Differential Forms: A Complement to Vector Calculus (ISBN 0127425101)

A good introduction to forms:
Fortunately there are several books, at an introductory level suitable for undergraduate students, on how differential forms constitute a "new" powerful mathematical technique that surpasses the outdated vector calculus. This book by Steven H. Weintraub is a very good example among others -- such as: (i) "Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards (Birkhäuser, Boston, 1994); (ii) "Vector Calculus, Linear Algebra, and Differential Forms" by John H. Hubbard and Barbara Burke Hubbard (Prentice Hall, NJ, 2nd ed., 2002). As far as I know, it was in "Gravitation" -- by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (Freeman, San Francisco, 1973) -- that a pictorial representation of forms was clearly presented to physicists for the first time. These authors went even further, explaining how "forms illuminate electromagnetism, and electromagnetism illuminates forms" (p. 105). However, until now, it seems that in engineering forms have been disregarded -- despite early attempts by George A. Deschamps (see, e.g., his paper "Electromagnetism and differential forms", Proc. IEEE, Vol. 69, pp. 676-679, 1981), not to mention Harley Flanders's book ("Differential Forms with Applications to the Physical Sciences", Dover, NY, 1989). Perhaps the book by Ismo V. Lindell ("Differential Forms in Electromagnetics", IEEE Press/Wiley, NJ, 2004) will be able to change this sad scenario. It seems that the difficulty lies mainly in the fact that a proper understanding of k-forms, as antisymmetric (0,k) tensors in differentiable manifolds, requires the study of technical demanding subjects such as de Rham cohomology. However, this book shows that it is possible to make an introduction to forms without mastering such concepts in topological and smooth manifolds -- although there is an extensive bibliography on this subject out there (the books by John M. Lee on manifolds are my favorite). For more advanced readers, the book by Friedrich H. Hehl and Yuri N. Obukhov on the "Foundations of Classical Electrodynamics" (Birkhäuser, Boston, 2003) is, in my opinion, the most elegant exposition on the relation between electromagnetism and forms.

Intuitive Book:
This is the only book on the subject I know of that gives me a feel for what is going on. To many other books are long on notation and short on insight. Occasionally the book seems to plod but that is a minor problem compared with the dense presentation of other books. Highly recommended.

good introduction:
The language of differential forms presented at the level of this student-friendly text provides a refreshing outlook on vector analysis. And with a view towards more advanced courses, this book hints at the remarkable computational prowess it bears on differential geometry at large. In light of the author's heuristic approach, the book does well in setting the stage for the applications he has in mind (casting Stokes' theorem in its true form, for example). One should then go on to read books like Do Carmo, written in a similar vein, but this time, delineating the algebraic machinery needed to set up the theory in a more rigourous framework. Have fun!

Very confusing....:
This book was a very confusing book on a very confusing topic. I am looking forward to the day that someone can write an understandable treatise on this subject.

Ok for an introduction:
I don't see what the hype is about differential forms. They are just antisymmetric tensors. Big deal. Reading books on forms turns out to be a waste of time for most physicists and engineers (sorry to spoil your excitement). It is mostly meant for mathematicians, as a tool to prove theorems and study cohomology. Engineers can get most of what they need in classical electromagnetism using vector calculus - don't waste your time with Forms unless you have time on your hands and want to get a nice geometric view of how things work (otherwise, there is no computational advantage). That being said, this book is well written. (All typos should have been corrected by now.) The book takes a very friendly approach to teaching the topic from the grounds up. I learned forms from Spivak's Comprehensive Intro to Diff Geo and from Madsen and Thornehave (from Calculus to Cohomology), both books are truly excellent but probably too advanced for the average person (i.e. those who are not math majors). The book by Weintraub fills the gap and should give you a very good start. The I recommend the book by Harley Flanders (for engineers). The logical path if you want to read Madsen and Thornehave you need to learn some basic analysis (e.g. Spivak's Calculus on Manifolds is the quickest route).