A risk-neutral pricing argument equates two things: the expected payoff (an integral over the interior of the state space) and the replicating portfolio's cash flows (integrals along the boundary of the state space). Green's theorem is the two-dimensional theorem that formalises this exchange of perspectives — converting a line integral along a closed boundary into a double integral over the enclosed region, and vice versa. It is the plane-geometry ancestor of Stokes's theorem and the Divergence theorem, and its proof technique (reduce to the Fundamental Theorem of Calculus) is the blueprint for both.
Learning objectives. By the end of this note you should be able to:
- state Green's theorem and identify the orientation convention (positive = counterclockwise);
- apply Green's theorem to convert a line integral to a double integral and vice versa;
- use Green's theorem to compute enclosed area via ;
- extend Green's theorem to regions with holes (multiply connected domains).
In this lesson
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- Statement of Green's Theorem
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- Applications
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- Converting Double Integrals to Line Integrals
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- Regions with Holes (Multiply Connected Domains)
Before you start
- Divergence, Curl, and the Del Operator
- Double Integrals over General Regions and Changing the Order of Integration
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