Green's theorem, Stokes's theorem, and the Divergence theorem each say the same thing in different dimensions: the integral of a derivative over a region equals the integral of the original quantity over the region's boundary. This meta-principle has deep consequences in quantitative finance. The Black-Scholes PDE can be derived from the Divergence theorem by requiring that the hedged portfolio has no net flow (conservation law). Exact differentials and conservative vector fields formalise the path-independence properties that underpin no-arbitrage arguments. And the Feynman-Kac formula — which prices derivatives by turning a PDE into an expectation — rests on the same analytical machinery as the classical integral theorems.
This topic develops the full toolkit in order of increasing dimension: vector fields and line integrals in the plane, Green's theorem connecting boundary line integrals to double integrals, surface integrals in three dimensions, Stokes's theorem connecting circulation to curl flux, and finally the Divergence theorem connecting surface flux to volume divergence.
By the end of this topic you should be able to:
- classify a vector field as conservative/gradient and find its potential function;
- evaluate scalar and vector line integrals along parameterised curves;
- apply Green's theorem to convert a boundary integral to a double integral and vice versa;
- compute divergence, curl, and the Laplacian using the del operator;
- set up and evaluate surface integrals for area and flux;
- apply Stokes's theorem to relate circulation to curl flux;
- apply the Divergence theorem to relate surface flux to volume divergence;
- recognise all three integral theorems as special cases of a single higher-dimensional fundamental theorem.
In this lesson
- Contents
- The big picture
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