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I'm looking for recommendations for a multivariable calculus book at a somewhat sophisticated level; somewhere between Stewart's Calculus and Munkres' Analysis on Manifolds. I'll have a background in single variable calculus and the typical material from a basic "proofs" class (set theory, logic, proof techniques, some topics in discrete math). This will be my first formal exposure to multivariable calculus beyond some reading I've done for fun. Note that, although I'll have some mathematical maturity and some background in proof-writing, I'll have learned single variable calculus from Stewart, obviously not a very rigorous book. Let me know if you think it's really necessary that I read a more sophisticated calculus text (like Spivak's Calculus) before moving on to multivariable calculus at the level that I'm describing.

I think a book like Spivak's Calculus on Manifolds or Apostol's Calculus, Vol. 2 would be what I'm looking for. Of these two, I think I'd slightly prefer using Spivak since I'm interested in differential geometry and I like the idea of introducing manifolds in multivariable calculus.

I hope this gives some idea of the kind of book I'm looking for. I don't want something crazy rigorous, but definitely something more sophisticated than a typical computational calculus book.

Josh M
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    seek for Hubbard-Hubbard: "Linear Algebra, Vector Calculus and Differential Forms" – janmarqz May 25 '21 at 04:35
  • If you'd like a midpoint to what you're describing, Lax-Terrell's Multivariable Calculus with Applications is essentially a more rigorous version of a standard Calc 3 text. No manifolds, but it does talk about derivative matrices, and covers just the fundamental theorems in 2 and 3 dimensions. – TomKern May 25 '21 at 05:17
  • Have you looked at the suggestions here and here? – Hans Lundmark May 25 '21 at 06:19
  • "introducing manifolds in multivariable calculus"... I'd suggest getting at least a grasp of multivariable calculus in $\mathbb R^n$ before introducing manifolds. I do not think that you require more "rigorous" analysis at this stage, you can go back to it if needed. – Miguel Aug 23 '21 at 21:47

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You've learned single-variable calculus from Stewart, but you seem fairly confident in your ability to do real mathematics.

In this situation, I wouldn't recommend a book like Spivak or Apostol to redo single-variable calculus. Instead, it would make sense to fill the gaps in your preparation with a book like Burkill's A First Course in Mathematical Anaysis. It takes you close to the level you'd be at after Spivak, but it's concise, it doesn't over-emphasize the mechanical aspects of calculus which you know well, and it doesn't venture into the more abstract territory of analysis books like Apostol and Rudin.

With this kind of preparation, you would probably be fine starting with any level of rigour in multivariable calculus (Munkres, Spivak, even Loomis-Sternberg). Without it, I think you'll be more limited in your choice, with probably even Volume II of Apostol being difficult.

By the way, you're mistaken in thinking that Spivak's Calculus on Manifolds is easier than Munkres' Analysis on Manifolds. Quite the opposite. They cover similar material with similar approaches, but Spivak has fewer examples and expects you to work much more out for yourself.

If you do want to go straight into multivariable calculus with less preparation, there are some books from the U.S. aimed at your demographic - people coming in from high school who want or need multivariable calculus right away, and who are mathematically able but only have something like non-rigorous BC calculus. I've heard the books by Shifrin and Hubbard are good for this, but I don't know them well - you'll find more information elsewhere on the site. What I have been able to see of them is that eventually there will be a good deal you need to relearn in a more mature way.

The bottom line, I suppose, is this. There are people who need multivariable calculus now because of physics, course requirements or something else. But if what you're aiming for is an optimal math education, my advice is, don't do it. Build your mathematical maturity first, especially in analysis (at the level of Burkill or Spivak/Apostol), and then learn multivariable calculus at a high level of rigour.

Edit. Another option, which I didn't mention above, is the two-volume book by Zorich. This would be good for someone who already has some acquaintance with single-variable calculus and is looking to cover single- and multi-variable calculus theoretically and in depth. However, the level of difficulty is very high, particularly in some of the problems. So this is a book for a reader highly confident in their own ability.

Anonymous
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I have tried several calculus books over the past years, and indeed Spivak and Apostol sometimes tend to be overwhelming, as an introduction to analysis. I would like to recommend you "Vector Calculus, Linear Algebra and Differential Forms - A Unified Approach", as it's not a formal analysis course nor a pure calculative book. If you are interested into geometry and manifolds, this can be a good choice for you, and also as transition text for a more rigurous treatment of calculus.

Nicolás Gómez
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