Green's theorem

English

Etymology

Named after the mathematician Template:W.

Pronunciation

• Template:Audio

Noun

1. Template:Lb 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

   $\underset{R}{\iint }(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy={\oint }_{\partial R}Pdx+Qdy$.

2. Template:Lb Letting $\overrightarrow{G}=(P,Q)$ be a vector field, and $d\overrightarrow{l}=(dx,dy)$ this can be restated as

   $\underset{R}{\iint }\mathrm{\nabla }\wedge \overrightarrow{G}dxdy={\oint }_{\partial R}\overrightarrow{G}\cdot d\overrightarrow{l}$

   where $\wedge$ is the wedge product, or equivalently, as

   $\underset{R}{\iint }\mathrm{\nabla }\cdot \overrightarrow{G}dxdy={\oint }_{\partial R}\overrightarrow{G}\wedge d\overrightarrow{l}$,

   with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem.