Green's theorem

Etymology
Named after the mathematician.

Noun

 * 1)  A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or
 * $$ \iint_R \left( {\partial Q \over \partial x} - {\partial P \over \partial y}\right) dx \, dy = \oint_{\partial R} P\, dx + Q\, dy $$.
 * 1)  Letting $$ \vec G = (P, Q) $$ be a vector field, and $$ d\vec l = (dx, dy) $$ this can be restated as
 * $$ \iint_R \nabla \wedge \vec G \ dx \, dy = \oint_{\partial R} \vec G \cdot d\vec l $$
 * where $$\wedge$$ is the wedge product, or equivalently, as
 * $$ \iint_R \nabla \cdot \vec G \ dx \, dy = \oint_{\partial R} \vec G \wedge d\vec l $$,
 * with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem.