For obvious reasons, many authors are inclined to exaggerate their descriptions of the intended readership of their books, for example, by an understatement of the necessary pre-requisite mathematical knowledge. Then there are those self-formulated recommendations of the book's scope and scholarly qualities, perhaps arising from the encouragement of the publisher. In the case of this book, however, none of these factors apply, which means that the task of reviewing it is altogether simpler and more pleasant. Portions in italics below are quotes.

**Intended readership:**

*Pure and applied mathematicians, physicists, engineers, economists, biologists and statisticians. The text may be used as a supplement to a course on single variable analysis by concentrating on the first three chapters (Vectors and Volumes, Metric Spaces, Differentiation). A second use is for a semester-long course introducing students to manifolds and differential forms.*

In the opinion of this reviewer, this is a very accurate description.

**Highlights and key features:**

*Systematic exposition supported by numerous examples and exercises from the computational to the theoretic.*
*Brief development of linear algebra in* **R**^{n}.
*Review of metric space theory.*
*Treatment of standard multivariable analysis; differentials as linear maps etc.*
*Lebesgue integration introduced in a concrete way rather than via measure theory.*
*Extension ideas from* **R**^{n} *to calculus on manifolds (wedge product, differential forms, generalised Stokes' Theorem.*
*Bibliography and comprehensive index.*

All this is absolutely true but omits any statement attesting to the high quality of the writing the high level of mathematical scholarship. So, go and order a copy of this attractively produced, and nicely composed, scholarly tome. If you're not teaching this sort of mathematics, this book will inspire you to do so.

**P.S.**

It's a bit of cheek to produce a review based on the authors' self-recommendation, but why reinvent a perfectly good wheel? (Anyway, I'm not getting paid for this work!)

Peter Ruane is retired from university teaching.