After calculus, then what? Very often, a course called "Advanced Calculus". The title is not as descriptive as it could be because such courses are regulated at neither the state nor national level and what they include is variable. Most of them do elementary calculus over again, doing it right this time, but then they go all over the place. The gamma function can pop up, or Raabe's test, or even (as I found in one table of contents), area in polar coordinates. Functions of several variables always appear, but with more or less emphasis.
Professor Guzman's book is about nothing but functions of several variables, and there is a lot to be said for using it as a text for the course after calculus. For one thing, it contains a mere 246 pages of exposition, so it can be gotten through in a semester. For another, doing derivatives and integrals of functions of several variables right includes doing the single-variable case right. Students will not be bored. On page 2 we have the definition of the derivative — this time it's a linear transformation. For a third, it is written very well. It contains many helpful examples and it has good exercises, with complete solutions provided for all. It ends properly (i.e., in the same way as the text from which I learned advanced calculus) with the theorems of Green, Stokes, and Gauss. It is a text both satisfactory and satisfying.
Of course, no text is perfect. The dots that Professor Guzman uses in dot products are much too big.
Underwood Dudley has retired from DePauw University and is now living in Florida.
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