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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.

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    There are quite a few advanced undergraduate level texts for what you're asking for, as the answers and comments demonstrate. What seems to be missing, however, are suggestions for texts at the (U.S.) second year level that include an introduction to differential forms. Here are two examples for those who might be interested: John B. Fraleigh, **Calculus. A Linear Approach**, Volume 1 (1971) and Volume 2 (1972); David M. Bressoud, **Second Year Calculus. From Celestial Mechanics to Special Relativity** (1991). [I realize, from a comment, that the OP is a little beyond this level.]2012-04-02

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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

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    @HanAltae-Tran, is this it : http://math.stanford.edu/~eliash/Public/177-2015/52htext-2015.pdf?2016-09-22
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If you like the way Terence Tao writes, then I would recommend Tao's Analysis I and II.

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    I don't really know the scope of a standard multivariable calculus course. But, at the very least, I know there are topics like multiple integrals and differential forms, which are not present in the book.2012-03-31
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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.

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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.

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    it seems great, than$k$s for the suggestio$n$.2012-03-31
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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.