(I have posted a similar question hours ago, but I deleted that and slightly modified the question and reposting it here.)
So far, I have studied all undergraduate courses only except Green's theorem and Stoke's theorem.
As far as I know, to prove the most general form of Stoke's theorem, one must know some sophisticated algebras. (Tensor algebra, Exterior algebra and etc) I like this approach but it seems like I have to learn so many things to achieve this. Anyway, what are the rigorous texts introducing this general form of Stoke's theorem?
Moreover, while the general form of Stoke's theorem requires lots of prerequisites, Green's theorem seems much easier to prove than Stoke's theorem in abstract context.
I'm not asking for a very top generalized setting of Green's theorem. Just like one learns Cauchy's Integral formula with relation of homotopy theory in graduate school while one learns it for a simple closed curve (without knowledge of homotopy theory), I'm curious whether there is a text treating Green's theorem and related multivariable calculus in graduate-level abstract setting.
Say $\alpha$ is a differentiable simple closed curve one is applying Green's theorem on. And consider a rectifiable simple closed curve $\beta$ which is homotopic to $\alpha$ rel $\{0,1\}$. Such $\beta$ can be chosen very close to $\alpha$ so I think the area difference of interiors of $\alpha$ and $\beta$ can be made arbitrarily small or zero. I think to make this assertion precise, one needs lebesgue measure to control areas.
Even though a text does not cover what I just described, what is an abstract text treating multivariable calculus that you know? Thank you in advance.