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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, 2014 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, 2014 at 20:57
  • $\begingroup$ Just to clarify, I was thinking of the Bendixson-Dulac theorem. $\endgroup$
    – J W
    Apr 4, 2014 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, 2014 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, 2018 at 19:53

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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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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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  • $\begingroup$ Where do you need line/surface integrals or stokes theorem in the schrödinger equation? $\endgroup$
    – Michael Bächtold
    Sep 23 at 9:19
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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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  • $\begingroup$ With surface integrals and flow of fields through surfaces? $\endgroup$
    – Michael Bächtold
    Sep 22 at 15:23
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I'm just only learning about this stuff but so feel free to fix/improve my answer:

If you build a neural network to do some machine learning on predicting financial/economic entities (stock prices, true/false acceptance of loans, etc), you are setting up a function $\mathbb{R}^L \to \mathbb{R}^O$ where $L$ is the total number of components of your weight matrices $\theta$ and $O$ is the total number of outputs you're trying to predict. When you test your predicted outputs against the real data, you obtain an error function $J(\theta)$ which is a multi-variable vector-valued function!

The gradient $\nabla_{J(\theta)}$ which can be approximated numerically, tells us which direction we should alter our components of our weight matrix to maximize (locally) $J(\theta)$. We want to minimize error, $J$, so we walk our $\theta$ in the direction of $-\nabla_{J(\theta)}$.

This results in the linear approximation to a path flowing along the vector-field $-\nabla_{J(\theta)}$ on $\mathbb{R}^L$ given by $\theta_{t+1} = \theta_t - \epsilon\cdot \nabla_{J(\theta)}$ where $\epsilon$ is our length of straight-line walking/flowing. The assumptions that theoretically everything is actually a smooth higher dimensional surface make this method sensible, and so that the smaller $\epsilon$ is the close our path is to a flow along the field.

Now the line integral along this path $\theta_t$ for the vector field $-\nabla_{J(\theta)}$ should represent the total change in error, $J$, as we move from $\theta_a$ to $\theta_b$.

Stokes' theorem would say that two different paths we take, flowing along the gradient vector field, if they end at the same place, should end up in equal reductions of the error if our situation is "nice".

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