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Concept
Green's identities
Summary
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 ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X ) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ Rd, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Using the product rule above, but letting X = ∇φ, integrate ∇⋅(ψ∇φ) over U. Then
where ∆ ≡ ∇2 is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal to the surface element dS and dS = ndS 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 = ψΓ,
If φ and ψ are both twice continuously differentiable on U ⊂ R3, and ε is once continuously differentiable, one may choose F = ψε ∇φ − φε ∇ψ to obtain
For the special case of ε = 1 all across U ⊂ R3, then,
In the equation above, ∂φ/∂n is the directional derivative of φ in the direction of the outward pointing surface normal n of the surface element dS,
Explicitly incorporating this definition in the Green's second identity with ε = 1 results in
In particular, this demonstrates that the Laplacian is a self-adjoint operator in the L2 inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero.
Green's third identity derives from the second identity by choosing φ = G, where the Green's function G is taken to be a fundamental solution of the Laplace operator, ∆.
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