Tensor Analysis on Manifolds (Dover Books on Mathematics)

Richard L. Bishop; Samuel I. Goldberg

ISBN 10: 0486640396 / ISBN 13: 9780486640396

Editore: Dover Publications (edition Later Printing), 1980

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Codice articolo 0486640396-8-1

Dati bibliografici

• Titolo: Tensor Analysis on Manifolds (Dover Books on Mathematics)
• Editore: Dover Publications (edition Later Printing)
• Data di pubblicazione: 1980
• Lingua: Inglese
• ISBN 10: 0486640396
• ISBN 13: 9780486640396
• Rilegatura: Paperback
• Edizione: Later Printing.
• Condizione: Very Good

Descrizione articolo

Riassunto

si passa da argomenti generali a argomenti speciali, compresi capitoli sull'analisi vettoriale delle varietà e sulla teoria dell'integrazione.

Contenuti

Chapter 0/Set Theory and Topology 0.1. 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 Relations 0.2. 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 Paracompactness Chapter 1/Manifolds 1.1. Definition of a Mainifold 1.2. Examples of Manifolds 1.3. Differentiable Maps 1.4. Submanifolds 1.5. Differentiable Maps 1.6. Tangents 1.7. Coordinate Vector Fields 1.8. Differential of a Map Chapter 2/Tensor Algebra 2.1. Vector Spaces 2.2. Linear Independence 2.3. Summation Convention 2.4. Subspaces 2.5. Linear Functions 2.6. Spaces of Linear Functions 2.7. Dual Space 2.8. Multilinear Functions 2.9. Natural Pairing 2.10. Tensor Spaces 2.11. Algebra of Tensors 2.12. Reinterpretations 2.13. Transformation Laws 2.14. Invariants 2.15. Symmetric Tensors 2.16. Symmetric Algebra 2.17. Skew-Symmetric Tensors 2.18. Exterior Algebra 2.19. Determinants 2.20. Bilinear Forms 2.21. Quadratic Forms 2.22. Hodge Duality 2.23. Symplectic Forms Chapter 3/Vector Analysis on Manifolds 3.1. Vector Fields 3.2. Tensor Fields 3.3. Riemannian Metrics 3.4. Integral Curves 3.5. Flows 3.6. Lie Derivatives 3.7. Bracket 3.8. Geometric Interpretation of Brackets 3.9. Action of Maps 3.10. Critical Point Theory 3.11. First Order Partial Differential Equations 3.12. Frobenius' Theorem Appendix to Chapter 3 3A. Tensor Bundles 3B. Parallelizable Manifolds 3C. Orientability Chapter 4/Integration Theory 4.1. Introduction 4.2. Differential Forms 4.3. Exterior Derivatives 4.4. Interior Products 4.5. Converse of the Poincaré Lemma 4.6. Cubical Chains 4.7. Integration on Euclidean Spaces 4.8. Integration of Forms 4.9. Strokes' Theorem 4.10. Differential Systems Chapter 5/Riemannian and Semi-riemannian Manifolds 5.1. Introduction 5.2. Riemannian and Semi-riemannian Metrics 5.3. "Lengeth, Angle, Distance, and Energy" 5.4. Euclidean Space 5.5. Variations and Rectangles 5.6. Flat Spaces 5.7. Affine connexions 5.8 Parallel Translation 5.9. Covariant Differentiation of Tensor Fields 5.10. Curvature and Torsion Tensors 5.11. Connexion of a Semi-riemannian Structure 5.12. Geodesics 5.13. Minimizing Properties of Geodesics 5.14. Sectional Curvature Chapter 6/Physical Application 6.1 Introduction 6.2. Hamiltonian Manifolds 6.3. Canonical Hamiltonian Structure on the Cotangent Bundle 6.4. Geodesic Spray of a Semi-riemannian Manifold 6.5. Phase Space 6.6. State Space 6.7. Contact Coordinates 6.8. Contact Manifolds Bibliography Index