Green's Theorem

Trading a closed boundary integral for a double integral over the region it encloses: the three hypotheses each doing work, the choice of field that turns the region integrand into a constant and computes an area from its boundary alone, and the enclosed singularity that voids the theorem.

Definition

Green's theorem. For a positively oriented simple closed curve $C$ bounding a region $R$ , with $P$ and $Q$ having continuous partials on $R$ ,

\oint_{C} P d x + Q d y = \iint_{R} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A .

Taking $P = - y$ , $Q = x$ makes the integrand $2$ , so $\frac{1}{2} \oint_{C} x d y - y d x$ is the area of $R$ . A region integral computed entirely from its boundary.

Assumptions and scope

Green's theorem requires a simple closed curve, positive (counterclockwise) orientation, and continuous partial derivatives on the enclosed region. A field with a singularity inside the region (the punctured-plane example again) fails the hypothesis, which is why it evades the conclusion.
These statements are for the plane. The analogous three-dimensional condition involves the curl rather than a single cross-partial comparison, and the corresponding theorems are those of Stokes and the divergence theorem, which these units do not cover.
All figures in this unit come from exact evaluation of the stated integrals, with each conservative case checked twice (once by direct parametrisation along every stated path, and once as a difference of potential values), and with both sides of Green's theorem computed independently rather than one being inferred from the other.

Green's Theorem

Assumptions and scope

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