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I will be teaching a course focusing on multivariable integration soon, for the millionth time. The most difficult topic in such a course is certainly Vector Calculus, by which I mean line and surface integrals of vector fields. It is essential to present good applications of these so that students are motivated, but all examples I've ever used are standard physical ones. My course will have many economics/finance majors, and I would love to have some examples I could present along these lines. However my knowledge in these fields are lacking, so I ask: what are some common applications of vector calculus to economics and/or finance, ones which will keep students in these fields motivated? If possible, these applications should be understandable by someone who has (or will have) only an undergraduate background in economics.

I could probably come up with some ad-hoc example where I model the flow of resources from one sector of an economy to another using a vector field and then come up with some interpretation of a line or surface integral involving this, but I would really like some examples which actually come up in practice. Bonus if they use Stokes' or Gauss' Theorem.

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    $\begingroup$ This should probably be a comment instead of an answer, but here's a Math Overflow question along the same lines. mathoverflow.net/questions/123227/… I would be interested in seeing some explicit examples, assuming that some exist. $\endgroup$ – Ross Sweet Apr 3 '14 at 20:06
  • $\begingroup$ Green's Theorem turns up in the study of dynamical systems, so an opening could be to introduce appropriate examples of dynamical systems from economics, assuming this doesn't take you too far off course. $\endgroup$ – J W Apr 3 '14 at 20:57
  • $\begingroup$ Just to clarify, I was thinking of the Bendixson-Dulac theorem. $\endgroup$ – J W Apr 4 '14 at 14:27
  • $\begingroup$ ...that is to say, Green's theorem is used in the proof of the Bendixson-Dulac theorem. $\endgroup$ – J W Apr 4 '14 at 14:37
  • $\begingroup$ Most of the applications advised so far are too advanced ("used in the proof" and such). Rather than emphasizing line and surface integral financial examples, I would just think of the part of the course having to do with partial derivatives. Finance and economics are very multifactorial and many analyses rely on "ceteris paribus". $\endgroup$ – guest Nov 23 '18 at 19:53
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As far as I know, vector calculus is applied by financial analysts in exotic derivatives pricing. The Black-Scholes Model is actually a special form of Schrödinger equation. Thus, if you want to establish high precision models to price exotic derivatives, you will have the chance to apply vector calculus. However, these kinds of applications presumes students have solid background in economics and finance.

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The most important applications of multivariable integration to economics and finance are in statistics, especially expectations with multivariate probabilities.

Many colleges have enough economics and finance majors to support a multivariable calculus class designed on this basis. It would assume a course in probability and statistics as a prerequisite. It would skip integrals of vector fields entirely.

So I can't help you for this year. You could ask an appropriate professor of economics to write a letter requesting such a class from the math department, and that might pave the way for the future.

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I'm not totally sure the scope of your course but I think there is a wealth of applications from constrained optimization problems, gradient descent, dynamical systems, etc. Check Economic Dynamics: Methods and Models by Giancarlo Gandolfo as a starting place.

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