References for the "extended" Green and Stokes' theorem. I was watching these videos from MIT's series: Green, Stokes.
And I didn't understand the justification: their "extended version" of the theorems.
I looked up on google and couldn't find many references (or they explained things with differential forms, which haven't yet studied).
Does anyone know some references about these versions? I'm not looking for very rigorous/proofy style notes, just some explanation and examples.

I am following the videos of 18.02 course from MIT too, and also got confused while understanding the extended version of Stokes theorem.
I've found this link Stokes' theorem for surfaces with holes.
Although explained for the case of a surface on the plane, we can extend it easily to 3-space.
For example consider the surface of a paraboloid z=9-x^2-y^2 that lies above the plane z=0 and below the plane z=4, as in this graph. Now let S be the surface between the closed curves C1 and C2. If we slit the surface S from one point in C1 towards a point in C2 then we get a surface between a closed curve C=C1+C2+(lines in opposite directin just at the slit). So now applying Stokes' Theorem we can see how as the slit approaches to zero the work along lines in opposite direction cancel each other so only the works through C1 and C2 remain.
In general, let S be the surface between C1 and C2 (for C1 and C2 closed curves), for S, C1, C2 compatibly oriented, then by Stokes' Theorem:
$$
\iint_S (\nabla \times \vec{F}) \cdot \hat n \,dS = \oint_{C_{1}} \vec{F} \cdot d \vec{r} + \oint_{C_{2}} \vec{F} \cdot d \vec{r}
$$
So this is the way I think about the extended Stokes' Theorem in 3-space, even though I couldn't find any formal reference for this more than couple examples where it is applied. Please let me know if my reasoning is wrong.
