If is a singular -cube and is a -form on the image of , then we know how to define the integral of over :
On its face, this formula depends on the function used to parameterize the region of integration. But does it really? What if we have a different function with the same image? For convenience we’ll only consider singular -cubes that are diffeomorphisms onto their images — any singular -cube can be broken into pieces for which this is true, and we’ll soon deal with how to put these together.
Anyway, if , then given our assumptions there is some diffeomorphism such that . If is everywhere positive, then we say that is an “orientation-preserving reparameterization” of . And I say that the integrals of over and are the same. Indeed, we calculate:
where we use our expression for the integral of over the image in passing from the third to the fourth line. Thus the integral is a geometric quantity, depending only on the image and the -form rather than on any detail of the parameterization itself.