This is a self-contained introduction to differential geometry and the calculus of differential forms. It is written primarily for physicists. The material complements the usual mathematical methods, which emphasize analysis rather than geometry. The reader is expected to have the standard physics background in mechanics, electrodynamics, and mathematical methods. The mathematically knowledgeable can skip directly to the heart of the book, the calculus of differential forms, in Chapter IV.

This book falls between the usual mathematics and physics texts. On the one hand, proofs are given only when they are especially instructive. On the other hand, definitions, especially of mathematical structures, are given far more carefully than is the usual practice in physics. It is very dangerous to be sloppy in your definitions. I have taken considerable care to give many physical applications and to respect the physical subtleties of these applications. Indeed, my operational rule was to include no mathematics for which I could not produce a useful example. These examples form nearly half the book, and a large part of your learning will take place while reading and thinking about them. I feel that we learn far more from carefully chosen examples than from formal and unnatural deductive reasoning. Most of these examples were originally problems. I wish that I had been left with still more problems for the reader.

I call this a geometric treatment. What do I mean by geometry? One connotation is that of diagrams and pictorial representations.