Why is it that stokes theorem relates a line integral of a cross product to a double integral, where as divergence theorem relates a double integral of F dot product S to del dot F dV?
By this I mean what pattern exists in these two theorems, and is there any "easy" way to visualize the pattern?

I know they are both a specific form of Gauss's theorem, but I am not so sure I am familiar enough with mathematics to understand his theorem as commonly stated.

Is there a simple relation from a differentiable manifold of R^n and R^(n-1)? or is it more complicated than this and requires more depth?

Question

Why is it that stokes theorem relates a line integral of a cross product to a double integral, where as divergence theorem relates a double integral of F dot product S to del dot F dV?
By this I mean what pattern exists in these two theorems, and is there any "easy" way to visualize the pattern?

I know they are both a specific form of Gauss's theorem, but I am not so sure I am familiar enough with mathematics to understand his theorem as commonly stated.

Is there a simple relation from a differentiable manifold of R^n and R^(n-1)? or is it more complicated than this and requires more depth?