# Applications to Green's Theorem in Economics?

I was wondering about possible of application of integration to economics (other than welfare), more specifically, how might Green's theorem be useful for an economist?

Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C . If G and H are functions of (x,y) defined on an open region containing D and have continuous partial derivatives there, then

$$\oint_C{(G\ dx + H\ dy) = \int\!\!\!\int_D {\left({{\partial H} \over {\partial x}} - {{\partial G} \over {\partial y}}\right)\ dx\ dy} }$$

where the path integral is traversed counterclockwise.

The idea behind this theorem is that if you have a line integral in two dimensions, then Green's theorem can be used to compute the integral: Green's theorem transforms the line integral around a simple closed curve $$C$$ into a double integral over the plane region $$D$$ bounded by $$C$$.

## 1 Answer

Here and here are two application of the theorem to finance.

Here is an application to game theory.

This is an application of the theorem to complex Bayesian stuff (potentially useful in econometrics).

Also, the Green's theorem is used in this proof, which relates to dynamics system, and therefore could be related to economic models (source of this here)

Last but surely not least, this book might have relevant stuff. At least Chapter 12 does use it.

• Lucho, what about for the Stokes' theorem? ;) – An old man in the sea. Nov 8 '17 at 16:41