Green's theorem

== English == === Etymology === Named after the mathematician George Green. === Pronunciation === === Proper noun === Green's theorem (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 ∬ R ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y = ∮ ∂ R P d x + Q d y {\displaystyle \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 G → = ( P , Q ) {\displaystyle {\vec {G}}=(P,Q)} be a vector field, and d l → = ( d x , d y ) {\displaystyle d{\vec {l}}=(dx,dy)} this can be restated as ∬ R ∇ ∧ G → d x d y = ∮ ∂ R G → ⋅ d l → {\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 ∬ R ∇ ⋅ G → d x d y = ∮ ∂ R G → ∧ d l → {\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. ==== Translations ====