This is an introduction to methods of applied mathematics. When it was first published in 1956 its approach must have been quite novel. At a time when it appeared that pure and applied mathematics were on widely divergent tracks, the author aimed for a modest reconciliation. He says in the preface that he wrote the book “to show how the powerful methods developed by the abstract studies can be used to systematize the methods and techniques for solving problems in applied mathematics”. The book emerged from a first year graduate course for students of physics, engineering and applied mathematics that the author taught at New York University. Recommended prerequisites are linear algebra and complex integration.

Fully a third of the book is devoted to vector spaces and operators defined on vector spaces. Except for a few short sections the author’s extensive presentation of the spectral theory of operators is developed only in finite dimensions. This makes the treatment more concrete, avoids the technical complications arising in infinite dimensions, and seems to work reasonably well in an introduction to the subject. The author unfortunately chooses a notation for inner products that lies somewhere between usual mathematical practice and Dirac’s bra and ket approach. He says he does this to “avoid overworking parentheses”, but ends up with a confusing proliferation of pointy brackets that is worse than either alternative.

A good deal of the remaining part of the book deals with Green’s functions. The treatment relies extensively on the operator theory presented earlier. It largely focuses on using Green’s functions to solve ordinary and partial differential equations. The author introduces the theory of distributions early, mostly to enable a more rigorous interpretation of the Dirac delta function. (Distributions had been described by Laurent Schwartz only a few years before this book was first published — another indication of how forward-looking it was at the time.)

The remaining chapters are also heavily dependent on the operator theory of the early chapters. One chapter investigates eigenvalue problems for ordinary differential equations using the spectral representation theory for linear differential operators. The last chapter uses operator methods to study the Laplacian, and thereby solve the common partial differential equations of mathematical physics: the potential equation, heat equation and wave equation.

While it feels a bit dated this remains a solid introduction to classical methods of applied mathematics.

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.