Despite the imposing title, this book is "designed as a first encounter with rigorous, formal mathematics for students with one year of calculus". I think the authors succeed quite well in this goal in first half of the book, which treats the usual topics of introductory analysis in the real line in a leisurely way that includes many details. They are less successful in the second half of the book, which treats calculus in two dimensions, including line integrals and Green's theorem, and introductory complex analysis, up to Cauchy's integral formula. Although the authors strive to keep the leisurely pace in that second part, they succeed only partially in their treatment of calculus in the plane (their motivation for generalizations of the Fundamental Theorem of Calculus is good) and mostly not in their introduction to complex analysis: they do a lot of preparation and very few applications, except for a proof of the Fundamental Theorem of Algebra and the computation of one improper real integral. One cannot really do justice to complex analysis in fifty pages.
On the positive side, the prose is pleasant throughout the book. The first chapter, which discusses the real number system, is particularly nice, if probably a bit challenging for the intended audience. It starts with a discussion of irrational numbers, with the mandatory proof of the irrationality of the square root of 2 (including a proof of the Fundamental Theorem of Arithmetic), and then surprisingly moves directly to a construction of the reals via Dedekind cuts. The authors then consider the real numbers again, but this time axiomatically as a complete ordered field, with the least upper bound axiom as the expression of completeness. Dedekind cuts are no longer mentioned. The chapter ends with a proof of the Heine-Borel theorem, which, while not particularly difficult, will probably seem mysterious and unmotivated at that point. (It won't be used until the end of the second chapter when discussing uniform continuity and the boundedness of continuous function in compact intervals.)
The book is useful as a starting point for beginners in analysis and can be used for independent study and reference, but cannot really (and does not intend to) compete with Spivak's Calculus or Abbott's Understanding Analysis. On the other hand, the Dover price does add to its attractiveness.
Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.
|1. The Real Number System|
|2. Functions, Limits, and Continuity|
|3. Differentiation and Integration|
|4. Sequences and Series|
|5. Calculus in Two Dimensions|
|6. Line Integrals and Green's Theorem|
|7. Complex Analysis|