I'm looking for a book to learn multivariable calculus that is rigorous, but not overly technical, and also provides meaningful insight. Standard calculus texts like Stewart and Thomas are too sketchy. I've also skimmed through some texts in analysis, e.g. Rudin and Pugh, but they are not so readable due to unpleasant notation (which is probably inevitable) and lack of intuitive motivation.
I came across Terence Tao's article on differential forms. I like his way of explaining the analogues and intuitions behind the definitions and theorems. This kind of writing is what I'm looking for.
Please advice me some reference. Thanks in advance.
Best Answer
I would highly recommend using the text Eliashberg uses to teach Math 52H at Stanford University. He has a rigorous development of differential forms from linear algebra and uses these to derive change of variables, integration on manifolds, etc. It is not completely necessary to understand all of the theorems to use them, so I think you might enjoy this: Multilinear Algebra, Differential Forms, Stokes Theorem