Green's theorem

Named after the mathematician George Green.

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Green's theorem

1. (calculus) 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

   ${\displaystyle \iint _{R}\left({\partial Q \over \partial x}-{\partial P \over \partial y}\right)dx\,dy=\oint _{\partial R}P\,dx+Q\,dy}$.

2. (calculus) Letting ${\displaystyle {\vec {G}}=(P,Q)}$ be a vector field, and ${\displaystyle d{\vec {l}}=(dx,dy)}$ this can be restated as

   ${\displaystyle \iint _{R}\nabla \wedge {\vec {G}}\ dx\,dy=\oint _{\partial R}{\vec {G}}\cdot d{\vec {l}}}$

   where ${\displaystyle \wedge }$ is the wedge product, or equivalently, as

   ${\displaystyle \iint _{R}\nabla \cdot {\vec {G}}\ dx\,dy=\oint _{\partial R}{\vec {G}}\wedge d{\vec {l}}}$,

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