Mathematical Methods: For Students of Physics and Related Fields Sadri Hassani Springer-Verlag, New York, 2000. $69.95 (659 pp.). ISBN 0-387-98958-7
Sadri Hassani’s Mathematical Methods is the latest addition to the already long list of textbooks for an undergraduate course on mathematical methods for students of physics, engineering, and related fields. Four of these books (not including this one) were recently discussed in an excellent comparative review by Donald Spector (American Journal of Physics , volume 67, page 165, 1999). In terms of target audience, pace, and field coverage, Hassani’s text is most nearly comparable to Mathematical Methods in the Physical Sciences, second edition, by Mary L. Boas (Wiley, 1983).
The topic coverage is fairly standard—infinite series, vector analysis, eigenvalues and eigenvectors, complex analysis, and ordinary and partial differential equations. There is an entire chapter on the Dirac delta function, including two- and three-dimensional versions, that I particularly like, and a chapter on nonlinear dynamics and chaos that is also well done. Topics covered in other books at this level but not covered here include the calculus of variations, probabilities, tensors, integral transforms, and the use of computers either for symbolic computing or numerical analysis. The first I don’t particularly miss; students can learn the calculus of variations, when the need arises, in a classical mechanics course. The rest, however, I do miss to varying degrees.
The book has many strengths. For example: Each chapter starts with a preamble that puts the chapter in context. Often, the author uses physical examples to motivate definitions, illustrate relationships, or culminate the development of particular mathematical strands. The use of Maxwell’s equations to cap the presentation of vector calculus, a discussion that includes some tidbits about what led Maxwell to the displacement current, is a particularly enjoyable example. Historical touches like this are not isolated cases; the book includes a large number of notes on people and ideas, subtly reminding the student that science and mathematics are continuing and fascinating human activities.
Sometimes, the author deviates from the traditional subdivisions to stress connections among different topics and techniques. For instance, he unifies the discussion of finite-dimensional vector spaces and orthogonal polynomials so that, when Fourier series or special functions are later used to solve ordinary differential equations, the notion of a function space with an orthogonal basis set is no longer entirely new. There are numerous detailed examples that students should find helpful, and many and varied end-of-chapter problems.
On the other hand, the first two-and-a-half chapters review material that students taking a mathematical methods course should already know. The chapters are not just straightforward review, however. It seems as if the author has identified certain problems and misconceptions that he is determined to exorcise, but the exorcisms can sometimes lead to more confusion. One of his candidate misconceptions is the notion that a derivative is “just a rate of change.” His solution is to assert (in a statement enclosed in a box for emphasis) that a derivative is just “the ratio of two physical quantities which are defined locally.” This may work if the quantities are functionally related, but it does not quite exclude the possibility of trying to define a derivative in the absence of a function.
The book contains numerous unwarranted generalizations (“in all cases of physical interest the separation of variables works”). It also has too many footnotes (18 in chapter 3, 31 in chapter 8—I find them distracting), too many acronyms (PDE, ODE, FODE, SODE, HSOLDE). Additionally, considering the large number of historical notes, some people are bound to take exception to one or another of them. My list includes such things as stopping at Julian Schwinger and Richard Feynman in a list that should have included Sin-Itiro Tomonaga, or describing a positron as a particle with negative energy—which, as P. A. M. Dirac himself cautioned, “would make the dynamical relations all wrong.”
I would find the book more attractive if the first couple of chapters were shorter, or had avoided “exorcisms,” or were omitted altogether. Then Hassani might have made room for some discussion of probabilities, tensors, and integral transforms. Furthermore, at a time when the physics community is becoming more aware that linear, or separable, or exactly soluble problems do not exhaust the complexity of the physical universe, and the use of computers is more ubiquitous, some discussion of the use of computers for numerical analysis would have been welcome. All in all, however, this book could provide Boas with some serious competition, especially if her forthcoming third edition, scheduled for January 2002, does not come forth soon.
