Mathematical Analysis of Physical Problems

This text on mathematical physics succeeds by presenting a careful balance of mathematical technique and physical application. First published in 1972, republished by Dover in 1984 and reissued in 2011, this is a classic. It is not exactly “methods of mathematical physics,” for it engages with physical problems in a way that many books on that subject do not. Neither does it have the scope of Hilbert and Courant’s well known text. The author’s aims are more modest, and he is largely successful in achieving them.

The author writes that a book on mathematical physics “…tries to construct workable (and necessarily approximate and incomplete) models of aspects of physical reality.” His book incorporates descriptions of mathematical structures that provide framework for the development of theories describing physical phenomena. The test he prescribes for the adequacy of the mathematical framework is first, whether it provides an effective description of the physics, and then whether it enables answers to physical problems that can be tested by observation.

The selection of topics is dictated by the author’s desire to “link classical and modern physics through common techniques and concepts.” A significant component of this is the theme of wave phenomena as a testing ground for concepts and methods of wide relevance across physics. The most “modern” subject treated here is quantum mechanics, and the author has chosen not to include any group theory.

The book begins with a treatment of the vibrating string and the one-dimensional wave equation. This is followed a few chapters later by an extended treatment of propagation and scattering of three-dimensional waves. In between the author fills in background on vector spaces as well as Fourier and Laplace transforms. There is also an intervening chapter on potential theory. Each chapter gives the author the opportunity to introduce mathematical concepts and techniques in the context of physical systems. Other chapters include extended discussions of diffusion and attenuation and probability with stochastic processes. The final three chapters provide a quite readable introduction to quantum mechanics.

Each chapter begins with a prelude that outlines the chapter’s contents and suggests the background that’s required. There are many exercises scattered throughout, and these are intended to be an integral part of the text. Some of them test the reader’s understanding and others provide opportunities to use just-described techniques on related problems. The book is aimed at advanced undergraduates and beginning graduate students in mathematics and physics, but it is accessible to anyone with some background in differential equations and multivariable calculus. The book as a whole has aged well; it continues to be a valuable introduction to mathematical physics.

It is worth noting, however, that this is a book that uses mathematics in the style of a physicist. So, for example, functions are always integrable and series always converge. The author approvingly repeats the physicist Landau’s comment that mathematical rigor has no relevance to physics. This by no means reduces the value of the text, but readers should be aware of the bias.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.