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Mathematical Physics: An Introduction

Derek Raine
Mercury Learning & Information
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

This introduction to mathematical physics is aimed at undergraduate students in physics. The lead author notes in his introduction that he and his colleagues (at the University of Leicester) found forty years ago that their traditional approach using lectures, graded homework and exams did not work very effectively. They then apparently produced prototypes of the current book as lecture notes and redesigned the class to focus on weekly workshops and small group tutorials. This book was prepared as an “outward facing version” for a larger audience.

The title is a bit misleading; the book is really an introduction to mathematics for physics students beginning with very basic calculus. There is surprisingly little physics in the book, even when its presence might have provided some desirable motivation.

The book has several unusual features that mirror the teaching approach that the authors propose. They use examples to introduce the underlying theory and present the material in very short sections followed by exercises. Routine exercises are avoided in favor of ones designed to be more diagnostic and more clearly aimed at testing understanding. The authors note that their book, while it could be used for independent study, is specifically intended to be a course text.

Very roughly speaking, the text is broken into parts that include the basic elements of calculus, linear algebra, ordinary and partial differential equations, multivariable calculus, and Fourier series. Each chapter is focused on teaching students the mechanics of computing things — derivatives, integrals, solutions of linear systems and differential equations, Fourier series and so on. Discussions of theory are minimal. For example, while Maclaurin and Taylor series are discussed and computed, nothing like the Mean Value theorem appears. L’Hôpital’s rule appears but without any justification or cautions. Infinite series are briefly considered and convergence is touched on, but there’s little of substance. Nothing is proved anywhere in the book, though the author indicates, for example, that the chain rule could be proved from the definition of the derivative.

My sense is that this is a very efficient way to teach students the mechanical elements of the mathematical techniques that they need to do basic physics. But this careful step-by-step approach with examples followed by simple exercises is monotonous and drill-like. The chapter on vector calculus is typical. It discusses vector fields in about three pages, introduces the gradient, shows two basic examples and poses an exercise. Level surfaces and normals to a surface get similar treatment. Divergence and curl of a vector field are treated next and in the same way. Although this would be a natural place at least to allude to applications in physics, the author adds very little by way of motivation.

It could be that the author’s intention is that all these topics would be expanded and fleshed out in succeeding courses. But the content of the current book is forbiddingly dull. It is like learning carpentry by studying individual tools one-by-one but never using them together to build something. Nowhere does a student get a sense of the breadth of the whole subject or the depth of any single part.

The author and his colleagues have clearly given their approach a lot of thought, and the book is carefully and logically assembled. Several of the end-of-chapter exercises are quite good. Nonetheless this is not a book I’d recommend to anyone.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

The table of contents is not available.