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Tensor Analysis on Manifolds

by Samuel I. Goldberg

  • ISBN: 9780486640396
  • ISBN10: 0486640396

Tensor Analysis on Manifolds

by Samuel I. Goldberg

  • List Price: $16.95
  • Binding: Paperback
  • Publisher: Dover Pubns
  • Publish date: 09/01/1981
  • ISBN: 9780486640396
  • ISBN10: 0486640396
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Description: Chapter 0/Set Theory and Topology0.1. SET THEORY 0.1.1. Sets 0.1.2. Set Operations 0.1.3. Cartesian Products 0.1.4. Functions 0.1.5. Functions and Set Operations 0.1.6. Equivalence Relations0.2. TOPOLOGY 0.2.1. Topologies 0.2.2. Metric Spaces 0.2.3. Subspaces 0.2.4. Product Topologies 0.2.5. Hausdorff Spaces 0.2.6. Continuity 0.2.7. Connectedness 0.2.8. Compactness 0.2.9. Local Compactness 0.2.10. Separability 0.2.11 ParacompactnessChapter 1/Manifolds1.1. Definition of a Mainifold1.2. Examples of Manifolds1.3. Differentiable Maps1.4. Submanifolds1.5. Differentiable Maps1.6. Tangents1.7. Coordinate Vector Fields1.8. Differential of a MapChapter 2/Tensor Algebra2.1. Vector Spaces2.2. Linear Independence2.3. Summation Convention2.4. Subspaces2.5. Linear Functions2.6. Spaces of Linear Functions2.7. Dual Space2.8. Multilinear Functions2.9. Natural Pairing2.10. Tensor Spaces2.11. Algebra of Tensors2.12. Reinterpretations2.13. Transformation Laws2.14. Invariants2.15. Symmetric Tensors2.16. Symmetric Algebra2.17. Skew-Symmetric Tensors2.18. Exterior Algebra2.19. Determinants2.20. Bilinear Forms2.21. Quadratic Forms2.22. Hodge Duality2.23. Symplectic FormsChapter 3/Vector Analysis on Manifolds3.1. Vector Fields3.2. Tensor Fields3.3. Riemannian Metrics3.4. Integral Curves3.5. Flows3.6. Lie Derivatives3.7. Bracket3.8. Geometric Interpretation of Brackets3.9. Action of Maps3.10. Critical Point Theory3.11. First Order Partial Differential Equations3.12. Frobenius' TheoremAppendix to Chapter 33A. Tensor Bundles3B. Parallelizable Manifolds3C. OrientabilityChapter 4/Integration Theory4.1. Introduction4.2. Differential Forms4.3. Exterior Derivatives4.4. Interior Products4.5. Converse of the Poincar Lemma4.6. Cubical Chains4.7. Integration on Euclidean Spaces4.8. Integration of Forms4.9. Strokes' Theorem4.10. Differential SystemsChapter 5/Riemannian and Semi-riemannian Manifolds5.1. Introduction5.2. Riemannian and Semi-riemannian Metrics5.3. "Lengeth, Angle, Distance, and Energy"5.4. Euclidean Space5.5. Variations and Rectangles5.6. Flat Spaces5.7. Affine connexions5.8 Parallel Translation5.9. Covariant Differentiation of Tensor Fields5.10. Curvature and Torsion Tensors5.11. Connexion of a Semi-riemannian Structure5.12. Geodesics5.13. Minimizing Properties of Geodesics5.14. Sectional CurvatureChapter 6/Physical Application6.1 Introduction6.2. Hamiltonian Manifolds6.3. Canonical Hamiltonian Structure on the Cotangent Bundle6.4. Geodesic Spray of a Semi-riemannian Manifold6.5. Phase Space6.6. State Space6.7. Contact Coordinates6.8. Contact ManifoldsBibliographyIndex
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