Seismological Research Letters
 © 2002 by the Seismological Society of America
Many of us who studied physics as undergraduates were encouraged to take a course in mathematical methods. The one I took, at MIT, was called Advanced Calculus for Application and was taught by Francis Hildebrand using his textbook of the same title (PrenticeHall, 1962). Here we first probed the mysterious Bessel and Legendre functions, learned to steer integration paths around the complex plane, and learned to transform x to k (and t to omega) and back again with abandon. We all found this stuff rather mysterious and intimidating. Oddly, I used rather little of it in subsequent physics courses (which were much more into different but equally mysterious quantum mechanical operations, such as those relating to spin). Only when I started to learn seismology as a grad student did the math methods find a real niche in my brain. Bessel and Legendre functions described the free oscillations of the Earth (or at least would if the Earth had the good sense to be homogeneous). Steering contours through branch cuts—ah! branch cuts!—were necessary to pull the firstarriving head wave out of a synthetic seismogram. And Fourier analysis was indeed at the basis of everything that we did. Two paradoxical lessons emerge: The first is that connecting the methodology and the physics makes for better learning. The second is that sometimes connections can be made only in hindsight.
Herein lies both the strength and weakness of A Guided Tour of Mathematical Methods for the Physical Sciences by Roel Snieder. The book covers essentially the same mathematical methods as my old Hildebrand text (and the many more recent texts of similar scope). But it does a much better job of connecting the math to the physics. Not only are many examples and problem sets provided, drawn from diverse areas of applied physics, but each method is introduced in a way that strongly associates it with some fundamental physical principle. The discussion of Gauss's Theorem, for example, makes an immediate connection to patterns of gravitational field lines and their sources. Elsewhere in the book connections are made to acoustics, continuum mechanics, Coriolis forces, electrostatics, fluid mechanics, magnetic induction, mantle convection, normal modes of vibration, quantum mechanics, radiogenic heating, soap films, thermal conduction, nuclear fission, satellite geodesy, stress and strain, surface waves, tomography, wingtip vortices, etc. I find some of these connections really fascinating. I had never realized, for instance, that Schrödinger's equation embodies the same conservation laws as does a moving fluid. The diversity, however, is also a liability. Someone with a less broad physics background than I might find the rapid shifts from one area of physics to another rather intimidating.
In the introduction, the author indicates that the book is meant for upperlevel undergraduate students or lowerlevel graduate students in physics and geophysics. The inclusion of geophysics into the spectrum of the subdisciplines of physics really does come across in the book and will be appreciated by all of the geophysics students who use it. Furthermore, the choice of material very much corresponds to what I would consider to be the most important methods for geophysicists to know. Subjects with wide geophysical application, such as Fourier analysis and perturbation theory, get a great deal of attention. Subjects that seem to have little application, like group theory—and why did I study group theory, anyway?—get none. All of the material is explained very well. For instance, this book has the best discussion of Green's functions that I know of. So I would certainly steer students toward this book if they were struggling with that subject, or for that matter, most of the other subjects covered by the book. In contrast, I don't much like the section on Cartesian tensors. Such a term is rather an oxymoron, isn't it? The whole point of tensors is their coordinate invariance. But overall, I find the book quite appealing.
On the other hand, I don't think that I could use the book as the primary text in a upperlevel undergraduate or lowerlevel graduate geophysics course. The diversity is part of the problem. My experience in the classroom is that students have a hard time coping with the diversity of solidearth geophysical phenomena (seismic waves, geomagnetism, gravity, heat flow) alone. Adding quantum mechanics (and etc.), when most of my students have probably never even taken an introductory course in it will just make matters worse. But another problem is that the book really does live up to its title of being a “guided tour.” It moves very rapidly though a rather substantial body of material, quite a bit more material than I would feel appropriate for a single course directed at students who had never seen any of it before. The book would be fantastically suitable for a “review course” taught to students who had already seen most of the material in some other context. Alas, we never find the time to teach such courses at Columbia (and if we did, our students would never find the time to take them). And it will certainly find a place as a selfstudy guide for aspiring geophysicists, especially those who might be preparing for Ph.D. qualifying exams. It's a fascinating and noteworthy book (but keep in mind that it's not exactly light reading).
Footnotes

Roel Snieder Cambridge University Press, 2001 Hardcover ISBN 0521782414; paperback ISBN 0521787513