Multivariable Calculus

Reviewed by Andy Rich, Professor of Mathematics, PALNI, Manchester University on 12/19/19

Comprehensiveness rating: 5 see less

Has all the usual topics and then some. I liked the development of differential forms towards the end and having chapter 11 as a teaser for higher level stuff. The development was clear enough that I hope most students at this level could get it.
Some of my favorite examples were missing: e.g., the cycloid and deriving Kepler's laws from Newton's laws, but everybody has their own favorites so I am okay with that.
In discussing multivariable continuity, it would have been nice to pull out the two path discussion which appears in the text and highlight it as a theorem, but these are all minor points. I still give this text a 5 for comprehensiveness.

Content Accuracy rating: 5

My only complaint here is in the discussion of torsion. He describes torsion as "wobbling" which to me gives the wrong idea. Wobbling means going back and forth which can happen in the plane. Torsion is "twisting" out of the plane.
The mathematics is all correct and the author is honest about where things are being swept under the rug, e.g., in the proof of Green's Theorem.

Relevance/Longevity rating: 5

This is not much of an issue for math texts.

Clarity rating: 5

Excellent! Good clear explanations of the ideas behind the theory, e.g., Lagrange multipliers which are sometimes presented as magic, but here are motivated with geometrical ideas. Nice treatment of multidimensional chain rule via matrix multiplication with one section on the conceptual picture and one on computations.
Good diagrams throughout, present whenever needed to help with understanding, e.g., showing the relationship of two different parametrizations when proving that the value of the integral is independent of the parametrization.

Consistency rating: 5

No problems that I saw.

Modularity rating: 5

The division into parts, chapters, and sections made sense and the sections were not too long.

Organization/Structure/Flow rating: 5

Good. The preface and table of contents outline the organization and make it easy to find topics.
It might be helpful to introduce polar coordinates earlier.

Interface rating: 5

No problems that I saw. It easy to navigate jumping around with the bookmarks pane in Adobe. Good color illustrations showing geometric pictures when helpful.

Grammatical Errors rating: 5

Fine.

Cultural Relevance rating: 5

Not really relevant.

Comments

I really liked this text. It is more theoretical, more proof-based than what I usually teach, and I might skip some of that if I were teaching from this text, but I think it is good to include the proofs in the text.
I liked the presentation of differential forms towards the end. I think one addition to chapter 11 as a further teaser and to try to show the relationship of the various theorems, would be to set up the deRham sequence in R^3 and show how gradient, curl, and divergence come into that single unifying sequence. (This tiptoes up to topology of the underlying space which the author has already alluded to at various places.)
The conversational style was great: "trying to find the maximum of a function like ... is silly" for example. Humor is scattered throughout, for example, including a picture of a porcupine as an example of a mammal with an orientation.
There seemed to be plentiful exercises.
Overall, I think this is an excellent text for multivariable calculus.