Mathematics for Physics: A Guided Tour for Graduate Students

Mathematics for Physics: A Guided Tour for Graduate Students , Michael Stone and Paul Goldbart , Cambridge U. Press, New York, 2009. $90.00 (820 pp.). ISBN 978-0-521-85403-0

Without textbooks, the education of scientists is unthinkable. Textbook authors rearrange, repackage, and present established facts and discoveries—along the way straightening logic, excluding unnecessary details, and, finally, shrinking the volume of preparatory reading for the next generation. Writing them is therefore one of the most important collective tasks of the academic community, and an often underrated one at that. Textbooks are not easy to create, but once they are, the good ones become cornerstones, often advancing and redefining common knowledge. That is why I welcome the recent appearance of Mathematics for Physics: A Guided Tour for Graduate Students by Michael Stone and Paul Goldbart.

At the end of World War II, a scientist’s standard mathematical arsenal consisted of calculus, differential and integral equations, and complex analysis, as presented, for instance, in Methods of Theoretical Physics (McGraw-Hill, 1953) by Philip McCord Morse and Herman Feshbach. Even now, those topics remain the core of mathematics taught to scientists. But many physicists felt the need to include in the compulsory scope such modern topics as Lie algebras and groups, differential geometry, and topology. Some “mathematics for physicists” authors have made meaningful attempts to include the new topics; those additions are discussed in a number of recent textbooks. Still, only a minority of physicists are using those modern topics in their practice.

Stone and Goldbart’s “Guided Tour” has the potential to change that. The authors have drawn from material they’ve taught for many years to first-year graduate students at the University of Illinois at Urbana-Champaign. That experience has resulted in a text that is unique in its construction and approach and filled with high-quality examples. The book begins with traditional material, including the calculus of variations, Sturm–Liouville theory, and integral equations. But even here, the authors manage to introduce original pedagogical elements and examples. Subsequent sections discuss modern topics such as differential and integral calculus on manifolds, differential topology, and Lie algebras and groups. The book ends with an original complex analysis that combines the ideas of geometry and topology with purely analytical material. Some of the aforementioned topics are found in other texts, but in Mathematics for Physics they are combined into a single, and relatively thin, volume. That achievement may well become the foundation of a modern mathematical minimum for such textbooks.

Remarkably, the authors have mastered the pedagogical art of making their material both inspiring and digestible. Stone and Goldbart are both theoretical physicists, and their presentation is closer to the manner in which physicists communicate mathematics: They pay less attention to proofs and place more emphasis on concepts and examples. One important element of the book is the large number of exercises. The authors’ refined judgment and broad knowledge of physics allow them to bring to their exercises material from distant areas of physics. I have no doubt that readers will be inspired in their study of nontrivial mathematics once they are exposed to such physics applications as solitary waves on shallow water, faceting of crystals, electron energy levels of the C₆₀ molecule, and supersymmetric quantum mechanics.

The amount of material in Mathematics for Physics is definitely more than enough for two single-term courses; that provides a potential lecturer considerable flexibility. It also gives students an opportunity to go beyond what is presented in the lectures. The many features that make the book valuable to students and teachers also represent a substantial step toward making modern mathematics a part of the working arsenal of practicing physicists. I strongly recommend it to those who feel the need to upgrade their mathematics repertoire.