The Geometry of Physics: An Introduction

by Theodore Frankel

• ISBN13: 9780521387538
• ISBN10: 0521387531
• Format: Paperback
• Copyright: 1999-04-13
• Publisher: Cambridge University Press

Summary

This book is intended to provide a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism, thermodynamics, the deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should also be of interest to mathematics students. This book will be useful to graduate and advanced undergraduate students of physics, engineering and mathematics. It can be used as a course text or for self study.

Table of Contents

Preface

Part I. Manifolds, Tensors and Exterior Forms: 1. Manifolds and vector fields

2. Tensors and exterior forms

3. Integration of differential forms

4. The Lie derivative

5. The Poincaré

Lemma and potentials

6. Holonomic and non-holonomic constraints

Part II. Geometry and Topology: 7. R3 and Minkowski space

8. The geometry of surfaces in R3

9. Covariant differentiation and curvature

10. Geodesics

11. Relativity, tensors, and curvature

12. Curvature and topology: Synge's theorem

13. Betti numbers and De Rham's theorem

14. Harmonic forms

Part III. Lie Groups, Bundles and Chern Forms: 15. Lie groups

16. Vector bundles in geometry and physics

17. Fiber bundles, Gauss-Bonnet, and topological quantization

18. Connections and associated bundles

19. The Dirac equation

20. Yang-Mills fields

21. Betti numbers and covering spaces

22. Chern forms and homotopy groups

Appendix: forms in continuum mechanics.