The How and Why of One Variable Calculus

This is not a typical calculus book. The title is a bit of a tip-off, although the emphasis should probably be more on the “why” than the “how”. The author says that his book is intended as a textbook for honors calculus and that high school mathematics provides an adequate background. He notes that the book might also be used as supplementary reading for a regular methods-based calculus course or as a text for a transition-to-analysis class.

The book begins with an extended treatment of the real numbers: field axioms, order axioms, the least upper bound property and a brief look at Dedekind cuts. Following this are chapters on sequences, continuity, differentiation, integration and series. All the classic theorems of basic calculus are carefully stated and proved. The usual methods and techniques of calculus are not absent, just downplayed.

The pace is very deliberate. The author lays the groundwork for theorems carefully with examples and exercises. Proofs are clear and full of detail. The author tells students that they need not fully understand the theorems on first reading. He tells them that they should review the examples and exercises that follow, try to understand why the theorems are important, and then go back and study the proofs.

The author includes several topics that don’t often appear in calculus books. An extended treatment of the real numbers goes well beyond the usual quick overview. Sequences appear early. The chapter on continuity discusses how continuous functions preserve convergent sequences and connectedness and why uniform convergence and uniform continuity are important. One consequence of this is that it takes about 125 pages to get to the derivative. The concluding chapter on series has material that is more sophisticated than the more basic treatment that once appeared late in first year calculus.

Both examples and exercises have a broad range of difficulty and sophistication. Included, for instance, is a painstakingly complete proof that constant functions are continuous, but there is also an extended treatment of the Cantor set and proofs that the Cantor set has zero “length” and that the indicator function of the Cantor set is Riemann integrable.

The exercises are plentiful, well-selected and well-constructed. Detailed solutions to all exercises are provided; they fill the last 150 pages of the book. The author admonishes students to avoid peeking at solutions until they have made serious efforts at solving them. Whether this is idealistic or naïve, it means that instructors may need to provide separate exercises of their own.

A lot of lovely mathematics appears here, presented in a way that many of us would have loved as students. So it puzzles me that a short introductory section called “Why study calculus?” mostly cites four applications that range from somewhat interesting to …well… dull. A student capable of using this book should be inspired by the wonderful mathematics. Perhaps the author could provide a few dazzling applications too.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.