# Multivariable Calculus Texts (Theoretical + Computational/Intuitive/Geometric Texts)

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.

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.

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.