |
5 Curves | 189 |
6 Vector Fields and Flows | 195 |
7 Lie Brackets | 200 |
Connections and Covariant Differentiation | 205 |
2 Rolling Without Turning | 207 |
3 Differentiating Sections | 212 |
4 Parallel Transport | 222 |
5 Torsion and Symmetry | 228 |
6 Metric Tensors and Connections | 232 |
7 Covariant Differentiation of Tensors | 240 |
Geodesies | 246 |
4 Geodesic Deviation | 324 |
5 Sectional Curvature | 326 |
6 Ricci and Einstein Tensors | 329 |
7 The Weyl Tensor | 337 |
Special Relativity | 340 |
2 Motion in Flat Spacetime | 342 |
3 Fields | 355 |
4 Forces | 367 |
5 Gravitational Red Shift and Curvature | 369 |
General Relativity | 372 |
2 What Matter does to Geometry | 377 |
3 The Stars in Their Courses | 384 |
4 Farewell Particle | 398 |
Existence and Smoothness of Flows | 400 |
2 Two Fixed Point Theorems | 401 |
3 Sequences of Functions | 404 |
4 Integrating Vector Quantities | 408 |
6 Inverse Function Theorem | 415 |
418 | |
Index of Notations | 421 |
424 | |
Outras edições - Ver todos
Tensor Geometry: The Geometric Viewpoint and its Uses C. T. J. Dodson,Timothy Poston Visualização parcial - 2013 |
Tensor Geometry: The Geometric Viewpoint and Its Uses C. T. J. Dodson,Timothy Poston Prévia não disponível - 1991 |
Tensor Geometry: The Geometric Viewpoint and Its Uses C. T. J. Dodson,Timothy Poston Prévia não disponível - 1991 |
Termos e frases comuns
4-momentum affine space algebra b₁ basis vectors bilinear form called chart choice components connection constant continuous contravariant converges coordinates Corollary covariant curvature curve Deduce defined Definition denote derivative differential dimensions eigenvectors embedded energy equation equivalent Euclidean exactly example Exercise finite flat spacetime formula function f G₁ geodesic geometry given gives hence identity inner product integral isomorphism Lemma length Levi-Civita connection linear map Lorentz manifold matrix metric tensor metric tensor field metric vector space neighbourhood non-degenerate non-zero notation null open set operator orthogonal orthonormal basis parallel transport parametrised particle physics plane Proof prove real numbers Ricci Ricci tensor Riemannian scalar sequence Show smooth spacelike spacetime subset surface symmetric tangent space tangent vector tensor field tensor product Theorem theory timelike topology unique vector field vector space velocity zero
Passagens mais conhecidas
Página 432 - And further, by these, my son, be admonished : of making many books there is no end ; and much study is a weariness of the flesh.
Página 1 - Therefore is the name of it called Babel ; because the Lord did there confound the language of all the earth: and from thence did the Lord scatter them abroad upon the face of all the earth.
Referências a este livro
Theory and Practice of Geometric Modeling Wolfgang Strasser,Hans-Peter Seidel Visualização parcial - 1989 |
The Geometry of Spacetime: An Introduction to Special and General Relativity James J. Callahan Visualização parcial - 2000 |