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Definition 2024

Green 's theorem ‎(uncountable)

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