A First Course in Analysis

This is a very good text in the theory of calculus of one and several variables. Its strength is in the multivariate calculus part. There’s nothing wrong with the single-variable treatment, but for that material there are lots of good books out there already. (Ross’s Elementary Analysis: The Theory of Calculus is an especially good one). It’s much harder to find a rigorous but understandable treatment of multivariate calculus, and this book gives just such a treatment. The mathematical approach for multivariate calculus is similar to Rudin’s in his Principles of Mathematical Analysis (they’re especially close on differential forms), but Conway is much easier to understand. The book is also extremely reasonably priced at about 50 dollars for a modern, all-new hardcover text.

This book provides a rigorous view of what the student has already studied in first-year calculus, and generally speaking does not introduce new topics. In particular it does not go in new directions such as the Lebesgue integral or function spaces. There are a few exceptions: there’s a good bit on metric spaces, and there’s some coverage of the Stieltjes integral. There’s a whole chapter on differential forms (needed to make surface integrals rigorous) which will be new to most students.

The book is fairly conventional in most ways. The author states (p. x) that his guiding principle is to go from the particular to the general, and the book sticks to this, and hardly ever works in any more generality or abstraction than needed. The construction of the reals is not done very thoroughly but is done via Dedekind cuts, which I think is a good choice. Limits of sequences are defined first, then limits of functions (using the usual \(\epsilon\)–\(\delta\) method,) but then immediately proving an equivalent form in terms of sequences. The book deals only with metric spaces and not with general topological spaces. It deals with sets of measure zero and not with Lebesgue measure. The Banach Fixed Point Theorem is stated and proved in full generality, which is not very complicated, and is immediately put to good use in the Inverse Function Theorem and Implicit Function Theorem. It includes the beginnings of manifolds (it covers simplexes and chains but doesn’t go all the way to manifolds).

There are a few gems that you won’t find in most books. One is Darboux’s theorem that derivatives have the intermediate value property (one of my favorite calculus theorems). Another is a simple but very clever proof that \(\sqrt{x}\) can be uniformly approximated on \([0,1]\) by polynomials; this result is used in proving the Stone–Weierstrass Theorem. The proof is based on a recursively-defined sequence of polynomials; this construction is familiar in operator theory but I’ve never seen it in a calculus context.

The book has a good selection of exercises, most not very difficult; there are no answers or hints. A Very Good Feature is the inclusion of many short biographies of the mathematicians involved; they are unfortunately printed in footnotes in tiny type (about 8 points on 10 point leading) and so are a little hard to read.

Bottom line: A well-done “theory of calculus” text, that is especially useful if you need the theory of multivariate calculus.

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