## The Geometry of Physics: An IntroductionThis 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. |

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### Contents

xix | |

5 | |

37 | |

Integration of Differential Forms | 95 |

The Lie Derivative | 125 |

The Poincare Lemma and Potentials | 155 |

Holonomic and Nonholonomic Constraints | 165 |

R3 and Minkowski Space | 191 |

11 | 529 |

22 | 535 |

26 | 541 |

48 | 547 |

60 | 553 |

6a Instantons | 559 |

Forms and Homotopy Groups | 583 |

Homotopies and Extensions | 591 |

The Geometry of Surfaces in | 201 |

8 | 215 |

Gausss Theorema Egregium | 228 |

6b The Intrinsic Derivative and the Geodesic Equation | 234 |

Geodesics | 269 |

Relativity Tensors and Curvature | 291 |

Synges Theorem | 323 |

Betti Numbers and De Rhams Theorem | 333 |

Harmonic Forms | 361 |

Lie Groups | 391 |

Vector Bundles in Geometry and Physics | 413 |

Fiber Bundles GaussBonnet and Topological Quantization | 451 |

Connections and Associated Bundles | 476 |

The Dirac Equation | 491 |

YangMills Fields | 523 |

2a Homotopy | 602 |

Appendix A Forms in Continuum Mechanics | 617 |

A f Concluding Remarks | 627 |

Laplacians and Harmonic Cochains | 633 |

Symmetries Quarks and Meson Masses | 640 |

e A Reduced Symmetry Group | 648 |

e The Symmetric Traceless 3 3 Matrices Are Irreducible | 658 |

E b Application of Botts Extension of Morse Theory | 665 |

671 | |

675 | |

677 | |

679 | |

681 | |

684 | |

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### Common terms and phrases

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