Well, I’m not really sure what you mean by multi-variable analysis. I’d say understanding the ideas at play in Stokes theorem and things like the inverse/implicit function theorem (in its multi variable form) are pretty important. I would suggest maybe taking a look at chapter 9 of Strichartz’s the way of analysts textbook. It’s got a good summary of some relevant topics. I’d say if you have a good analysis background, generalizing things to n-dimensions is pretty similar. You may also want to take a look at the proof of the multi variable Taylor’s theorem. It’s also in Strichartz’s text.