The Geometry of Physics
An Introduction

Provides a working knowledge of tools that are of great value in geometry and physics and in engineering.

Theodore Frankel (Author)

9781107602601, Cambridge University Press

Paperback / softback, published 3 November 2011

748 pages, 260 b/w illus. 205 exercises
24.8 x 17.4 x 3.3 cm, 1.44 kg

'… contains a wealth of interesting material for both the beginning and the advanced levels. The writing may feel informal but it is precise - a masterful exposition. Users of this 'introduction' will be well prepared for further study of differential geometry and its use in physics and engineering … As did earlier editions, this third edition will continue to promote the language with which mathematicians and scientists can communicate.' Jay P. Fillmore, SIAM Review

This book provides 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 (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.

Preface to the Third Edition
Preface to the Second Edition
Preface to the revised printing
Preface to the First Edition
Overview
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 nonholonomic 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 A. Forms in continuum mechanics
Appendix B. Harmonic chains and Kirchhoff's circuit laws
Appendix C. Symmetries, quarks, and Meson masses
Appendix D. Representations and hyperelastic bodies
Appendix E. Orbits and Morse–Bott theory in compact Lie groups.

Subject Areas: Mathematical physics [PHU], Geometry [PBM]