This is cross-posted from math.stackexchange and stats.stackexchange. Probably there is no great answer to this question, but I thought I'd give it a shot here.
I'm teaching a class on integration of functions of several variables and vector calculus this semester. The class is made up most of economics majors and engineering majors, with a smattering of math and physics folks as well. I taught this class last semester, and I found that a lot of the economics majors were rather bored during the second half. I was able to motivate multiple integrals by doing some calculations with jointly distributed random variables, but for the vector analysis part of the course the only motivation I could think of was based on physics.
So I'm wondering if anybody knows a statistical/probabilistic interpretation of any of the main theorems of vector calculus. This might require having such an interpretation of div, grad, and curl, and it's not so obvious what it might be. Anyone have any ideas?