Real Analysis: Series, Functions of Several Variables, and Applications

This is the second volume of a two-volume set; please also see our review of the first volume, Real Analysis: Foundations and Functions of One Variable. This second volume deals with multivariate calculus and with series of numbers and of functions. It is not closely integrated with the first volume (there are few cross-references or explicit dependencies), but it does assume the reader has already had a rigorous course in single-variable calculus. These books were published originally in Hungarian and now in English translation. By American standards these books are closer to being “calculus with proofs” than they are to real analysis.

The present book has a thorough treatment of multivariate derivatives, viewing them both as linear transformations and as partial derivatives, and has rigorous proofs of the inverse function theorem, implicit function theorem, and Lagrange multipliers. It has good coverage of multiple integrals on fairly arbitrary (i.e., Jordan-measurable) sets and a good discussion of how to evaluate integrals by repeated integration and the theory and practice of changing variables (integration by substitution) using Jacobians.

The treatment of line and surface integrals is skimpier. The book deliberately omits a lot of the details of curves and surfaces and focuses more on the mechanics of evaluating these integrals. There’s a warning on p. 175 to consider this part of the book “a popular introduction”, although that is too harsh.

The book has good coverage of infinite series, both for series of numbers and of functions. Most books would have covered this earlier in the single-variable portion, and I think the authors felt they could stretch out some since they don’t cover it until near the end of the second volume. Convergence of sequences was already covered in the first volume and this book assumes and builds on this for convergence of series. There are some unusual topics such as Cesàro and Abel summation, multiplication of series, and more depth than usual on convergence criteria.

The last chapter gives a wide variety of applications to other areas of mathematics, such as numerical analysis and number theory. The book is weak on applications outside of mathematics, but there are a few physics applications in the line and surface integral sections.

The exercises are reasonable for a calculus book but weak for an analysis book, as most of them ask for calculations regarding specific functions rather than for proofs of theorems. Many of the exercises has solutions in the back, and some also have hints.

Bottom line: the book is well-done within its limitations, but it seems awkwardly-placed for the American curriculum. It’s too rigorous (and weak on applications) for a calculus course, it’s not general enough for an analysis course, and it skimps on the calculus-on-manifolds details needed by those who really want to understand line and surface integrals. A good recent book with similar coverage (single and multivariate) that is slanted toward analysis but stays concrete is John B. Conway’s A First Course in Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.