Like most mathematicians, my first brush with differential forms happened when I took (the equivalent of) vector calculus. They were, at that point, fairly mysterious things. A year or two later, I worked through Spivak's Calculus on Manifolds and was fascinated by the way differential forms allowed one to dramatically unify the Gauss-Green-Stokes theorems.
When it came to be my turn to teach vector calculus, I wanted to do it "the right way," namely, using forms. But it never worked out. Setting up the multilinear algebra took far too long, and students never quite "got" what was going on. It was frustrating. I suspect the experience is fairly common, because many authors have tried to write books aimed at undergraduates that "do things right." David Bachman's A Geometric Approach to Differential Forms is one more attempt, based on the course he has been teaching for some years. The result is not perfect, but I think he has produced a usable textbook.
Bachman's idea is to use geometric intuition to alleviate some of the algebraic difficulties. In an introductory "Guide to the Reader," he explains the basic approach: to use low-dimensional cases that can be visualized to support the general ideas. The emphasis is on understanding rather than on detailed derivations and proofs. This is definitely the right approach in a course at this level. One can always keep a copy of a more formal book at hand for those students who want to see more of the proofs.
In order to minimize the pre-requisites, the book opens with a chapter on basic multivariable calculus: vectors, functions of several variables, partial derivatives, multiple integrals. This is too short to serve well if students have never seen this material, but will work as a review. In particular, students who have never had linear algebra will have a hard time learning enough from section 1.1 to carry them through the book.
A chapter on parametrizations follows. It is, I think, much too brief. For many students, learning to parametrize and to understand parametrizations is the toughest part of a vector calculus course. A little more help should have been provided here.
Next we get four solid chapters on forms. These begin with a very nice intuitive introduction that tries to explain why we should understand a 1-form as a linear functional. It clearly sounds the fundamental message: what is inside an integral is not a function, but rather a differential form.
Then we get down to fairly serious business. The treatment of forms is done well, but I think these chapters will be heavy going for most students. We get statements like
Every 2-form projects the parallelogram spanned by V1 and V2 onto each of the (2-dimensional) coordinate planes, computes the resulting (signed) areas, multiplies each by some constant, and adds the results.
Well, ok… Maybe it's the algebraist in me, but this seems like a very roundabout way of describing the basis of the space of 2-forms. I would have a hard time resisting the impulse to go a little heavier on the linear algebra in order to avoid having to say such things. It may well be, however, that my students would prefer going Bachman's way.
When we get to differential forms in chapter 5, things get (necessarily) harder. A glance at page 55, for example, shows that Bachman's "geometric approach" does not mean we do not get hairy computations. On that page he is justifying the formula for integrating a 2-form over a surface patch (not a term he uses) by setting up a kind of Riemann sum and checking that it can be translated to a Riemann sum on the plane. The result, of course, is that one integrates the form by integrating its pullback, but that idea is not introduced until much later in the book. (The reason is that the derivative is nowhere described as a linear transformation; if one can't push vectors forward, one also can't pull forms back.)
The treatment here is arguably geometric, but it is nevertheless pretty messy. On the other hand, it allows Bachman to derive the change of variable formula for multiple integrals from the theory of forms, which is kind of nice.
Chapter 6 introduces the exterior derivative. I was a little bit frustrated that Bachman decided to use the gradient and the directional derivative to define the derivative of a 1-form. I've always thought that the gradient is a poor substitute for the differential, so this feels backwards to me.
Once the theory of forms is well set up, the rest of the book proceeds in a fairly brisk fashion. We get Stokes' theorem (on a cube, with the general case asserted but not yet proved) and then derive the three big theorems from it. The books wraps up with chapters on Maxwell's equations and on manifolds (including the proof of Stokes' theorem on n-cells and then on manifolds with boundary). Only in this last chapter do we get a definition of "pullback" or see the term "partition of unity."
The book still shows its roots as a set of lecture notes. At several points I wished for a little more explanation, for more words to supplement the geometry and the derivations. At many points I felt that there simply weren't enough exercises to help students internalize the material.
Will it work as a textbook? Given the right students, I think so, provided the professor does some work to supplement what is missing, particularly in the chapter on parametrization. Students used to the massively detailed, colorfully illustrated, and exercise-laden textbooks used in calculus course would have to learn how to use a book that is brief and to the point and that expects them to be thinking as they read. It might actually do them some good.
The first mathematics class Fernando Q. Gouvêa ever taught was a vector calculus course for engineering students at the Universidade de São Paulo. Alas, he has never had a chance to teach this material at Colby College, where he is now the Carter Professor of Mathematics.
Preface.- Guide to the Reader.-Multivariable Calculus.- Parameterizations.- Introduction to Forms.- Forms.- Differential Forms.- Differentiation of Forms.- Stokes' Theorem.- Applications.- Manifolds.- Non-linear Forms.- References.- Index.- Solutions.