When I was a freshman undergraduate, my basic approach to multivariable calculus was to ignore it and hope it would go away. It didn’t, of course, and so I spent a semester tortured by horrible triple integrals requiring evaluation, as well as three-dimensional diagrams that I was hopelessly bad at visualizing, much less constructing on my own. The ideas that I found interesting in single-variable calculus seemed conspicuously absent now, replaced by tedious calculation. Even when, a few semesters later, I took a course in advanced calculus, nothing much changed, since that course was rooted solidly in the one-variable theory. It was not until much later, when I started hearing about things like manifolds and differential forms, that I began to appreciate that what had bored me then was really just a special case of things that were actually kind of interesting.

A book like Shirali and Vasudeva’s would not have helped me much as a freshman, because it is intended for a more sophisticated audience, but I would have enjoyed looking at it in my advanced calculus days. This is a textbook for a rigorous multivariable analysis course, intended for an undergraduate audience of juniors and seniors that has already taken introductory multivariable calculus and perhaps also some has background in one-variable elementary analysis (while basic topological notions like compactness are defined in the text, some prior exposure to them would be helpful). So, for example, the author, in his discussion of extrema, begins by stating that he assumes the reader to be adept at doing calculations with Lagrange Multipliers in specific situations. Worked-out numerical examples are of course given in the text, but are usually in aid of illustrating some theoretical point (the necessity of a given condition, for example) rather than as mere busy-work.

The book contains detailed discussions (with proofs) of the usual topics in multivariable analysis: derivative of a function from one Euclidean space to another as a linear transformation, implicit and inverse function theorems, constrained optimization and extrema from a more sophisticated standpoint than usually covered in third-semester calculus, multiple integrals, (including Fubini’s theorem) and differential forms and the generalized Stokes’ Theorem.

The book also includes some material not easily found in the existing literature. The discussion of the inverse and implicit function theorems, for example, struck me as more detailed and elaborate than one usually finds in other texts (multiple versions of the implicit function theorem are given, for example), and in the next chapter Shirali and Vasudeva provide not just a discussion of necessary conditions for extrema but sufficient conditions as well. This is another topic not generally covered in textbooks, although Shifrin’s book, mentioned below, does have an exercise on this subject. In addition, Shirali and Vasudeva include a section on double sequences and series (i.e., sequences and series indexed by ordered pairs of natural numbers), but unfortunately no indication is given as to *why* one would be interested in these things, and nothing in this section is used in the rest of the book.

Principal competitors for this text would include Shifrin’s *Multivariable Mathematics: Linear Algebra, Multivariable Calculus and Manifolds*, and Hubbard and Hubbard’s *Vector Calculus, Linear Algebra and Differentiable Forms: A Unified Approach*. Both of these books are pitched at a lower level than Shirali and Vasudeva’s text and do not assume prior background in multivariable calculus (although both have enough material — in the case of Hubbard and Hubbard, an Appendix of more than one hundred pages of proofs — to make the book suitable for post-calculus analysis courses for students who do have this background), and both have some features not found in Shirali and Vasudeva: considerably more emphasis on linear algebra, use of manifold terminology, and, in the case of Hubbard and Hubbard, a discussion of the Lebesgue integral.

On the other hand, Shirali-Vasudeva offers, as noted before, some novel features of its own and topics not readily available elsewhere in other texts, and also approaches some things in a different way than do the books by the Hubbards and Shifrin: Shirali and Vasudeva, for example, unlike Shifrin and the Hubbards, eschew the use of multilinear functions in their discussion of differential forms, and defines them the way Rudin does in *Principles of Mathematical Analysis* (as a function defined on a set of surfaces which assigns to every surface in its domain a number obtained by integration). Shirali and Vasudeva’s text also contains detailed solutions to many of its exercises; in fact, almost one-quarter of the book is devoted to solutions. (The exercises themselves range from fairly routine to quite unusual.) The ready availability of “back of the book” solutions may not please instructors who will not be able to assign these problems for homework, but at the same time may make the book very attractive to people interested in self-study, as well as to instructors using different books who are looking for a source of additional problems for their classes.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.