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Calculus: A Rigorous First Course

Daniel J. Velleman
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This new text from Dover’s Aurora Series offers a take on introductory calculus that is quite different in spirit from most current calculus books. The author’s intention is to treat the usual topics of first-year calculus but to do so in a mathematically rigorous way while keeping the focus on solving problems. He emphasizes that his is a rigorous calculus book and not an analysis book.

At first glance this looks like a typical first-year calculus book. The usual topics appear: limits, derivatives, integrals and their applications, and infinite series. The first indication that something is different is an extended chapter on limits. It sets the tone for the remainder of the book. The author includes a section called “What Does ‘Limit’ Mean?” in which he provides the formal definition of the limit of a function with explanation and motivation using pictures, formulas and words. He then provides several examples of finding limits of functions and then proving the result using the definition. Finally he states and proves the standard general limit theorems.

By the end of the chapter on limits the student will have seen not only formal definitions of continuity and of the limit of a sequence of real numbers but also the completeness of the real numbers via the nested interval theorem (without proof) and its consequence the Intermediate Value Theorem. Plausibility of the nested interval theorem (any nested sequence of closed intervals whose lengths approach zero has a unique real number that is in all the intervals) is illustrated by figures. This theorem is used a number of times throughout the book.

The remainder of the book follows in this fashion: careful definitions, theorems precisely stated, and proofs of pretty much everything. The author’s intention to focus on using calculus to solve problems means that we also see many of the usual applications and examples as well as plenty of practice of the typical “find the derivative” and “evaluate the integral” exercises.

This is a well-written and well-designed text that consistently follows the author’s standards of rigor. He does indeed succeed at keeping his book from becoming another analysis book. It is clearly not a book for every calculus student but it might work well for an honors course. Certainly students finishing this book would be well prepared for analysis.

The word rigor is used in a number of ways by mathematicians and is rarely defined. The author doesn’t exactly define it either, but he says this (italics are the author’s) in his preface for students:

Our standard will be certainty: when reasoning about a problem, our goal will be not just to determine the answer, but to become certain of the answer. 

Certainty is a fine thing, but is that really why we value rigor in our arguments and proofs? It seems to me that rigor is a means of trying to avoid error, perhaps of attempting to achieve certainty. Many distinguished mathematicians have offered apparently rigorous proofs that were wrong, and for a whole variety of reasons. Certainty, such as it is, can take a long time in coming and often relies on review of results by colleagues and the community of mathematicians. So, is certainty what we’d want students to expect? Maybe it would be better to encourage in students at least a modest level of skepticism and a healthy appreciation of counterexamples.

Bill Satzer ([email protected]) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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