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I'll be finishing Calculus by Spivak somewhat soon, and want to continue into linear algebra and multivariable calculus afterwards. My current plan for learning the two subjects is just to read and work through Apostol volume II; is this a good idea, or would it be better to get a dedicated Linear Algebra book? Are there better books for multivariable calculus? (I don't think I want to jump right into Calculus on manifolds.)

EDIT: I'd like to add, as part of my question, something I mentioned in a comment below. Namely, is it useful to learn multivariable calculus without differential forms and the general results on manifolds before reading something like Calculus on Manifolds or Analysis on Manifolds? That is, do I need to learn vector calculus as it is taught in a second semester undergraduate course before approaching differential forms?

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Try Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds by Theodore Shifrin. It shows how the two subjects are related.

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If you want to start with abstract linear algebra, try Axler's Linear Algebra Done Right.

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I don't think Apostol has enough for a first course in Linear Algebra, why don't you try: Linear Algebra by Hoffman and Kunze. The linear algebra bits in Artin's Algebra is also very good.

For multivariable calculus, I found Munkres' Analysis on Manifolds much easier to digest than Spivak.

If you want something requiring less mathematical maturity then you could try: Hubbard, Vector Calculus, Linear Algebra and Differential Forms. This is a fantastic text.

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  • $\begingroup$ Thanks for the suggestions. So, it seems that Analysis on Manifolds and Calculus on Manifolds treat multivariable calculus immediately through differential forms (or this is my understanding, at least). Is it worth it at all to learn multivariable in a more basic way before looking at differential forms. (It is not that I am worried that I cannot handle learning it through differential forms; I am just wondering if it is instructive or useful to learn the subject without them first, and then read something on manifolds afterwards.) $\endgroup$
    – Arpon
    Jan 1, 2011 at 20:38
  • $\begingroup$ I actually really love that last book you recommended as well! It makes the transition to more abstract differential geometry really nice. $\endgroup$
    – Matt
    Jan 1, 2011 at 23:47
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I LOVED Advanced Calculus: A Differential Forms Approach by Edwards. Great book.. It's great to move to the next level in geometry.

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