It’s fun to collect answers to the question “What is calculus?” The preface to Kenneth Kuttler’s *Calculus: Theory and Applications, Volume I *begins with an interesting one: “Calculus consists of the study of limits of various sorts and the systematic exploitation of the completeness axiom.” That opening sentence very effectively sets the tone for this introductory text. It is an earnest effort to resolve the content demands of traditional freshman calculus with a desire to include some real mathematics.

Kuttler gets right to business. This book is not about calculator explorations, colorful diagrams, or vague intuitive arguments. The narrative is tightly written and the author is committed to stating results precisely and making sure they are proved.

Coverage includes material common for a one-year calculus sequence, taking the reader through elementary differential equations and a development of vectors and three-dimensional coordinate systems (but coming short of any discussion of calculus in three dimensions; that’s for the next volume). As you might expect from the opening line, there is considerable discussion of sequences and the notion of completeness. Some topics trending toward a more formal analysis course are flagged as optional. The “applications” in the title refers primarily to classical mechanics. The use of computer algebra systems is mentioned but not covered in any detail.

The book could be easier to navigate. There are few visual cues to what you are seeing and the top header features only the chapter name and not the section. Exercises are found not at the end of sections or chapters, but interspersed as sections themselves. For example, Chapter 2 is titled “Functions” and Sections 2.2, 2.6, 2.8, 2.10, 2.12, 2.14, and 2.16 are all titled “Exercises.” Other irritants include vague chapter and section titles, separate numbering for different theorem-like structures, and a refusal to indicate the names of definitions and theorems in any obvious way.

Generally speaking, any book that wants to be rigorous about calculus will face the problem of interspersing the symbol-pushing mechanics of differentiation and integration with formal proofs that can be tough for freshmen inexperienced with the subtle complexity of the real line. Throw in some applications and you have a lot of differently shaped balls in the air. Kuttler’s approach demands a dexterous reader, as he does not compartmentalize the theoretical, computational, and applied ideas like many texts do — he allows the material to develop as he feels it must. As such, the organization takes some getting used to (which exacerbates some of the aforementioned navigational issues).

There is a healthy number of exercises of varying difficulty and interest, with answers to most found in an appendix. The author offers additional exercise worksheets at his website in files organized nicely by chapter but not by section.

This text may be worth considering for a calculus course oriented toward prospective mathematics or possibly physics majors, or generally for a course looking for more rigor than most. There are some nice things in this book, but the organization and point of view are sufficiently peculiar that the prospective adopter should review it with care.

Bill Wood is an assistant professor of mathematics at the University of Northern Iowa.