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I took calculus I, II, and III a long time ago, I barely remember anything, so I want to review it. Also I would like it if this time around I learn more about the why of things, since when I took those calculus classes it was all very computational (compute this integral) and we didn't prove a single theorem. I've heard that Spivak is the perfect book for this, self-contained and goes into most of the nitty-gritty, perhaps not as deep as an actual analysis book, but close enough for people like me, so I've decided to read it. Unfortunately it seems the book only covers single variable calculus, and I would like to also learn multivariable calculus. I'd like to hear your suggestions for a good multivariable calculus book to read after Spivak, with roughly the same level of rigor, and that doesn't require any other knowledge besides what I learn with Spivak, so if the book assumes one already knows linear algebra, it's no good for me, since I'd have to read a linear algebra book too.

I did some research and honestly the only books I could find that were kind of close were:

-Apostol Calculus vol. II, this one seems good but I would rather avoid reading a book that is a sequel of another one, since there might be things that Apostol assumes you know, since they were on his first volume, that maybe Spivak didn't cover.

-Vector Calculus, Linear Algebra, and Differential Forms_ A Unified Approach, by Hubbard and Hubbard. This one also seems good, however I'm unsure if this book can be read knowing only what Spivak teaches. I read the preface and it says students at Cornell University use it when taking Math 2230, I did some research and Math 2230 has as prerequisite a year of calculus, a year of calculus means single variable and multivariable calculus, so maybe the book assumes one already knows multivariable? I'm not sure.

So I guess my question is kind of a double one, could I read Hubbard right after Spivak without problems? If the answer is no, then what book would you suggest? Thanks.

zlaaemi
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  • well, times do change; still, when I was a TA, calculus in one variable was one year, the second year was linear algebra, O.D.E, finally multivariable. I should emphasize this: in high school I had a few weeks of vectors in the plane and (three)-space, dot and cross product calculations. That made a big difference. These days, I would suggest a bit of time with quadratic forms before doing Lie groups, same idea. – Will Jagy Nov 29 '23 at 04:41
  • I suggest the two-volume book by Zorich, containing both one and several variables. See https://math.stackexchange.com/questions/1003791/vladimir-zorich-vs-rudin-pugh-abbott – GReyes Nov 29 '23 at 07:39
  • Maybe consider one of the standard advanced calculus texts that used to be covered (U.S. perspective) in a year-long course usually taken by 3rd or 4th year undergraduate students (sometimes 2nd year, if the student was very advanced) that revisited Calculus 1, 2, 3 (diff, integral, multi) from a more advanced perspective and also covered additional topics (the additional topics changed a little over the years, i.e. 1930s-1950s as compared to the 1960s-1980s+). Some of these texts are given in this MSE answer. – Dave L. Renfro Nov 29 '23 at 08:33
  • @GReyes Do you think someone with my level of experience can jump straight into Zorich? I believe that's a real analysis book. – zlaaemi Nov 29 '23 at 12:25
  • From the preface of Hubbard Hubbard: "Chapters 1 through 6 of this book cover the standard topics in multivariate calculus and a first course in linear algebra." I think it's a very nice book. You can supplement with Duistermaat & Kolk I,II, and Apostol Adv Calc if you want. I would just skip the Spivak - you'll probably start remembering what you need as you read HubbardHubbard. – p.co Dec 13 '23 at 19:53

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