on 25 July 2009
The book is nicely, and concisely, organized. Based on the topics covered and the explanations presented, I would place it somewhere between typical calculus service course texts and introductory texts on real analysis. As a result it can be used for self-study in preparation for a more applications-oriented calculus course, a prelude to a real analysis course, or a short refresher for those who've had calculus in the past.
There is a brief introductory chapter (35 pages) before the chapter on Differential Calculus begins. Subsequent chapters are Differentiation as a Tool, Integral Calculus, and Integration as a Tool. The text concludes with a chapter on Functions of Several Variables.
Although this Dover edition includes additional answers not available in the original, the solution section was not retyped. The additional answers are not integrated into the original answer section; they are included, in a different font, as a separate section after the original. This is only a minor issue, but it does require some extra 'page flipping' as answers are checked. However, the additional answer section is of real value for self-study, probably the most likely reason this book will be purchased.
Questions range from easy to hard, with harder questions marked by asterisks. Asterisked questions ask the reader to solve problems that are only peripherally related to the section's topic, or where no problem solving exemplar had been previously presented. For example, the reader is asked to find the rational number representations for a set of repeating decimal numbers when the procedure to do this had not yet been discussed. Fortunately, in many of these cases problems have been sequenced so that the results of earlier questions can provide insight into solving later ones.
This book was written before the "questions proliferation" juggernaut of recent times. So, although there is the occasional section with thirty or so questions, most sections contain twenty of less well chosen questions reinforcing the section's core concepts, with asterisked questions asking the reader to consider new material. Thus, readers can consider all problems and still complete the book within a reasonable period.
In proofs, the author often takes the interesting approach of returning to basic definitions rather than using theorems previously proven. For example, the section on inequalities presents theorems with algebraic proofs that multiplying both sides of an inequality by the same positive (negative) quantity does not (does) change the sense of the inequality. However, in the solutions section, the author often returns to basic definitions for proofs, e.g., a > b means a - b > 0, even where the use of previously proven theorems would have produced more concise results.
There are some minor additional issues: In the section on sets, Venn diagrams are unfortunately missing. These usually prove helpful to those new to this topic. Occasionally, solutions in the "hints and answers" section do not provide either answers or hints. For example, a problem asks the reader to "verify" a statement. The answers section provides the unhelpful 'solution', "Trivial, but give details anyway". In a similar vein, occasionally a question such as "(why?)" is asked within the text, with the answer posed for the reader but with no further explanation presented.
Although, the back cover states this can be "ideal as a primary text...", it does not cover many topics now typically included in introductory calculus courses. Thus, this will not be its main application. However, its conciseness combined with the excellent style of presentation, means it can be highly recommended as a self-study refresher for those who previously studied calculus, as a supplement to those currently in a calculus course, or as a comparatively short prelude for courses in calculus or real analysis.