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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.2012-03-31
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    Did you really find Rudin unpleasant?2012-03-31
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    Also, take a look at the notes in this website: http://www.math.princeton.edu/~gunning/2012-03-31
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    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.2012-03-31
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    Oh, I see. Yes, we used Rudin for real analysis, but never used it for multivariable calculus. Anyway, Robert Gunning's notes (link to his website above) seems good.2012-03-31
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    By [searching](http://math.stackexchange.com/questions/tagged/reference-request+multivariable-calculus) in tags [tag:reference-request]+[tag:multivariable-calculus] I found these two similar questions: [References for the multivariate calculus](http://math.stackexchange.com/questions/43290/references-for-the-multivariate-calculus) and [Need Help: Any good textbook in undergrad multi-variable analysis/calculus?](http://math.stackexchange.com/questions/97261/need-help-any-good-textbook-in-undergrad-multi-variable-analysis-calculus). Maybe you can find something useful there.2012-03-31
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    This is your first time in MSE, but you should know that duplicate questions are closed. Your question does seem to be a duplicate of the questions whose links are given above.2012-03-31
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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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