This is a moderately-abstract treatment of multivariable calculus and of manifolds, done in the context of Banach spaces rather than Euclidean spaces. It is not what most people would call “advanced calculus” or “multivariable calculus” today, because it does not (explicitly) deal much with partial derivatives, multiple integrals, or line or surface integrals. The book is based on lectures in a undergraduate honors course in calculus at Harvard in the1960s. It probably wouldn’t work as a calculus course today, but would still be good for a senior or first-year graduate course in analysis.

This is a 2014 unaltered reprint of the 1990 edition from Jones and Bartlett. The 1990 edition in turn is a lightly-revised version of the 1968 Addison-Wesley edition; some clarifications were squeezed in without re-typesetting; most of the pages were unaltered, and no pages were added. So this is essentially a 46-year-old book; is it still useful? I say Yes. The material is still all relevant, and the subject matter has not changed a lot in the past 50 years. The present book is positioned at a comfortable level of abstraction, and no more recent books have come out to compete with it at this level.

The first half of the book deals with derivatives in normed linear spaces, and the second half deals with manifolds in Banach spaces and integration in Euclidean spaces and on manifolds.The pace is leisurely, with many side comments to tell us why we are doing this and where we are going. The book develops the needed background in linear spaces and topology. Integration in Euclidean spaces is done using Jordan content rather than the Lebesgue measure.

The level of abstraction is adjusted up or down throughout the book, balancing generality with ease of understanding. For example, there is a chapter on linear spaces in general, followed by a more specialized one on finite-dimensional spaces. Most derivative work is done in normed linear spaces, but there are many examples from \(\mathbb{R}^2\) and \(\mathbb{R}^3\), and some results are proved that require a complete normed linear space, for example the Implicit Function Theorem. Manifolds are defined and their geometric properties developed in Banach spaces, but the integration work is done in finite-dimensional spaces.

The book closes with two fairly detailed chapters of applications, one on potential theory and one on classical mechanics. There is also a chapter on differential equations, which is not really an application because they aren’t used for anything, but the chapter does prove all the key existence theorems and shows the use of Fourier series.

One minor flaw of this edition is that it is a photographic reproduction, with some of the pages being slightly faint; some of the fine lines in the type are eroded. Most of the faintness appears to have originated in the 1990 reprint, which seems to be a photographic reproduction with some modified pages. Even the faint pages are still quite readable, but it’s not the high-quality reproduction we are used to from Dover or Chelsea.

There are a number of books that do a good job of covering the same material for Euclidean spaces, for example, Munkres’s *Analysis on Manifolds*. There’s probably no single book that takes a more advanced look at the material in both halves of the present book. Lebesgue integration in several variables and in more general measure spaces is covered in a number of books, such as Apostol’s *Mathematical Analysis* and Hewitt & Stromberg’s *Real and Abstract Analysis* (the latter book is also very strong on differentiation). The present book recommends Serge Lang’s 1962 *Introduction to Differentiable Manifolds* for more abstract coverage of manifolds, but Lang’s book was completely rewritten in 2002 to be more elementary and cover only finite-dimensional spaces. A good alternative might be Lang’s *Fundamentals of Differential Geometry* (Springer, 1999).

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