Callahan's Advanced Calculus: A Geometric View vs Hubbard's Vector C, L A, and Di Forms vs Ad- Calculus: A Differential Forms Approach by Edwards

Question

My friend has given me the chance to get one of the books mentioned in the title of this question for free. I've learned single variable calculus. Which one of these books do you think would best serve my purposes, which are learning deeply multivariable calculus from scratch with physical intuition?

Answer 1

You should get the book by $Hubbard^2$ or Callahan. Then you should buy Edward's text which is way cheaper. I have both Hubbard and Callahan and each is excellent in its way. For Hubbard, you get breadth, a bit of manifold theory and a wealth of nice examples, of course linear algebra is used throughout. For Callahan, you also have the use of linear algebra throughout. It focuses on $\mathbb{R}^n$ so lacks some of the abstractness of the other two, however, it derives the basic lemmas of Morse theory which sets it apart from just about any other advanced calculus text I own.

So, the obvious question, asked by timur some weeks ago, is what are your goals in this study?

I read Edward's this summer, personally, I got annoyed by the lack of function-theoretic thinking. The text uses equations in a much more classical style than most modern authors allow. It does patiently try to develop a geometric intuition for differential forms, the only other place I've seen that sort of discussion was in Misner Thorne and Wheeler's Gravitation with its bings and bongs and egg-crates to understand differential forms.