All the Mathematics You Missed: But Need to Know for Graduate School, T.A. Garrity,
Cambridge University Press, 2002, pp: 337, ISBN 0-521-79707-1 (pbk), 0-521-79285-1 (hc); Price: $25.00/55.00.

There is simply not enough time during an undergraduate physics degree to learn all the mathematics one really should. Compounding the problem of limited time is the fact that modern texts are often becoming less and less careful with their mathematical rigor. The end result is that many students find themselves unaware of what might be called "a certain level of mathematical culture". Thomas A. Garrity's All The Mathematics You Missed is a text designed to give students a rough idea of what they should really know before beginning graduate school in mathematics. Likewise, it is also a fairly reasonable summary of what a physics student should know after completing their undergraduate degree. While the text is far from rigorous in its treatment of any topic, it does succeed in presenting most of the mathematics a physics graduate student should know, all couched in a certain level of mathematical culture.

The text covers each topic in 16 brief chapters of about 20 pages. Each chapter is initiated with a summary box that outlines the basic objects and goals of the chapter. Each chapter ends with a set of exercises and a useful annotated bibliography.

The book begins with a chapter on linear algebra that is followed by several on the calculus of real and vector functions. A gentle introduction to differential forms is presented before continuing to complex analysis. Groups and rings are presented, but the words "Lie algebra" are nowhere to be found. The Lebesgue measure is covered in a succinct 11 pages. Fourier analysis is established during a search for the basis vectors of periodic functions. Ordinary and partial differential equation techniques are discussed and during one of the rare moments when physics is discussed, the wave equation of an ideal string is derived. Combinatorics and Probability are squeezed into the penultimate chapter. The book finishes with a chapter on algorithms that includes a few pages on graphs and the P=NP question. The style of the prose is candid and relaxed making reading enjoyable. However, one is sure to find plenty of definitions, theorems and proofs. There are a surprising number of typos, but a bigger disappointment is the poor quality figures.

Overall, the text is useful as a reference for forgetful minds or people looking to catch the essence of a topic their studies did not provide. The text is neither rigorous nor broad in scope, but it is sufficiently attentive to leave the reader feeling that they are in safe enough hands to learn some of the mathematics that they missed in their undergraduate years but need to know for graduate school.

Benjamin J. Sussman
Ottawa