I am trying to find some reference regarding some generalizations of Stokes Theorem. I could simply state it as follows:
Assume $D_1$ is a $C^1$ 2-disk with a piecewise $C^1$ boundary $S_1$ and assume you glue another disk $C^1$ 2-disk $D_2$ with a piecewise $C^1$ boundary $S_2$ along their boundaries. If $D_1$ and $D_2$ do not intersect then the new object bounds a 3 sphere $X$ and then for any smooth differential 1-form $\alpha,$ we have $\int_{D_1 \cup D_2} d\alpha = \int_{X}dd\alpha=0$ by Stokes theorem.
Now what happens if $D_1$ and $D_2$ have intersections? Are there anologues of Stokes theorem for this case?
I did some research and there are various other generalizations but for this case I could only find a single source which claims it works without reference. So I probably dont know the correct key words (I searched Stokes theorem for manifolds with self intersection).
I guess in case the intersections are "simple" you can always chop up things and work with simpler objects but I am not sure how bad the intersections can look like. Also if there are intersections what is the corrrect way to define the interior of $D_1 \cup D_2$? I guess field of normals to the surface would still be defined which you can use to define the interior but I am lacking basic tools to understand this issue formally. So I would be very grateful for any reference on Stokes theorem for this generality so I can learn the tools of trade.