Once upon a time textbooks on mathematical methods were focused almost entirely on physics and engineering. Of these perhaps the best-known examples are Courant and Hilbert’s *Methods of Mathematical Physics* and Morse and Feshbach’s *Methods of Theoretical Physics*. Primary topics included differential and integral equations, the calculus of variations and special functions. Now, with many disciplines expecting more extensive mathematical skills from graduate students, the scope and character of “mathematical methods” books have been changing. The current book is a good example of that trend.

The book is aimed at students in the earth sciences, but particularly those in geophysics, atmospheric, oceanic, planetary and space physics, as well as astronomy. The author notes that there is significant overlap between the needs of these students and those in physics and applied mathematics. At the same time there are notable differences in the preparations of students in the earth sciences. Accordingly a presentation of some topics in a different sequence and with motivating examples closer to their scientific interests is desirable.

The author also comments that the background of first and second year graduate students in these fields can be quite variable, so he feels that there are advantages in providing derivations to show students how important relevant equations arise and how their solutions might be addressed. The general area of geophysics now relies on a broader collection of mathematical methods — including probability, statistics, numerical methods, nonlinear dynamics and chaos— and students need some competence in working with them.

The book begins with a short chapter on mathematical preliminaries that includes everything from notation and coordinate systems to the Gauss-Green-Stokes theorems, matrix and tensor representations and the Dirac delta function. Then the next chapter addresses ordinary differential equations in pretty much the usual way: first and second order linear equations, then nonlinear equations. The variations are interesting — greater emphasis on driven oscillators and resonance, earlier treatment of Green’s functions, and examples more relevant to geophysics. The first example of a nonlinear equation is based on Bullard’s homopolar dynamo, a model of the earth’s dynamo roughly analogous to an LRC electric circuit in which a conductor moves steadily in a constant magnetic field and produces a direct current.

A chapter on evaluation of integrals and integral transform methods follows. In about fifty pages it includes quite a few topics: steepest descent approximations, contour integration, Fourier, Radon and Abel transforms, as well as some inverse theory, calculus of variations and integral equations. Whew!

The fourth chapter introduces the fundamental partial differential equations most relevant to geophysics. These include the wave, gravitational potential (relating a planet’s mass distribution to its potential) and diffusion equations. The author presents complete solutions in three dimensions of these equations while developing Green’s function methods. He introduces the student to spherical harmonics to solve the gravitational potential equation. An extended example develops elements of perturbation theory with a simple nonlinear partial differential equation arising in the propagation of sound waves. A final section introduces the ideas of self-similarity and scaling and applies them to turbulence to present a derivation of the Kolmogorov spectrum.

The final chapter treats mostly standard topics of probability and statistics with a view toward equipping geophysics students with a better understanding of the goals and practices of data analysis. This chapter also includes a treatment of basic techniques in numerical analysis.

The book is intended for a one semester or one quarter long course. A great deal of material is introduced in somewhat less than 250 pages. Students who use the text will be exposed to a lot of mathematics, but it is not clear how much of it that they could assimilate. The author calls his book a “whirlwind survey”, and it is. Many sections are clearly written with an adequate level of detail, but more than a few touch on their subjects very briefly. While this would be an attractive book to accompany a lecture course, it would not be ideal for self-study.

Bill Satzer ([email protected]) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.