Tensor Geometry: The Geometric Viewpoint and its Uses

Christopher Terence John Dodson, Timothy Poston (auth.)038752018X, 9783540520184, 9780387520186, 354052018X

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Table of contents :
Front Matter….Pages I-XIV
Fundamental Not(at)ions….Pages 1-17
Real Vector Spaces….Pages 18-42
Affine Spaces….Pages 43-56
Dual Spaces….Pages 57-63
Metric Vector Spaces….Pages 64-97
Tensors and Multilinear Forms….Pages 98-113
Topological Vector Spaces….Pages 114-148
Differentiation and Manifolds….Pages 149-204
Connections and Covariant Differentiation….Pages 205-245
Geodesics….Pages 246-297
Curvature….Pages 298-339
Special Relativity….Pages 340-371
General Relativity….Pages 372-417
Back Matter….Pages 418-434