
- Author: Peter R. Mercer
- Edition: 1
- Series: Compact Textbooks in Mathematics (CTM)
- Publisher: Springer
- Publication Date: 11/13/2024
- Number of Pages: 168
- Format: Paperback
- Price: $64.99
- ISBN: 978-3-031-43916-2
- Category: textbook
[Reviewed by Jer-Chin Chuang, on 11/12/2024]
This book is a recent addition to Springer's Compact Textbooks in Mathematics series aimed at advanced undergraduates and master's students. Some of the topics in the book under review have appeared previously in the author's More Calculus of a Single Variable.
That book more closely follows the order of a traditional calculus sequence whereas the present book is more topical in structure. In 15 short, focused chapters the book explores various non-standard topics which are often glimpsed by students only among exercises at the tail end of calculus problem sets. Off the main highway through the calculus curriculum, these more challenging scenic by-ways thread through beautiful ideas and arguments that showcase an ingenious blend of elementary techniques accessible to students with the techniques of single-variable calculus.
The book is divided into three parts separated by two interludes: the first begins with the Viète and Wallis formulae for pi, defines the Gamma function, and computes the volume of high-dimensional spheres. The second section looks at geometric probability, convexity and its various implications such as Jensen's Inequality, the AM-GM Inequality, and the Minkowski norm. The third section combines previous topics to derive Stirling's formula and the Euler sine product formula and to compute the zeta function at even positive integers. The final pages hark back to the book's opening by exhibiting the Viète and Wallis formulae as a special cases in a family of formulae linked by the Euler sine product.
The book is very well written. Its clever proofs are clearly explained with sufficient detail for undergraduates to follow, keeping the book very readable. The topics all rather concrete and well within the scope of undergraduate students. Biographical snippets also introduce the many famous former travelers along these roads. Most chapters conclude with ten problems of various types: verification of a previous claim, a variation of the preceding argument, or derivation of a result from a journal article. Many are non-routine. Students will likely find them a good mental workout even if stumped on a question here or there. Also, each chapter has a helpful bibliography pointing to sources and relevant literature for solutions, extensions, background, and further explorations. Most are articles in MAA journals or the Mathematical Gazette, and thus encourage readers to regularly explore these journals.
I enjoyed reading this book. The book could be used as a supplement to an honors section of a second-semester calculus course or for a topics course. But I suspect any motivated, inquisitive calculus student who enjoyed working the challenge problems in calculus texts will find much pleasure and profit from reading it.
Jer-Chin Chuang is a Lecturer at the University of Illinois at Urbana-Champaign.