Physical Mathematics Kevin Cahill Cambridge University Press, 2013 666pp., USD xxx, Hardcover ISBN: 9781107005211 undergraduates and above Dr. Manuel Vogel, TU Darmstadt and GSI Darmstadt,
[email protected] This book represents a comprehensive collection of mathematical tools as needed and commonly applied in physical sciences. While covering a broad range, individual sections generally start with a concise and direct presentation of a certain mathematical topic and then give an example of its application, mostly to a physics problem. The presentation is concise in the sense that no lengthy derivations are given, but definitions and their consequences as well as connections between the topics and some historic background are provided. There are comparable books on the market which treat the borderline between mathematics and physics, sometimes as physical mathematics, sometimes as mathematical physics, depending on the point of view, and normally this does not matter in the end, as long as the material is well-selected and well-presented. This book is indeed very modern with regard to notation, language, style of presentation, but most importantly in the selection of topics. It treats Linear algebra, Fourier series, Fourier and Laplace transforms, Infinite series, Complex-variable theory, Differential equations, Integral equations, Legendre functions, Bessel functions, Group theory, Tensors and local symmetries, Forms, Probability and statistics, Monte Carlo methods, Functional derivatives, Path integrals, The renormalization group, Chaos and fractals and Strings. This selection and hence the book as a whole go back to a series of lectures the author has taught at the University of New Mexico and at Fudan University in Shanghai. It basically represents all the common mathematics you need to know when studying physical sciences at a University and taking courses, for example, in mechanics, electrodynamics, quantum mechanics or statistical mechanics. Its structure and style of presentation also make it a valuable reference for researchers. There is a link on the Cambridge University Press website (www.cambridge.org/cahill) about this book which contains a list of errata as part of the resources (also mentioned in the preface), and particularly a list of links to video material showing these lectures in full length and making the connections to the corresponding sections in the book. This site also contains solutions to the end-of-chapter problems which are accessible to instructors (registered users). Within the book, no solutions to the problems are given, but further reading is suggested. Throughout, the text is referenced to other books and journal articles, the references are given in the backmatter, together with a comprehensive index. The quality of the paper, the print, and of the bookmaking is excellent. Overall, this is a valuable source of well-selected and well-presented material building a bridge between mathematics and physics. It is accessible to advanced undergraduates and graduate students in physical sciences and makes a good preparation for lectures about to be attended. It also serves well as a reference for researchers who wish to be reminded about certain mathematical topics.
