A COURSE IN MATHEMATICS FOR STUDENTS OF PHYSICS: VOLUME I
Paul Bamberg and Schlomo Sternberg, 403 pp, Cambridge UP
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Change vector calculus? What an unlikely place for a revolution. Yet over the
last few decades there have been important changes in calculus. I quote the
authors of this book: "It is generally accepted in the mathematics community,
and gradually being accepted in the physics community, that the most suitable
framework for geometrical analysis is the exterior differential calculus of
Grassman and Cartan." You can find about a dozen books on exterior calculus,
that is, the calculus of differential forms, written for physicists. The one
reviewed here is the most elementary. It is excellent, and provides a solid
foundation. Until now one only learned this material in graduate school, where
much of the new work in theoretical physics, and of course general relativity,
is done in this language.
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.
William Burke