This is a very general quesiton and I am asking for only a brief feedback about whether I'm thinking in the right direction.
Exact and closed k-forms have something to do with how we do the integration. While exact form is like a boundary of a manifold with boundary, I'm not sure what closed form is like, perhaps it's a 'closed up' boundary (e.g. a curve $c$ with its head and tail matched up and makes a circle) that is not necessarily differentiable everywhere;(or a closed form is like a curve $c^*$ starting from a point and repeatedly extending along itself). If so, the impact of such a difference is perhaps similar to that integrations over a contour, of complex function with or without singularity, are non-zero and zero respectively.
Not every closed k-form is exact. de Rahm cohomolgy vector space perhaps deals with to what extent the size of the set of closed k-forms on a manifold is larger than that of exact k-forms. For example, for $c$ as a closed form, the corresponding exact form a circle $S^1$. And the size of the closed form is $|S^1|$ times the size of exact form, since starting point of $c$ can be any point in $S^1$.
In other words, the question is about what a closed form is like, what de Rahm cohomology is for?
The question is complicated so I break it down, one part of the question is about difference between 'exact' and 'closed'.
From John Lee's book, ch4, I guess 'exact' is that we can apply fundamental theo of calc, (the definition of 'exact' is not necessary, though it does imply, that integral over a contour is 0; since 'exact' is a concept about k-form, 'the integrand', not one about where we integrate a function over) or we can say it’s that integral $fdx$ or $dF$ over line doesn't depend on path but on the two ends.
An example of 'closed and not exact' k-form is $\frac{xdy-ydx}{x^2+y^2}$, or $d\theta$, we see if we go from a point to another point counterclockwise and clockwise, we get different integration of $d\theta$.
An example of 'exact' is $\frac{xdx+ydy}{x^2+y^2}$ (NOT $\frac{xdy+ydx}{x^2+y^2}$), namely $2dlog(r)$, which doesn’t depend on path. (Notice in both examples, for $adx+bdy$, $\frac{\partial a}{\partial y}=\frac{\partial b}{\partial x}$, which is the definition of 'closed'.)
So basically 'exact' is $df(r)$, integral that doesn’t depend on path but two (end) points of a curve; 'closed' is $df(\theta)$, integral that can be written as substraction of values of a function, though not single valued. Perhaps ‘exact and closed’ originated from (two kinds of derivatives of functions with domain of dim >1) $df(r)$ and $df(r, \theta)$, for $f(r)$ behaves very much like $\mathbb{R}$ to $\mathbb{R}$ funciton, while $f(r, \theta)$ does not. (That's also why the distinction between 'exact' and 'closed' doesn't appear until we go to functions involving space of dim >1, just as the distinction between 1-form and derivative of a function does, as discussed in comments here Analog of 2-form for $f: \mathbb{R}^2\rightarrow \mathbb{R}$ .)
Or say 'exact' is a form which can be written as $df$ (for 1-form; for k-form the notation may be more complicated, perhaps like $d(df)$, which I know not yet how to expand to sth with more obvious meaning), where $f$ is a single valued function, 'closed' is a form which can be written as $df$, where f is a function (single valued or multi valued). (That is, 'closed' k-form $\omega$ can also be $d\eta$, just now $\eta$ is not single valued. Previously what confuses me is that since the definition of 'closed' is as stated above, how can it not be 'exact' (a derivative), now I find an explanation.) In other words, both imply we can exactly or roughly apply fundamental theorem to the integration, but the value of integration will be 'exact' (single valued) only when the integrand (k-form) is exact.
Again, for both we have, e.g. in 1-form (from $\mathbb{R}^2$ to $\mathbb{R}$), $adx+bdy$, $\frac{\partial a}{\partial y}=\frac{\partial b}{\partial x}$.
Is my understanding correct?
