| Contents Pretace CHAPTER Ⅰ Differentlal Calculus 1. Categories 2. Topological Vector Spaces 3. Derivatives and Composition of Maps 4. Integration and Taylor's Formula 5. The Inverse Mapping Theorem CHAPTER Ⅱ Manitolds 1. Atlases, Charts, Morphisms 2. Submanifolds, Immersions, Submersions 3. Partitions of Unity 4. Manifolds with Boundary CHAPTER Ⅲ Vector Bundles 1. Definition, Pull Backs 2. The Tangent Bundle 3. Exact Sequences of Bundles 4. Operations on Vector Bundles 5. Splitting of Vector Bundles CHAPTER Ⅳ Vector Fields and Ditterential Equatlons 1. Existence Theorem for Differential Equations 2. Vector Fields, Curves, and Flows 3. Sprays 4. The Flow of a Spray and the Exponential Map 5. Existence of Tubular Neighborhoods 6. Uniqueness of Tubular Neighborhoods CHAPTER Ⅴ Operations on Vector Flelds and Diffterential Forms 1. Vector Fields, Differential Operators, Brackets 2. Lie Derivative 3. Exterior Derivative 4. The Poincare Lemma 5. Contractions and Lie Derivative 6. Vector Fields and l-Forms Under Self Duality 7. The Canonical 2-Fonn 8. Darboux's Theorem CHAPTER Ⅵ The Theorem ot Frobenlus CHAPTER Ⅶ Metrlcs CHAPTER Ⅷ Covariant Derlvatlves and Geodesics CHAPTER Ⅸ Curvature CHAPTER Ⅹ Volume Forms CHAPTER Ⅺ Integratlon of Differentlal Forms CHAPTER Ⅻ Stokes' Theorem CHAPTER ⅩⅢ Appllcatlons of Stokes' Theorem APPENDIX The Spectral Theorem Bibliography Index |
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