The geometry of physics: an introduction

"This book is intended to provide 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 deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers, quarks, and the quark model for mesons. Before a discussion of abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should be of interest also to mathematics students. This book will be useful to graduate and advance undergraduate students of physics, engineering or mathematics. It can be used as a course text or for self-study. This third edition includes a new overview of Cartan's exterior differential forms. It previews many of the geometric concepts developed in the text and illustrates their applications to a single extended problem in engineering, namely the Cauchy stresses created by a small twist of an elastic cylindrical rod about its axis"--

Overview: an informal overview of Cartan's exterior differential forms, illustrated with an application to Cauchy's stress tensor -- t I. Part I: Manifolds and vector fields -- Tensors and exterior forms -- Integration of differential forms -- The Lie derivative -- The Poincare Lemma and potentials -- Holonomic and nonholonomic constraints -- Part II: Geometry and topology -- R³ and Minkowski space -- The geometry of surfaces in R³ -- Covariant differentiation and curvature -- Geodesics -- Relativity, tensors, and curvature -- Curvature and topology: Synge's theorem -- Betti numbers and De Rham's theorem -- Harmonic forms -- Part III: Lie Groups -- Vector bundles in geometry and physics -- Fiber bundles, Gauss-Bonnet, and topological quantization -- Connections and associated bundles -- The Dirac equation -- Yang-Mills fields -- Betti numbers and covering spaces -- Chern forms and homotopy groups.