A Course in Mathematics for Students of Physics
by Paul Bamberg and Shlomo Sternberg
Cambridge University Press, 1988

Reviewed by Danny Goroff

Bamberg and Sternberg provide an offbeat and intriguing example of a text that covers linear algebra, multivariate calculus, and many other topics for students in a client discipline. Assuming only familiarity with one variable calculus as a prerequisite, it launches into a resolutely twentieth century approach to mathematics. The notions and notations therefore do not always look the same as those in traditional texts. For example, the authors place great emphasis on the role of coordinate transformations and the power of coordinate free representations. Though conceptually deep in this way, the treatment is not particularly formal. Rather, it relies for motivation on very physical considerations. In the first chapter, for example, their discussions about space and time cause the authors to give more attention to affine spaces than to linear ones. The next few chapters develop properties of linear transformations and matrices in the 2x2 case motivated by problems concerning Markov processes and linear differential equations. The chapter on scalar products also takes up normal modes and special relativity. The middle of the first volume develops calculus using the language of differential forms. Linear algebra returns in a chapter on Gaussian optics. This is followed by one that develops the abstract theory needed for studying electrical circuits, including duals spaces, quotient spaces, and adjoint transformations. The same machinery is used to introduce some algebraic topology, especially its relation to differential forms. Thinking about electrostatics from this point of view allows for an actual derivation of Maxwell's equations by simply transforming under a Lorentz transformation suitably coded versions of Ohm's Law and Kirkoff's Law. This seems like a remarkable result for undergraduates to be able to appreciate. While daring, inspiring, and full of good problems, the text itself is not very polished and has not found widespread use in introductory courses. Some similar ideas have begun to appear in more down to earth books, however, including those by Strang, Bressoud, and Burke.