Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one.
I'll just state what I understand from it and I'd like someone to correct me where I'm wrong.
So, I know Green's theorem is a special case of this one. In Green's theorem I had a two-dimensional surface along which I integrated $d\omega$. The theorem stated that I can also integrate over the one-dimensional edge of that suface the form $\omega$ and I will get the same result.
Stoke's theorem is supposed to be a generalization of that but the thing that first came to mind when I thought about a generalization doesn't actually match up with what my textbook is saying. I thought that in ST, the 2D surface is replaced by a kD surface, and the 1D edge is replaced by a (k-1)D edge. But then my textbook goes into proving it for kD cubes and basically starts with the fact that the edge of a kD cube is a set of $2^k$ (k-1)D cubes.
How can this be correct? A 3D cube has only 6 2D sides, right? It does have 8 edges but those are not 2 dimensional.
So what am I missing here?
Now, this is stretching the bounds of the question, but could someone try to explain where the order of the differential form being integrated comes into play? It seems like an extra number that doesn't really fit into my vision.