# Vector calculus in statistics

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: Green's theorem, Stokes' theorem, and the divergence theorem. Part of the problem is that vector fields don't seem to come up very often in probability theory, let alone divergence, gradient, or curl. I also posted this question on math.stackexchange a few days ago, but I'm still looking for more ideas.

• I have an MA in econ and I never recall needing divergence or curl (or the theorems you listed). If there are applications (perhaps to a DSGE), it would be at the PhD level. However, gradients are used in optimization, which is important for economics and portfolio management.
– John
Commented Feb 5, 2013 at 18:56
• Vector fields actually appear in a fundamental way within the answer at stats.stackexchange.com/questions/29121/…, showing that they can crop up in unexpected places. Moreover, the comment thread after that answer suggests (to me, at least) that a very well established statistician would have appreciated this theory better had he had more training and practice in geometric and analytic methods.
– whuber
Commented Feb 5, 2013 at 19:04
• It's very difficult to get even physicists interested in the theorems and their proof, even with Stokes' theorem, which is very important in physics. I have no recollection of the proof at all! Yet I remember how to use it. So, I wouldn't worry too much about making it interesting, it's basically impossible. Commented Jul 24, 2017 at 19:12