The Feynman-Kac formula relates a solution to a PDE on a domain to an expectation over the domain's boundary — exactly the same type of "interior-to-boundary" exchange as Stokes's theorem. Stokes's theorem is the three-dimensional generalisation of Green's theorem: it says the circulation of a vector field around a closed curve (a line integral) equals the curl flux through any surface bounded by that curve (a surface integral). The curl measures local rotation; Stokes's theorem says the total accumulated rotation over the surface equals what you "see" from the boundary. This connecting principle is why the Black-Scholes PDE can be derived from the no-arbitrage condition: the circulation around any closed trading path must be zero (no free money), which forces the price process to satisfy a curl-free condition.
Learning objectives. By the end of this note you should be able to:
- state Stokes's theorem and apply it;
- orient a surface consistently with its boundary using the right-hand rule;
- recognise Green's theorem as the planar special case of Stokes's theorem;
- use Stokes's theorem to evaluate difficult line integrals by switching to easier surface integrals.
In this lesson
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- Statement of Stokes's Theorem
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- The Orientation Convention
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- Applying Stokes's Theorem
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- Corollary: Conservative Fields
Before you start
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