This text is an introduction to some of the mathematical wonders of Maxwell's equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell's equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell's equations is endless. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell's equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.
This one was tough. It covers differential forms, k forms, bundles, and connections. It's not a light read and even though it introduced me to many mathematical concepts, I would not recommend it to a person wanting to thoroughly understand these concepts.
Some Typos and some missing explanations(subjective view). A good book to gain confidence by solving back of the chapter problems. It would be great if the problem set is improved to broaden the understanding of the text before it. Especially in the later chapters that deal with mostly mathematical ideas that underlie modern gauge theories. The solution for integral of the forms is explained in a bit complicated way. Its easier to deal with the transformations from the parameter space to the real space than going via the Jacobian inside the integrals especially for the curves.
I love this book, but I feel that it's coverage of QED and beyond is pretty lacking. On the other hand it really *does* feel like a physics book for mathematicians, which are very hard to find.