This is a book about mathematics written by a physicist for other physicists. The author, Walter Appel from École Normale Supérieure de Lyon, is fairly candid in his assessment of physicists’ thinking:

“There is a fairly fashionable current of thought that holds that the use of advanced mathematics is of little real use in physics, and goes sometimes as far as to say that knowing mathematics is already by itself harmful to true physics.”

The author proceeds to develop a staunch defense of the value of mathematics in physics. Given the deeply interwoven history of the two disciplines and renewed fruitful interactions between mathematics and physicists over the last two decades, it is hard to believe that mathematics needs defending. Perhaps we are all still paying for the sins of Bourbaki.

*Mathematics for Physics and Physicists*was a bit of surprise to me. I had expected a more or less standard “methods of mathematical physics” approach, and — while there is some of that — it goes deeper. A good example is the first chapter, which treats some elementary topics on the convergence of sequences and series. Appel provides several examples where paradoxes arise because of limiting operations — either where the limiting operation is hidden in a calculation or where the order of a double limit is switched without justification.

The majority of the book is a collection that might be called “all the mathematics that a physicist needs to know”, except that Appel has a broader view than one might expect. For instance, the first two chapters treat the theory of integration, including the Lebesgue integral and some measure theory. A couple of later chapters develop the theory of distributions (i.e., generalized functions), motivated by problems in electrostatics and elastic shock. Throughout the book Appel maintains a nice balance between rigorous mathematics and physical applications. He provides a splendid treatment of Green’s functions, a subject that often gets a very mysterious treatment in physics texts.

Other topics included here include an extended treatment of complex analysis (including conformal mappings), Hilbert spaces and Fourier analysis (with several good physical examples), some differential geometry, a modest amount of probability and random variables, a little bit of group representation theory, and a brief treatment of Dirac’s ket and bra notation in quantum mechanics in the context of operator theory. There is a lot of mathematics here.

Each chapter has a modest collection of exercises, and fairly complete solutions are provided for some of them. This is an attractive introduction to mathematical physics probably most suitable for first or second year graduate students.

This book was originally published in 2001 as *Mathématiques pour la physique… et les physiciens!* The current volume was translated by a mathematician and the mathematics comes across smoothly. In some places there are odd phrasings, unusual word choices and some spellings that retain a French flavor, but overall it reads quite well.

Bill Satzer ([email protected]) 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.