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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'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.] – Dylan Moreland Jul 9 '12 at 13:43
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. – David Speyer Jul 9 '12 at 13:48
@Siminore 7: could you please explain your use of the conjunction but in your phrase nice but rather easy ? – Georges Elencwajg Jul 9 '12 at 13:54
@DavidSpeyer, That seems to be a very good book but 1298 pages ?! I'm not sure I have the patience. – Inquest Jul 9 '12 at 17:03
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:-) – Georges Elencwajg Jul 9 '12 at 17:25
up vote 4 down vote accepted

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 it seems that many undergraduate level math textbooks from mainstream publishing houses now are.)

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