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*A Course in Mathematics for Students of Physics*

by Paul Bamberg and Shlomo Sternberg

Cambridge University Press, 1988

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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.