Stokes’ Theorem (proof part 2)
Of course, we only have to handle the case of a general singular cube, since we defined the boundary operator to be additive; if is a general chain — a formal sum of singular cubes — then is the same formal sum of the boundaries of these cubes. Since integration is also defined to be additive on the chain over which we integrate, everything works out:
where we have used the special case of singular cubes to pass from the second line to the third.
So let’s tackle this special case:
Basically, it all works out for the same reason parameterization invariance and the change of variables formula do. Passing from the boundary of the singular cube back to the boundary of the standard one transforms the integral one way; passing from the standard cube itself back to the singular cube undoes this transformation. And so Stokes’ theorem is proved.