Multivariable calculus.

*(English)*Zbl 0892.26002
New York, NY: Wiley. xv, 503 p. (1997).

This is the second volume of the Harvard Consortium’s much discussed “reform calculus”. The same two guiding principles are used as were in the first volume [D. Hughes-Hallett and A. M. Gleason: “Calculus. Single variable” (1st ed. 1994, 2nd ed. 1998; preceding review)]: 1) The Rule of Three: Every topic should be presented geometrically, numerically and algebraically. 2) The Way of Archimedes: formal definitions and procedures evolve from the investigation of practical problems. As to content, although the authors “…started with a clean slate, and compiled a list of topics that we thought were fundamental …,” a glance at the chapter headings would not particularly set this book apart from its traditional predecessors. However, within each chapter one is aware of the effects of the two guiding principles. The opening chapter (Chapter 11), Functions of Several Variables, discusses numerous examples of functions of two and three variables in accordance with principle 1) for 50 pages, and then discusses the idea of limit and continuity in examples for two pages before making the following definition: The function \(f\) has a limit \(L\) at point \((a, b)\), written \(\lim_{x\to (a,b)} f(x,y)= L\), if the difference \(| f(x,y)- L|\) is as small as we wish whenever the distance from the point \((x,y)\) to the point \((a,b)\) is sufficiently small, but not zero. The usual definition of continuity is then given and followed by examples where the definition is used to show failure of continuity at a point. The definition of partial derivative is the standard one, but before computing any partial derivatives algebraically they are discussed graphically and estimated from a contour diagram. The ten chapters are followed by seven appendices (24 pages) containing a brief review of one-variable calculus. Answers are given to the odd-numbered problems which have a “short answer”.

This book and its predecessor on single variable calculus have taken some rather severe criticism from some quarters for lack of rigor. My impression is that the book should indeed be more effective than traditional texts in bringing the ideas and applications of calculus to a larger audience. The proofs given for the Divergence Theorem, Stokes’ Theorem and the FTC for Line Integrals seem to be at an appropriate level of rigor for the beginning calculus sequence, and the more serious mathematics students should have no undue difficulty in adjusting to the higher levels of rigor to be expected in real and complex analysis courses to follow.

This book and its predecessor on single variable calculus have taken some rather severe criticism from some quarters for lack of rigor. My impression is that the book should indeed be more effective than traditional texts in bringing the ideas and applications of calculus to a larger audience. The proofs given for the Divergence Theorem, Stokes’ Theorem and the FTC for Line Integrals seem to be at an appropriate level of rigor for the beginning calculus sequence, and the more serious mathematics students should have no undue difficulty in adjusting to the higher levels of rigor to be expected in real and complex analysis courses to follow.

Reviewer: Gerald A.Heuer (Moorhead)