[Reviewed by Tushar Das, on 12/18/2014]
Peter Lax is a modern master. He writes with great verve. His textbooks are models of clarity, though a far cry from a pristine Bourbakian treatise. I find those I have read closer to transcriptions of brilliant, impassioned lectures — filled with insights that lead the reader to vantage points from where familiar faces of old theorems are seen in a new light.
Lax received the 2005 Abel Prize and gave an interview sometime before the award ceremony to Messrs. Raussen and Skau (R&S) in Oslo. In the course of candid reminiscences about an interesting life and a smorgasbord of mathematical contributions, Lax was asked about a few of his pedagogical forays. We quote his response regarding his first attempt at a “radical” calculus textbook, co-authored with Samuel Burstein and Anneli Lax: Calculus with applications and computing Vol. 1 (New York: Springer. 1976. 513 p. ISBN: 0387901795.)
R&S: You have also been engaged in the teaching of calculus. For instance, you have written a calculus textbook with your wife Anneli as one of the coauthors. In this connection you have expressed strong opinions about how calculus should be exposed to beginning students. Could you elaborate on this?
Lax: Our calculus book was enormously unsuccessful, in spite of containing many excellent ideas. Part of the reason was that certain materials were not presented in a fashion that students could absorb. A calculus book has to be fine-tuned, and I didn’t have the patience for it. Anneli would have had it, but I bullied her too much, I am afraid. Sometimes I dream of redoing it because the ideas that were in there, and that I have had since, are still valid.
Of course, there has been a calculus reform movement and some good books have come out of it, but I don’t think they are the answer. First of all, the books are too thick, often more than 1,000 pages. It’s unfair to put such a book into the hands of an unsuspecting student who can barely carry it. And the reaction to it would be: “Oh, my God, I have to learn all that is in it?” Well, all that is not in it! Secondly, if you compare it to the old standards, Thomas, say, it’s not so different — the order of the topics and concepts, perhaps.
In my calculus book, for instance, instead of continuity at a point, I advocated uniform continuity. This you can explain much more easily than defining continuity at a point and then say the function is continuous at every point. You lose the students; there are too many quantifiers in that. But the mathematical communities are enormously conservative: “Continuity has been defined pointwise, and so it should be!”
Other things that I would emphasize: To be sure there are applications in these new books. But the applications should all stand out. In my book there were chapters devoted to the applications, that’s how it should be — they should be featured prominently. I have many other ideas as well. I still dream of redoing my calculus book, and I am looking for a good collaborator. I recently met someone who expressed admiration for the original book, so perhaps it could be realized, if I have the energy. I have other things to do as well, like the second edition of my linear algebra book, and revising some old lecture notes on hyperbolic equations. But even if I could find a collaborator on a calculus book, would it be accepted? Not clear. In 1873, Dedekind posed the important question: “What are, and what should be, the real numbers?” Unfortunately, he gave the wrong answer as far as calculus students are concerned. The right answer is: infinidecimals. I don’t know how such a joke will go down.
(M. Raussen and C. Skau, “Interview with Peter D Lax,” Notices Amer. Math. Soc. (February 2006), 223–229.
The book under review, a little over 500 pages co-authored with Maria Terrell, is a first-approximation to Lax’s dream come true: a “thorough revision” of the 1976 Lax-Burstein-Lax. The overall tone of this unorthodox single-variable calculus textbook, as well as the topics covered, remain faithful to the previous edition. The changes are subtle and their motivations not easy to discern. For example, the introduction of real numbers as infinite decimals and the study of population dynamics via differential equations remain. However the primacy of uniform continuity (both in the 1976 edition and as advocated by Lax in 2005) has been replaced with the more traditional notion of continuity at a point. This reversion in the 2013 edition may have been one of the few places where Lax was unable to stand his ground against the irresistible tide of tradition. For a flavor of the style, here is the definition of “continuity” from the 1976 version:
[p. 64]…so we can determine f(x) if approximate knowledge of x is sufficient for approximate determination of f(x). Approximate knowledge of $x$ means we know all digits of x up to the $m$-th; this is the same as saying that we know an interval of length $10^{-m}$ within the domain of $f$ in which x lies. If the values which f takes in this interval of length $10^{-m}$ lie in an interval of length $10^{-k}$, this information about x suffices to determine all digits of $f(x)$ up to the k_-th._ This property of the function f can be expressed as a
Continuity criterion. In order for $f(x)$ and $f(y)$ to be so close that
$|f(x) - f(y)| < 10^{-k},$ it suffices for x and y to be so close that
$|x-y| < 10^{-m}$. The choice of m depends on k.
A function f which has this property for x [and y] in the domain of f is called continuous on its interval of definition.
Surely such concreteness must have benefits for the comprehension of an intuitively clear but famously difficult conception, that almost every beginning calculus student struggles with. This pedagogical move led naturally to certain theorems that are not to be found in more traditional calculus texts, e.g one now has: The product of two bounded continuous functions is bounded and continuous. One wonders whether such deviations from tradition may have had something do with renouncing Lax’s suggestion in the 2013 edition.
The radical (for the 1970s) inclusion of flowcharts and FORTRAN code for various algorithms have also, sadly, been done away with. Here is the list of routines, as it appeared at the end of the 1976 table of contents:
FORTRAN programs and instructions for their use
- P1. The bisection method for finding a zero of a function
- P2. A program to locate the maximum of a unimodal function
- P3. Newton’s method for finding a zero of a function
- P4. Simpson’s rule
- P5. Evaluation of $\log x$ by integration
- P6. Evaluation of $e^x$ using the Taylor series
- P7. Evaluation of $\sin x$ and $\cos x$ using the Taylor series
The reviewer remembers being pleasantly intrigued by these sections on his first exposure to them while still an undergraduate in the early 2000s. The 1976 edition included “concrete” bisection proofs of both the Intermediate Value Theorem (IVT) and the Extreme Value Theorem (EVT), followed up with algorithms for special cases, viz. to locate roots and to find the maximum for unimodal functions (see items P1 and P2 above). However the bisection proof of the IVT, that seemed more aligned with the author’s philosophy, was removed in the newer edition for reasons that are not clear to the reviewer. The new preface provides little evidence to warrant such an exclusion:
The word “computing” was dropped from the title because today, in contrast to 1976, it is generally agreed that computing is an integral part of calculus and that it poses interesting challenges. These are illustrated in this text in Sects. 4.4 [Approximating Derivatives], 5.3 [Newton’s Method for Finding the Zeros of a Function], and 10.4 [Numerical Solution of Differential Equations], and by all of Chap. 8 [Approximation of Integrals].
These portions of the 2013 edition, with the exception of Newton’s Method, are treated lightly and not given in-depth coverage as in the 1976 edition. Such material would certainly enhance the 2013 edition, especially if expanded upon and rewritten in pseudocode rather than in FORTRAN. These could also appear as guided-projects within the text that readers could easily implement on a variety of open-source mathematical software, e.g. using William Stein’s Sage project. In addition, the flowcharts seemed a very effective way of peering in to the mechanism/design of certain key algorithms and proofs. This reviewer will attempt to use them as a pedagogical tool when teaching single-variable calculus or introductory analysis in the future.
My criticisms and suggestions aside, this is an altogether excellent text. It is filled with beautiful ideas that are elegantly explained and chock-full with problems that will enchant both the experienced teacher and the curious novice. I recommend it strongly and look forward to an even better third edition! We also hope that Lax finds good collaborators to help him complete his sequel (Volume II was “in preparation” back in 1976) that would cover multivariable calculus and an introduction to analysis in $\mathbb{R}^n.$
Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.
[Reviewed by William J. Satzer, on 01/13/2016]
This is not a typical calculus book. The title is a bit of a tip-off, although the emphasis should probably be more on the “why” than the “how”. The author says that his book is intended as a textbook for honors calculus and that high school mathematics provides an adequate background. He notes that the book might also be used as supplementary reading for a regular methods-based calculus course or as a text for a transition-to-analysis class.
The book begins with an extended treatment of the real numbers: field axioms, order axioms, the least upper bound property and a brief look at Dedekind cuts. Following this are chapters on sequences, continuity, differentiation, integration and series. All the classic theorems of basic calculus are carefully stated and proved. The usual methods and techniques of calculus are not absent, just downplayed.
The pace is very deliberate. The author lays the groundwork for theorems carefully with examples and exercises. Proofs are clear and full of detail. The author tells students that they need not fully understand the theorems on first reading. He tells them that they should review the examples and exercises that follow, try to understand why the theorems are important, and then go back and study the proofs.
The author includes several topics that don’t often appear in calculus books. An extended treatment of the real numbers goes well beyond the usual quick overview. Sequences appear early. The chapter on continuity discusses how continuous functions preserve convergent sequences and connectedness and why uniform convergence and uniform continuity are important. One consequence of this is that it takes about 125 pages to get to the derivative. The concluding chapter on series has material that is more sophisticated than the more basic treatment that once appeared late in first year calculus.
Both examples and exercises have a broad range of difficulty and sophistication. Included, for instance, is a painstakingly complete proof that constant functions are continuous, but there is also an extended treatment of the Cantor set and proofs that the Cantor set has zero “length” and that the indicator function of the Cantor set is Riemann integrable.
The exercises are plentiful, well-selected and well-constructed. Detailed solutions to all exercises are provided; they fill the last 150 pages of the book. The author admonishes students to avoid peeking at solutions until they have made serious efforts at solving them. Whether this is idealistic or naïve, it means that instructors may need to provide separate exercises of their own.
A lot of lovely mathematics appears here, presented in a way that many of us would have loved as students. So it puzzles me that a short introductory section called “Why study calculus?” mostly cites four applications that range from somewhat interesting to …well… dull. A student capable of using this book should be inspired by the wonderful mathematics. Perhaps the author could provide a few dazzling applications too.
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
