This is a comprehensive introduction to the study of functions of several variables that includes several areas not commonly included in comparable textbooks. The authors have written three other books on various aspects of analysis. All of these have *Mathematical Analysis *as their titles, but they have different subtitles: Functions of One Variable, Approximations and Discrete Processes, and Linear and Metric Structures and Continuity. The current volume, as the authors acknowledge, is the most difficult because of its broad scope and intrinsic difficulty.

In some respects, this book reminds me of Wendell Fleming’s *Functions of Several Variables*, at least in approach and level of difficulty. The current book has a generally broader scope, except that it includes essentially nothing about manifolds. It begins with the definition of a directional derivative and roars on from there. By the end of the first chapter, the reader has seen all the basic theorems of the differential calculus of several variables in **R**^{n}, as well as a bit of differential calculus on Banach spaces.

The second chapter develops the Lebesgue integral (including the basic convergence theorems), does some measure theory and then proves the Gauss-Green theorem (no differential forms here). The next chapter, however, is all about curves and differential forms and it concludes with a proof of Stokes’ theorem in the plane.

Next comes complex analysis: a chapter on holomorphic functions moving from Cauchy’s theorem and the calculus of residues up through the Riemann mapping theorem. Whew! And there’s more. A very nice chapter on surfaces and level sets includes the implicit and inverse function theorems, Sard’s theorem and Morse’s lemma, as well as an extended section on curvature of curves and surfaces. The final chapter takes up systems of ordinary differential equations and includes a discussion of stability considerations and Poincaré-Bendixson theory.

This book leaves me with mixed feelings. There is a huge amount of mathematics here, presented carefully and with style. Yet the approach is occasionally rather eccentric. The authors choose to avoid discussing manifolds, yet they slip in a quick unmotivated definition of a submanifold of Euclidean space just before they want to use it. They build the machinery to prove a general version of Stokes’ theorem, but prove it only in the plane. The treatment of holomorphic functions here is nicely done, yet not really connected with the rest of the text. I suspect that it may be included primarily for completeness, to fill out the complex analysis component of the authors’ ambitious and comprehensive series. In the end, I find that this text would be an agreeable source for most of its individual topics, but it doesn’t quite hang together as a unified work.

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.