Paul Bamberg and Schlomo Sternberg, 403 pp, Cambridge UP
Some might find it surprising that this entails a shift from the abstract to the concrete, but AJP readers who follow the articles on Piaget will recognize the pattern: Excess abstraction appears in the early phase of learning when the subject is not really grasped. This pattern occurs also in the development of science itself. The gene, for example, was originally an abstract idea, manipulated mainly by jargon. Now it is a nuts and bolts item with even lawyers making a healthy living from it.
Shift your emphasis then from the representations of objects to the intrinsic properties of the objects themselves. As the authors say in the preface, "by the end ... the student should be thinking of a matrix as an object in its own right, and not as a square array of numbers." No longer is a vector to be thought of as a pile of numbers transforming properly; now it is the local linear approximation to a parametrized curve. A determinant is not an array of numbers with rules for its evaluation, but the ratio by which areas are modified by a linear transformation. (This makes the product rule obvious, by the way.) A similar shift occured earlier when Gibbs put the components of a vector together into a single geometric package.
With this attitude, objects with similar representations are not necessarily the same type of beast. One-forms, two-forms, and vectors have the same number of components, but they have different intrinsic properties. Recognizing this leads directly to the exterior calculus. Unlike vector calculus, this calculus extends simply to spaces beyond three dimensions, allows both Euclidean geometry and the Lorentz structure of relativity, and fits in easily with both the canonical structure of mechanics and the contact structure of thermodynamics. Exterior calculus simplifies by introducing the proper concepts much as the introduction of momentum conservation simplifies the paradoxes around action/reaction.
Mathematically these ideas are over 70 years old and date back to the seminal work by E. Cartan. The present book covers much the same material as Advanced Calculus by Loomis and Sternberg (1968) in a new presentation geared to the needs of physicists. The numerous examples are a fundamental part of the book, not tacked on as afterthoughts, and they are covered with considerable depth and physical accuracy. There is an entire chapter on matrix optics, for example.
Not that physicists have been neglecting these ideas. The course on theoretical physics put together by Thirring uses exterior calculus throughout. I myself teach a graduate course in Applied Differential Geometry, for which the present book will be an excellent prerequisite. What we have long needed is good support from the mathematicians at the elementary level. The present book exactly fills that need.
And it fills it very well indeed. Not only is the mathematics clean, elegant, and modern, but the presentation is humane, expecially for a mathematics text. Examples are provided before generalization, and motivation and applications are kept firmly in view. About equivalence classes, they say, "Before defining these classes, we should first see why something simpler will not suffice." This is first rate! They show admirable and at time incredible restraint. The first third and indeed most of the rest of Vol. I is restricted to two dimensions and 2x2 matrices. Their attitude is that once you get this much clear, any fool can see how it goes (my paraphrase). Physicists will like that. Speaking of restraint, the authors mercifully restrict their puns to the problems, of which there are about 150.
The preface promises even more goodies in the second volume. It will start with a "gentle introduction to the mathematics of shape, that is, algebraic topology. In my experience algebraic topology is exceedingly heavy going. They will teach it in the context of electrical circuit theory, and the preliminary version that I have seen looks excellent. It then goes on smoothly from this to electrodynamics in the language of differential forms. While I teach the Jackson course at UCSC using forms, it is a delight to see this done at the introductory level. Unfortunately, it does not appear that they will introduce twisted forms; thus one of the great simplifications of the exterior calculus, the elimination of right-hand rules, will be missed.
There are a few quibbles that I can bring in to make this review appear evenhanded. Their operational definition of mass uses rigid rods instead of clocks, which will cause problems later for the students in relativity. I expected conservation laws and Liouville's theorem to come up in the section on optics, especially since they discuss symplectic matrices. And the important Lie derivative has been left to Vol. II. The index should have been about three times as extensive, and should have included material from the examples. It would have been useful to provide pointers to further study. And of course, they are mathematicians. Thus we find separate plus signs for vectors and for numbers, but the solar system is discussed as if tidal friction didn't exist. And physicists will find it amusing to contemplate the elastic collisions of lead balls.
I don't expect these ideas to penetrate the curriculum soon. Div, grad, and curl are things that, like FORTRAN, we will be stuck with for quite a while. But those wishing to move on will find this the ideal introduction. It should be clear that I am a true believer that the material of vector calculus can be simplified, streamlined, and considerably improved by switching to exterior calculus. My first reaction to this book was to get together with some mathematicians to see how we can start up such a course here. How can I recommend it more.