Green’s Theorem

Green’s theorem. If U is an open subset of R2 and Γ is a closed curve such that Γ and Ω (the interior of Γ) both lie in U , then

∮

Γ M dx + N dy =

∫ ∫

( ∂N

∂x − ∂M

∂y

) dxdy

for any functions M and N with continuous first order partial derivatives on U .