The approach taken by the authors (and used in their classes at MIT for many years) differs from the conventional one in several ways, including an avoidance of the use of traditional index notation for tensors in favor of a semantically richer language of vector fields and differential forms. But the single biggest difference is the authors’ integration of computer programming into their explanations. By programming a computer to interpret a formula, the student soon learns whether or not a formula is correct. Students are led to improve their program, and as a result improve their understanding.
This second edition has been updated throughout, with revisions that reflect insights gained by the authors from using the text every year at MIT. In addition, because of substantial software improvements, this edition provides algebraic proofs of more generality than those in the previous edition; this improvement permeates the new edition.