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Applicable differential geometry /
This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitati...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge ; New York :
Cambridge University Press,
1986.
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| Colección: | London Mathematical Society lecture note series ;
59. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; CONTENTS; Preface; 0. THE BACKGROUND: VECTOR CALCULUS; 1. Vectors; 2. Derivatives; 3. Coordinates; 4. The Range and Summation Conventions; Note to Chapter 0; 1. AFFINE SPACES; 1. Affine Spaces; 2. Lines and Planes; 3. Affine Spaces Modelled on Quotients and Direct Sums; 4. Affine Maps; 5. Affine Maps of Lines and Hyperplanes; Summary of Chapter 1; Notes to Chapter 1; 2. CURVES, FUNCTIONS AND DERIVATIVES; 1. Curves and Functions; 2. Tangent Vectors; 3. Directional Derivatives; 4. Cotangent Vectors; 5. Induced Maps; 6. Curvilinear Coordinates; 7. Smooth Maps
- 8. Parallelism9. Covariant Derivatives; Summary of Chapter 2; Notes to Chapter 2; 3. VECTOR FIELDS AND FLOWS; 1. One-parameter Affine Groups; 2. One-parameter Groups: the General Case; 3. Flows; 4. Flows Associated with Vector Fields; 5. Lie Transport; 6. Lie Difference and Lie Derivative; 7. The Lie Derivative of a Vector Field as a Directional Derivative; 8. Vector Fields as Differential Operators; 9. Brackets and Commutators; 10. Covector Fields and the Lie Derivative; 11. Lie Derivative and Covariant Derivative Compared; 12. The Geometrical Significance of the Bracket
- 2. The Exterior Derivative3. Properties of the Exterior Derivative; 4. Lie Derivatives of Forms; 5. Volume Forms and the Divergence of a Vector Field; 6. A Formula Relating Lie and Exterior Derivatives; 8. Closed and Exact Forms; Summary of Chapter 5; 6. FROBENIUS'S THEOREM; 1. Distributions and Integral Submanifolds; Section 1; Section 2; 2. Necessary Conditions for Integrability; 3. Sufficient Conditions for Integrability; 4. Special Coordinate Systems; 5. Applications: Partial Differential Equations; 6. Application: Darboux's Theorem; 7. Application: Hamilton-Jacobi Theory