This is an interesting look at multivariable differential calculus, developed for functions on complete normed linear spaces rather than on **R**^{n}. The generality pays off in the last chapter, that develops differential calculus on manifolds. This is intended as an upper-division undergraduate text, and it has lots of examples and challenging exercises. The present book is a 2012 unaltered reprint of the 1976 Van Nostrand Reinhold edition.

The proofs are really not that different from the multivariable calculus found in careful textbooks such as Rudin’s Principles of Mathematical Analysis, but placing them in a more abstract context makes it easier to understand what’s going on. Happily, most of the examples are from **R**^{2} and **R**^{3}, and there are lots of pictures, so the level of abstraction is not overwhelming. The theory of partial derivatives gets developed along the way. Roughly the first third of the book develops the necessary theory of linear spaces, including a modest amount of functional analysis.

The book deals only with differential calculus, so the emphasis is on local behavior, extrema, and the inverse function theorem and implicit function theorem. Lagrange multipliers are presented through finding extrema on a manifold that satisfies the constraints.

There is no integration, and so no Stokes’s or Green’s theorems or any differential forms. The book has the same sort of “calculus from an advanced standpoint” approach as Spivak’s Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, although that book works only in Euclidean spaces, but does cover both integration and differentiation. The use of “applications” in the title is exaggerated; it means applications to other parts of mathematics, and not even a lot of those.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.