Is there a statistical interpretation of Green's theorem, Stokes' theorem, or the divergence theorem?

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?

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My calc professor gave us the following intuition on the flux-div formula.The flux of the vector field counts how many flow lines of the field go out through the surface minus how many go in. If there are no sources or sinks inside, this balance must be zero. If it's not zero,it must be equal to the number of flow lines created inside (wells) minus the ones that die inside (sinks). Div measures the intensities of these wells/sinks.To visualize flow line he asked us to think of iron powder on a sheet of paper with a magnet underneath. It's not probabilistic,but I bought this explanation. –  Liviu Nicolaescu Feb 28 '13 at 16:30
Please do not crosspost questions —specially between math.SE and MO; it introduces inefficiencies in the process. If you must, provide links from each Instance to the others. –  Mariano Suárez-Alvarez Oct 27 '13 at 19:01

I was told that a professor in our department puts it as follows: Your job is to determine the number of cars in a car park. One method is to go around the car park counting them. Alternatively if you know the number at one time then you can stand by the entrance/exit and adjust the number every time a car leaves or arrives.

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though the following is still not probabilistic it can be given a try.

consider a nation with a population of humans. these humans move around and their movement at any point of time describes a vector field over the nation. some humans are born at some point and then start moving depending on the flow lines at that point ... some humans perish too.

this vector field can then be operated as usual ... viz examined for grad and div, which will provide information about net immigration and emigration.

all the above can also be done for the flow of money. the source of money is the banking system which generates credit and injects money into the market. govt also creates money by borrowing. money is destroyed when the credit is repaid.

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The gradient and div appear in maximum likelihood models in statistics and in statistical mechanics which are all used in financial mathematics.

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Do you by any chance know any good references for this? –  Paul Siegel Mar 1 '13 at 14:27
"Multivariate Statistical Methods" by Donald Morrison, Professor at Wharton publ by McGraw-Hill. A more mathematical treatment is in "Multivariate Analysis" Mardia, Kent, & Bibby publ by Academic Press. –  awynjones Mar 6 '13 at 1:44

This doesn't quite answer the question but multivariable calculus is baby differential topology and there are a few topological theorems that economists quote frequently.

The most common one is probably Brouwer fixed point theorem, which is required for the existence of Nash equilibria of a pretty wide class of games. Game theory is the main ingredient of microeconomics. (Actually the relevant general theorem is Kakutani's fixed point theorem for compact- and convex-valued correspondences but that is a stretch for most econ folks.)

Also, the Hairy Ball theorem has an economic interpretation. In a market, for each corresponding price there is an excessive demand. If there are $n$ goods, then price and demand are vectors in $\mathbb{R}^n$. When excessive demand is zero, economics says the market clears and is in equilibrium. So Hair Ball theorem says that the market clears for some price.

If you cover Lagrange multipliers, economics students taking intermediate microeconomics and macroeconomics see that everyday. The econ view Lagrange multiplier is the "shadow value of money", meaning that if the budget constraint is relaxed by $\epsilon$ at the optimal bundle (in the direction of the gradient), consumer utility increases by $\epsilon \cdot \lambda$. The equation

$$\nabla u = \lambda \cdot \nabla g$$

describes the consumer substituting between goods in his basket as he compares marginal utilities (entries in $\nabla u$) and marginal cost (entries in $\nabla g$).

As for the gradient and Maximal Likelihood estimation procedure, at the very basic level it's just a calculation. Suppose you have observations $\{ x_i \}$ drawn independently from the probability space $(\mathbb{R}, f(x, \theta)dx)$, where $\theta$ lies in some compact parameter space $\subset \mathbb{R}^n$. To estimate $\theta$, you maximize the likelihood function

$$L(\theta) = \Pi_i f(x_i, \theta).$$

The gradient of $\log L$ is called the score function. On a deeper level, although I am sure how much of it you can mention to your undergrads, MLE is the beginning point of information geometry, where statistics and differential geometry interact.

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