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,
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. Before
discussing 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 also be of interest to mathematics students. This
book will be useful to graduate and advanced undergraduate
students of physics, engineering and mathematics. It can be used
as a course text or f