What is the FTC of a function from the perspective of Stokes' theorem? In the case of $\int_a^b f(x) dx=F(b) - F(a)$ this can be written as $\int_a^b \frac d {dx} F(x) dx=F(b)- F(a).$
Comparing to Stokes' theorem, $\int_{\partial\Omega} \omega=\int_\Omega d\omega$ where $d\omega$ is a differential form, I guess it would be something like
$$\int_{\Omega=[a,b]} dF=\int_{\partial\Omega=\{a,b\}} F$$?
So the manifold $\Omega$ is the compact interval $[a,b]$ over which the exterior derivative $dF$ is integrated, whereas the boundary consists of the set of the two limit points $\partial\Omega=\{a,b\}$ over which the differential form $F$ is integrated.

PS: Great explanation:


By the choice of F, dF/dx = f(x). In the vocabulary of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form, namely the function, F. In other words, that dF = f dx. The general Stokes theorem applies to higher differential forms ω instead of just 0-forms such as F.
A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. In the more general case, the manifold has to be orientable, and the form has to be compactly supported in order to give a well-defined integral.
The Orientation of the interval can be thought of as the ordering of the elements, the closed bounded interval [a,b] is compact as you learn in analysis.
The two points a and b form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds M with boundary. The boundary ∂M of M is itself a manifold and inherits a natural orientation from M. Intuitively, a inherits the opposite orientation as b, as they are at opposite ends of the interval. So, "integrating" F over two boundary points a, b is taking the difference F(b) − F(a).
This integrating over two points is really more of a degenerate case and maybe it is confusing as to why you are just plugging in the values.
You can understand it as integrating over a singleton using an atomic measure with the two point boundary. An atomic measure gives a non-zero measure to the singleton set so that taking an integral over a singleton makes sense in a way that is more than just hand waving.
