References

1. V.I. Arnold, Mathematical Methods of Classical Mechanics, Chap 7.

A definitive treatment. But some may find it hard to read. It should be noted that this chapter is quite independent of the previous six and a reader with no idea about classical mechanics will not be handicapped. The chapter can be regarded as a self-contained treatise on differential forms.

If you want to get started within a few days, this is the book to read. It explains key ideas efficiently, and includes a lot of applications presented in an organized fashion.

This book is actually about dynamics. The reason why it is listed here is that it has a short appendix which covers differential forms and explains briefly how differential forms provide a way of testing whether a transformation is canonical. This appendix covers the essential ideas concisely. But of course you will not be satisfied if you want details and proofs.

Another reason for including this book is because the author, Richard Rand, is one of my supervisors. He will be happy if more people read his books, and (I guess) even happier if more buy them. :)

A rigorous treatment can be found here. In particular, the author emphasizes the integration of differential forms and presents the materials in such a way that Stokes' Theorem comes out very naturally. However, readers looking for applications to physics, engineering and other branches of mathematics such as differential equations and differential geometry will have to try elsewhere.

In addition to the books mentioned above, there are some references in electronic form (well, here we see our old friend "form" again). For example, Bill Burke has something on this topic. He's also writing a book called Div Grad and Curl Are Dead. If you have read Section 3, you probably can guess what this book is about. Bill has also suggested many let me know.

If you are interested in getting a copy of the Mathematica notebook and packages containing the stuffs you see in these few pages, please drop me a mail.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).