Green's First Identity
This identity is derived from the divergence theorem applied to the vector field : Let φ and ψ be scalar functions defined on some region U in R3, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Then
where is the Laplace operator, is the boundary of region U and n is the outward pointing unit normal of surface element dS. This theorem 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 :
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