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Calculus: Multivariable

Brian E. Blank and Steven G. Krantz
Key College Publishing/Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
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First a disclaimer from the reviewer: I am not a party in the Calculus Reform wars. In fact, I have very limited knowledge of the phenomenon. If anyone reading this review wishes to let me in on some of the stories and whatnot related to this Great War of Our Times, I would certainly appreciate it and thank this person profusely.

So the question is: why another Multivariable Calculus text? The Stewart empire is still thriving, with various versions and many many editions. The Hughes-Hallet et al. aka. the Harvard Consortium text (Calculus: Multivariable, Fourth Edition, by McCallum, Hughes-Hallett et al.) is already in its fourth edition and is quite popular among many. And there are many other texts out there; one could plausibly assume that each and every instructor must already have his or her own favorite text. (Call this Conjecture A). Of course, these two very experienced teachers and authors must have a reason other than to add to the clutter. (Blank and Krantz "have a combined experience of more than 50 years teaching calculus.")

The table of contents of the Blank and Krantz text does not seem to differ much from the Stewart text on the reviewer's desk (Multivariable Calculus: Concepts and Contexts, Second Edition, which is the reprint of the last six chapters of the same author's Calculus: Concepts and Contexts , Second Edition). The material is divided into five chapters. (As expected from a text whose title implies emphasis on multivariable calculus, there is no chapter corresponding to Stewart's Chapter 8, on infinite sequences and series.) The first chapter, Chapter 11 (the numbers are following the numbering of the first volume, Calculus: Single Variable, which I have not seen), is on vectors. Vectors in the plane and vectors in three-dimensional space are introduced and various properties of and important facts about dot products and cross products are developed. Then in Chapter 12 we learn about vector valued functions. The usual calculus concepts (limits, continuity and derivatives) are defined, arc length and curvature are described, and the chapter ends with a discussion of vector valued functions in physics.

Chapter 13 is on functions of several variables. Once the usual calculus notions are defined, Blank and Krantz move on to discuss gradients and directional derivatives at length; the unsuspecting student might possibly see why vectors were introduced at the beginning when attempting to understand fully the idea of differentiability in the case of several variables. This chapter also includes a section on Lagrange multipliers to complete the whole package. At this point the student has completed the multivariable version of the single variable differential calculus course. It is natural to expect that integral calculus will follow differential calculus. And so next comes Chapter 14, on multiple integrals.

However as we all know, the story does not end there, and we go on to Chapter 15, on Vector Calculus, which is the climax of the typical multivariable course. As expected, this chapter is much more difficult, and somehow disappoints the reader who had grown accustomed to the well motivated discussion and very well organized development of the material that preceded it. Somehow, after four almost perfect chapters, this last chapter feels a bit too long and too heavy. Too many big ideas are to be digested in the same amount of time. However I do not know of a better way to do this myself, so I cannot hold it against the authors. At this point it is clear that Conjecture A is not correct; when one sits down to think about it, there may not yet be a perfect way to teach multivariable calculus out there, and it is a good thing that many experienced teachers, like Blank and Krantz among them, are trying to improve on the currently available texts.

While I do not always assume that my students will read every single line in their calculus text, I actually have read every single line in this book, and felt that this is really a text to be read. Many recent texts will include a note to the students urging them to read the chapters before attempting the problems. But many of these same texts seem to be written still keeping in mind the student who will ignore the written text and use the book only to search for a solved example that resembles the homework problem he is struggling with. Blank and Krantz do not seem to have much problem with writing on and on about ideas and motivation, which is a most welcome feature in a textbook of this level. The solved examples always seem to have a particular purpose that adds to the narrative itself; they are not only provided there to help the lazy student with his homework.

The most exciting feature of the text for this reviewer was the inclusion of brief  sections following each chapter, two to three pages long, entitled Genesis and Development. Many calculus texts will have a few pages on the history of the development of the relevant mathematics. But the way it is done in this particular book is much more readable; students who could not care less about mathematicians may be surprised to find out how much gossip can be found in a math text! These pages provide a lot more than pure gossip about long dead mathematicians, of course; these sections help place the math into its historical context as well and lead the interested student to further study in more advanced subjects.

Now a couple of negatives: I attempted to access the online sources using the access code at the back of my copy but was denied. This may very well be due to my copy being a review copy, but it definitely left an unpleasant aftertaste. Also the text claims to introduce diverse applications of multivariable calculus, but aside from the relatively standard ones from physics and engineering, this reviewer did not find many novel applications. The exhaustive (and very interesting) discussion of Kepler's third law was one application that may not be quite as standard in many texts, but once again it is an example from the world of physics.

Let us finish the review on a high note, though, because this book deserves it. The book overall is very well-written and the organization of the material is very logical. Since it is a first edition, it may take some time to see how it works in the classroom, but this reviewer can see it will certainly have its own, sizable, following.

Gizem Karaali is assistant professor of mathematics at Pomona College.

 Vectors.- Vector-Valued Functions.- Functions of Several Variables.- Multiple Integrals.- Vector Calculus.