The Geometry of Physics An Introduction

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.

Theodore Frankel explains 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. It can be used as a course text or for self study.

Theodore Frankel received his PhD from the University of California, Berkeley. He is currently Emeritus Professor of Mathematics at the University of California, San Diego.