The Geometry of Physics: An Introduction

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 Poincare 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.