Sorry for the delay, I’ve had a packed weekend.
Anyway, we’re ready to start getting into integration on manifolds. And we start with a simple case that everything else will be built on top of.
We let be the “standard -cube”. We know that the space of “top forms” — top because is the highest degree possible for a differential form on a differential form — has rank over the algebra of smooth functions. That is, if is a top form then we can always write
for some smooth function on the standard cube. Then we write
here we sorta pull a fast one, notationally speaking. On the left we’re defining the integral of a -form . In the middle we rewrite the form as above, in terms of a function and the canonical basis -form made from wedging together the basic -forms in order. And then on the right we suddenly switch to a -dimensional Riemann integral over the standard -cube. The canonical basis -form corresponds to the volume element , and top forms are often also called “volume forms” because of this correspondence. In fact, it’s not hard to see that they’re related to signed volumes. This is the starting point from which all integration on manifolds emerges, and everything will ultimately come back to this definition.