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. 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.
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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. 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 material is divided into five chapters. 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. 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. 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. 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.
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.
Many calculus texts will have a few pages on the history of the development of the relevant mathematics. 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.
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. Let us finish the review on a high note, though, because this book deserves it.
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.
Calculus: Multivariable, 2nd Edition
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Calculus: Single and Multivariable, 2nd Edition