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A friend of mine gifted me his copy of Spivak's Calculus on Manifolds. I was looking out for a good book to relearn MV Calculus to the extent of :

Multivariable Limits, Continuity and Differentiation Differential Calculus of Vector and Scalar Fields Multiple/Surface Integrals

My intention was to go through a nice rigorous text to prepare me for my research in Numerical Optimization.

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    I like Calculus on Manifolds, but it can be a bit terse in places (it covers everything you've listed, and more, in a very small book). It's also quite rigorous, which may be a good or bad thing depending on why you're relearning calculus.2012-07-09
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    @in_wolfram_we_trust, I'm re-learning to add rigour to the brainless and mechanical calculus I do.2012-07-09
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    If that's the case then Calculus on Manifolds is probably the book for you. Don't rush through - give yourself time to digest the ideas and do the exercises.2012-07-09
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    Try to grab a copy of Munkres, *Analysis on manifolds*. It is a nice but rather easy book.2012-07-09
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    I've heard from a few people — people who liked Spivak's _Calculus_ — that this book is too brief. Munkres' _Analysis on Manifolds_ is an alternative source. [I don't have much experience with either.]2012-07-09
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    @Siminore. I'd love to but I'm getting Spivak for free :P I don't want to spend more on MV books since I already have Apostol.2012-07-09
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    I don't think Spivak's book is a good choice for you for three reasons. (1) Spivak works on general manifolds, and it sounds from your list of topics like you would be satisfied with working in $\mathbb{R}^n$. (2) Spivak uses notation from the theoretical differential geometry community, and it sounds like you are going to be doing applied work. (3) In my opinion, Spivak is too terse and unmotivated (but I know many people who disagree with me). My suggestion, given your list of topics, would be Courant's Differential and Integral Calculus, vol. 2.2012-07-09
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    @Siminore 7: could you please explain your use of the conjunction *but* in your phrase *nice but rather easy* ?2012-07-09
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    @GeorgesElencwajg Actually I do not know why I wrote that. Probably because I tend to love books that start at a very abstract and general level. For instance, Munkres' book is essentially confined to submanifolds of the euclidean space, so my soul decided that a really *nice* book on analysis on manifolds should not be *so* elementary :-)2012-07-09
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    @DavidSpeyer, [That seems to be a very good book](http://www.amazon.com/Differential-Integral-Calculus-Classics-Library/dp/0471588814) but 1298 pages ?! I'm not sure I have the patience.2012-07-09
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    Caro @Siminore, my comment was not to be taken too seriously. Yet, I have the optimistic vision that a book can be profound, beautiful, instructive...and easy:-)2012-07-09

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For a textbook which presents multivariable calculus with an attention to rigor and geometric ideas, but which is not as abstract and terse as Spivak's book, I recommend my colleague Ted Shifrin's text Multivariable Mathematics.

(My only caveat is that it is extremely expensive...as it seems that many undergraduate level math textbooks from mainstream publishing houses now are.)