Reference for multivariable calculus

Solution 1:

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

Solution 2:

If you like the way Terence Tao writes, then I would recommend Tao's Analysis I and II.

Solution 3:

I learned multivariable calculus during my undergraduate studies using Marsden & Tromba, "Vector Calculus". I found it a bit "not too much rigorous" but clear and with lot's of examples taken from physics which are rather intuitive in the sense of Terence Tao's link you put above.

Solution 4:

I can understand why you'd find Rudin unpleasant,but Pugh I feel is much better written.You probably weren't ready for either of them, in which case you need something gentler. Try John H.Hubbard and Barbara Hubbard's Vector Calculus,Linear Algebra And Differential Forms:A Unified Approach. It's rigorous but gentle, beautifully written and has a legion of historical notes, references and applications to the physical sciences. I think you may find it just what you need.

Solution 5:

I really like Analysis on Manifolds by Munkres. A cheap Dover book is Advanced Calculus of Several Variables by C.H. Edwards. It is also pretty good from what little I've read of it. However it only has one section on differential forms, whereas Munkres devotes a whole chapter to them.