How much multivariable calculus do I need to read Evans’ PDE book? I am currently in a PDE class using Evans’ text covering roughly chapters 1-6 and have around two weeks off, in which I would like to further my understanding of prerequisite material if possible. I have some experience with functional analysis and have worked through Rudin’s first two books, save for chapter 10 in Rudin’s first text, a chapter on integration of differential forms. However, I have only taken a non rigorous course in multivariable calculus, so my lack of familiarity with multivariable calculus concepts in general, as well as those referenced in sections C.1-C.4. of the appendix in Evans, which cover Green’s formulas, integration by parts, and the coarea formula, have hindered my understanding of the text.
For those who have worked through the PDE text, which references would be most practical to supplement my weak areas well enough given the time constraints and background? I have seen similar questions asked and long textbooks that cover multivariable calculus as a whole but spend a long time building up elementary concepts are usually recommended. For now, would it be sufficient to just review chapter 9 (which covers some basic multivariable calculus) and read chapter 10 in Rudin’s first text?
 A: I'm sorry to see your question just now. But I will give my suggestion.
Your question may seem innocent or silly to some. However, I think it is important to reflect a little on this.
For a good PDE course that uses Evans' book (or a similar one) as a reference, having knowledge and familiarity with some topics is important.
As the book advances in some concepts, it is important to be familiar with Functional Analysis and Linear Algebra. For these topics, a good option is Rudin's books, as you mentioned. That is, for this part, I suggest:

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*Brezis, H. (2010). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer New York. https://books.google.com.br/books?id=GAA2XqOIIGoC


*Rudin, W. (1991). Functional Analysis. McGraw-Hill. https://books.google.com.br/books?id=Sh_vAAAAMAAJ


*Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill. https://books.google.com.br/books?id=NmW7QgAACAAJ
Regarding the content of Calculus, it is good to keep in mind some important concepts. Many passages made by Evans are omitted. I imagine he considers it trivial (besides not being the focus of the book). If you don't feel safe, it would certainly be good to do a review (including doing some exercises). Some books are interesting, such as:

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*Folland, G. B. (2002). Advanced Calculus. Prentice Hall. https://books.google.com.br/books?id=iatzQgAACAAJ


*Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill. https://books.google.com.br/books?id=kwqzPAAACAAJ


*Shifrin, T. (2005). Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds. Wiley. https://books.google.com.br/books?id=OVYZAQAAIAAJ
But if you have doubts about just a few aspects, you may want to continue with the course and review the specific points you need. This is also the style of each one. In mathematics, it is not uncommon for you to study a new topic and have to revise some things. I think that, to advance in mathematics, this is common. You learn advanced things and some "elementary" things not yet seen.
