Dictionary:Green’s theorem

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A form of Gauss's (divergence) theorem relating a volume integral to surface integrals.

If F and G are two scalar functions, then

${\displaystyle \iiint \left(F\nabla ^{2}G-G\nabla ^{2}F\right)dv=\iint \left(F\nabla G-G\nabla F\right)\cdot d\mathbf {s} }$,

where dv is a volume element and ${\displaystyle d\mathbf {s} }$ is a directed surface element, represented by ${\displaystyle \mathbf {\hat {n}} dS}$. Here ${\displaystyle \mathbf {\hat {n}} }$ is the inward directed normal vector to the surface.

Named for George Green (1793-1841), English mathematician.