Stokes and Gauss
So I was reading about Stoke's and Gauss's laws, and I was curious if there are analogous laws for other dimensions. For example, can you jump from a four dimensional integral to a five dimensional integral by using the proper combinations of curls and gradients? Most of my experience with vector calculus has been for more mundane things like electric charge density, but I can imagine such theorems would be useful in areas like statistics, economics, or quantum mechanics. Anyone deal with anything like that?