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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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Try Spivak's Calculus on Manifolds. It is a small gem. But, it is also somewhat terse. – Rankeya Mar 31 '12 at 6:38
Did you really find Rudin unpleasant? – Rankeya Mar 31 '12 at 6:40
Also, take a look at the notes in this website: – Rankeya Mar 31 '12 at 6:42
Rudin was extremely helpful when I took my first course in analysis. But I found the chapters on multivariable calculus notationally intimidating. Probably notations for multivariable calculus are inevitably horrible. That's why I'm asking for a more readable book. If none exists, I must persevere. – math Mar 31 '12 at 6:47
By searching in tags reference-request+multivariable-calculus I found these two similar questions: References for the multivariate calculus and Need Help: Any good textbook in undergrad multi-variable analysis/calculus?. Maybe you can find something useful there. – Martin Sleziak Mar 31 '12 at 6:55

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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+1. A very nice set of notes indeed and I was unaware of their existence! – Mathemagician1234 Mar 31 '12 at 7:43
yes, it is not well known since it is still in progress for a class specific to a university, but I think that the notes are very clear and make full use of linear algebra to explain multivariable calculus. – Han Altae-Tran Apr 2 '12 at 17:08
Can someone post it? It seems to have gone offline. – Sandeep Thilakan Sep 30 '14 at 8:37
@HanAltae-Tran: could you post a mirror for those notes? The link is broken. – user125763 Nov 30 '14 at 23:44

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

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I'm aware of his books, but he didn't write much about multivariable calculus, only a bit of derivatives in several variables. – math Mar 31 '12 at 8:01
How much are you looking for? – Holdsworth88 Mar 31 '12 at 12:53
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. – math Mar 31 '12 at 13:10

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, thanks for the suggestion. – math Mar 31 '12 at 8:13

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.

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