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The Geometry of Physics: An Introduction

Theodore Frankel
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

See our review of the second edition.

The major change for the third edition is the addition of an introductory chapter that offers a brief overview of the calculus of differential forms with applications to physics, with special focus on one specific example: the use of tensors in the theory of elasticity. The new chapter will probably prove useful to physics students, but most mathematicians are likely to do better skipping chapter 0 and only returning to it after reading chapters 1 to 3.

Preface to the Third Edition
Preface to the Second Edition
Preface to the revised printing
Preface to the First Edition
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.