ADVANCED MATHEMATICAL METHODS WITH MAPLE,
D. Richards, Cambridge University Press, 2001, pp: 853; ISBN 0-521-77981-2 (pbk), 0-521-77010-6 (hc); Price: $55.00/130.00.

I am attending a conference in Spain this year, and since reading this book, I may just add a short sidetrip while I am there, to visit a small northern city that figures in the book. No, not Hemmingway, and I won't be running with the bulls in Pamplona. Instead, having just read Advanced Mathematical Methods with Maple by D Richards, I now have to see for myself the remarkable swinging censer of the cathedral there, used by Richards to introduce parametric resonance in linear systems. He admits to some poetic licence: with a swing of amplitude 80 degrees, the swinging censer is not a linear system. For that matter, with a maximum velocity at the bottom of the swing of nearly 70 km/hr and a mass of 57 kg, it sounds every bit as intimidating as a Pamplona bull.

Some of you will be familiar with a previous book by Richards. Several years ago Percival and he wrote the nice, succinct Introduction to Dynamics. In contrast, his most recent effort is expansive, comprising a nearly complete survey of mathematical analysis techniques used by physicists, engineers, and applied mathematical analysts. Here I mean analysis in the sense of the mathematical subject that goes by that name. This book is not the place to find algebra, or even linear algebra.

The book stands apart from most texts on the subject in several regards. The first three chapters are dedicated to the use of Maple software to solve problems in applied mathematics. These chapters are an excellent introduction to and summary of Maple's capabilities. The material is more directly useful than what one finds in books dedicated exclusively to Maple, many of which (1) cost more than Richards' book in softcover, (2) teach less Maple, and (3) have none of Richards' mathematical content. Maple is integrated throughout the text, but the text is perfectly useable without Maple: some of the exercises require Maple, usually for graphing, but most do not.

Another distinction is the book's modern viewpoint. For example, there are several chapters on dynamical systems, including qualitative theory. And there are the wonderful little vignettes, such as the one about the swinging censer of Santiago. Some of these stories relate to modern research topics such as the discussion of the quantum-inspired Hermitian operator approach to the Riemann hypothesis in the chapter on series. Citations to the research literature are provided.

What is it about the bindings in Cambridge softcover books? My review copy threatened to fall apart soon after I opened it. The softcover edition is very reasonably priced, but $55 should be enough to entitle the purchasers to a decent binding.

Is the book suitable for a text in mathematical methods at the undergraduate level in physics? For a fourth-year course that focuses on analysis and does not cover topics in algebra, it would be an excellent choice. However, it would not serve as a text that could also be used in earlier, more elementary courses with broader focus. I found no background discussion of linear algebra, though there are a few condensed remarks in relevant sections but only enough to get by. There was also no real discussion of series solutions of differential equations. These are both topics that are usually covered in introductory courses before fourth year.

Somewhat oddly, the quite elementary topics of sequences, series, and the convergences thereof were covered, perhaps reflecting the analytical focus of the book. The book does cover the standard advanced analysis topics of Fourier series, Sturm-Liouville theory, special functions, Green functions, stationary phase and the WKB approximation. And as already mentioned, there is plenty of extra material on Maple and on dynamical systems, and there is a whole chapter on asymptotic expansions as well. There are more than enough good exercises throughout. If all this sounds like your course, then this book would be an excellent text.

E. Woolgar
University of Alberta