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I'm looking for suggestions for textbooks covering multivariable calculus. I am looking for two textbooks, one which covers the theory and the other which covers the computational aspects. I have already taken a (not so taught well) first course in multivariable calculus, but I'd ideally like to to keep a computational/intuitive text with me.

I have covered a first course analysis with a focus on point set topology. After revising a bit of the theory of integration and covering Lebesgue integrals which I haven't done thus far, I'd like to move on to cover the aforementioned texts.

My aim is the following: over the semester, I'll be enrolled in a course in advanced calculus. From what I gather, the course will cover manifolds etc. but it'll cover them from a very not so rigorous/differential geometric point of view. With that course and a course in GR, I'd like to get introduced to the rudiments of manifolds etc. With my out-of-class work on multivariable calculus, I hope to build up on the material of the courses in a few month's time -- before the term ends -- and use the material covered on both these fronts to start off with basic differential geometry. By then, I hope to have covered some algebra, topology etc. on my own as well.

With my motivation in mind, it'd be great if you could recommend some texts.

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Disclaimer: My path through multi was atypical, and somewhat incomplete. I had an intro at the end of my calc class last year, using a mix of Stewart and Spivak. Then in analysis last quarter, we spent a good bit of time on differentiation, but for integration, we just defined it in 5 minutes and jumped to change of variables, curves, and surfaces. This quarter, we did some differential forms in the plane and line integrals, but we've since moved on.

With that caveat out of the way, my short answer is Buck for computation, and either Pugh or something from NSS lineage. If possible, it may be a good idea to check out a dedicated book on forms, such as Do Carmo.

Long answer:

So, my analysis professor last quarter lectured out of 8.1-8.5 of Buck's Advanced Calculus, covering change of variables, curves and surfaces. We were also given a review assignment from chapter 7 (differentiation) the week before. From what I saw, it has good pictures, sticks to $\mathbb{R}^2$ and $\mathbb{R}^3$, and has mostly computational problems, while still proving its theorems. I was not terribly fond of its lack of generality and odd notation, but for a computational book it may be a reasonably good candidate. In previous years I've heard that we used Edward's Advanced Calculus of Several Variables, which apparently combines well integration and computation. I've heard mixed things about it.

One option for theoretical books is to learn the material from general analysis books. The advantage of this is that you get bonus material on analysis, with the tradeoff of having less of an eye toward differential geometry. My class last quarter bounced between Baby Rudin and Sally's Fundamentals of Mathematical Analysis mostly. Rudin is a standard analysis book, and is really good except in multi, where it's trash. A common alternative is Pugh's Real Mathematical Analysis, which is of a similar calibre but also better for multi. Sally's book is what we used for differentiation, which is alright, more advanced than the preceding books, but very niche (built around how my class used to be taught under him, and reeking of Moore method) and a bit error-ridden. Also, only really treats Lebesgue integration.

Books that are actually about theoretical/abstract/modern multivariable calc have mainly stemmed from Nickerson, Steenrod, and Spencer, who were pioneers in teaching the material to undergrads. Their material is contained in a book called Advanced Calculus, and there are a sea of other books by Fleming, Spivak, Munkres, Loomis & Sternberg, et al. They all seem pretty good, though the main one I've used is Spivak, which I can vouch for as being pretty good.

Finally, my professor this quarter recommended Differential Forms and Applications, by Do Carmo and Differential Forms by Cartan to supplement, since we stuck to the plane in order to get the major ideas without spending time to develop all the machinery. It's a pretty important topic, and a lot of the theorems of integration, such as change of variables, fall out of the sky nicely through the eyes of forms. Getting a dedicated book might be a bit overkill given the finance factor, since the material is treated in more general books on analysis/geometry, but if that's more of a moot point, it could be worth looking into.

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Goro Shimura recommended Michael Spivak "Calculus on Manifolds" and W. Fleming "Functions of Several Variables" for multivariable calculus book in his books in Japanese.

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