Mathematical Methods for Physicists: A Concise Introduction. Tai L. Chow. xvi + 555 pp. Cambridge University Press, 2000. $39.95.
Modern Mathematical Methods for Physicists and Engineers. Cyrus D. Cantrell. xx + 763 pp. Cambridge University Press, 2000. $120 cloth, $49.95 paper.
In most physics departments, "math methods" is a standard course for advanced undergraduates or graduate students. Such courses aim to develop students' fluency in the language of physics, so that they can then concentrate on understanding its substance. Clearly written, comprehensive and up-to-date texts on mathematical methods are therefore welcomed by those of us who teach such courses. Tai Chow's book, designed for intermediate-level undergraduate students in physics, engineering and mathematics, and Cyrus Cantrell's book, aimed at senior undergraduate and graduate students in the physical sciences and engineering, and at practicing engineers, are two new texts in mathematical physics. They compete with texts such as George B. Arfken and Hans Weber's Mathematical Methods for Physicists (5th ed., Academic Press, 2000) and Sadri Hassani's Mathematical Physics: A Modern Introduction to its Foundations (Springer-Verlag, 1999).
Chow's Mathematical Methods for Physicists aims to prepare majors for the core courses in physics, and it is therefore at a lower level than is usual for such texts. It includes vector and tensor analysis, ordinary and partial differential equations, matrix algebra, Fourier series and integrals, vector spaces, complex variables, special functions, calculus of variations, Laplace transforms, integral equations, group theory and once-over-lightly presentations of numerical methods and probability theory. Because the target audience is less mature, algebraic details are often presented, whereas they are usually omitted in books such as Arfken and Weber's, Hassani's or Cantrell's. Chow's text is therefore useful for second-year students and for students who are puzzled by "it can be shown that" references in other mathematical physics texts. It has many examples and exercises, as well as a complete index. There is no bibliography, which I consider a defect; however, there is a list of suggestions for further reading.
Cantrell's Modern Mathematical Methods for Physicists and Engineers is directed to students at a higher level and is more mathematically sophisticated than Chow's book. Its topics include foundations of computation; sets and mappings; evaluation of functions; groups, rings and fields; vector spaces; linear mappings and functionals; inner products; and special functions. The author provides many interesting examples and exercises. The book requires students to have at least a year more of mathematical maturity than does Chow's text. It is written from a mathematician's rather than from a physicist's mind-set, which may discourage its use in physics courses. Although the preface and the publisher's advertising emphasize computers and computation, very little in this book relates to the practical use of computers to solve problems in mathematical physics.
Unfortunately, something has gone awry with the indexing: Many of the page numbers in the index entries are incorrect, at least in the paperback version, although the terms incorrectly indexed can often be found within a couple of pages of the number cited.
Physicists and engineers often choose to train in those disciplines because they have a more concrete view of the world than do those trained only in mathematics. I therefore expected to find many figures in both books, especially since computer-generated figures are easy to prepare nowadays. Alas, I was sorely disappointed. For example, in Cantrell's text the chapter on special functions (which considers only cylindrical Bessel functions) has six figures, and the corresponding chapter in Chow's text (with eight special functions) also has only six figures.
Textbooks developed from authors' lecture notes commonly have a defect: The topics are often just those the author has found suitable for the local academic environment. For a text to be broadly useful, it must transcend local practice and include topics likely to be useful in other educational settings at the same level. In my opinion, this has not been done very well in either of these books. For example, in Chow's book the 20 pages each on numerical methods and probability theory may be useful at his institution but will not be relevant to a mathematical physics course in most physics departments. Cantrell's text has more than 20 pages on cylindrical Bessel functions but almost nothing on any other special function.
Overall, Chow's book will be useful for introductory courses in mathematical physics, whereas Cantrell's book may appeal to students who wish to delve deeper into the mathematics.—William J. Thompson, Physics and Astronomy, University of North Carolina at Chapel Hill