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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?

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Quantum mechanics might be your best bet. There, you can keep all the usual interpretations of physics but also use a lot of statistical/probabilistic scaffolding that people in your audience are more familiar with. In particular, the continuity equation for probability current would tell you that

$$\nabla \cdot j = - \frac{\partial \rho}{\partial t}$$

This tells you, more or less, that the rate at which the chance of finding a particle in a given spot changes according to the probability current flowing away from the point.

Quantum physics might break less-inclined people's brains, though. Still, you can talk about scalar probability distribution functions, which allows you to talk about the gradient as a generalization of the derivative of a distribution. Vector fields might be trickier. It's not obvious to me what use vector fields would be in a statistical context.

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  • $\begingroup$ Hmmm, this is a good idea. Quantum mechanics might indeed break my students, but on the other hand there is a classical counterpart to your idea, namely Brownian motion. Maybe that's a place to look. $\endgroup$ – Paul Siegel Jan 28 '13 at 21:38

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