Green's identities

In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem.

This identity is derived from the divergence theorem applied to the vector field $F = ψ \nabla φ$ : Let $φ$ and $ψ$ be scalar functions defined on some region $U \subset R d$ , and suppose that $φ$ is twice continuously differentiable, and $ψ$ is once continuously differentiable. Then

where $∆$ is the Laplace operator, $\partial U$ is the boundary of region $U$ , $n$ is the outward pointing unit normal of surface element $dS$ and $d$ $S$ is the oriented surface element.

This theorem is a special case of the divergence theorem, and is essentially the higher dimensional equivalent of integration by parts with $ψ$ and the gradient of $φ$ replacing $u$ and $v$ .

Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting $F = ψ Γ$ ,

If $φ$ and $ψ$ are both twice continuously differentiable on $U \subset R 3$ , and $ε$ is once continuously differentiable, one may choose $F = ψε \nabla φ - φε \nabla ψ$ to obtain

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