Functional Differential Geometry

Physics is naturally expressed in mathematical language. Students new to the subjectmust simultaneously learn an idiomatic mathematical language and the content that is expressed inthat language. It is as if they were asked to read Les Miserables whilestruggling with French grammar. This book offers an innovative way to learn the differentialgeometry needed as a foundation for a deep understanding of general relativity or quantum fieldtheory as taught at the college level.

The approach taken by the authors (and usedin their classes at MIT for many years) differs from the conventional one in several ways, includingan emphasis on the development of the covariant derivative and an avoidance of the use oftraditional index notation for tensors in favor of a semantically richer language of vector fieldsand differential forms. But the biggest single difference is the authors' integration of computerprogramming into their explanations. By programming a computer to interpret a formula, the studentsoon learns whether or not a formula is correct. Students are led to improve their program, and as aresult improve their understanding.

Gerald Jay Sussman is Panasonic (formerly Matsushita) Professor of Electrical Engineering in the Department of Electrical Engineering and Computer Science at MIT and the coauthor (with Hal Abelson) of Structure and Interpretation of Computer Programs (MIT Press).