Green's identities

Green's identities are a set of three identities in vector calculus. They are named after the mathematician George Green, who discovered Green's theorem.

First Green identity

If φ is twice continuously differentiable, and ψ is once continuously differentiable, on some region U, then:

\int_U \left( \psi \nabla^2 \phi\right)\, dV = \oint_{\partial U} \left( \psi{\partial \phi \over \partial n}\right)\, dS - \int_U \left( \nabla \phi \cdot \nabla \psi\right)\, dV

Second Green identity

If φ and ψ are both twice continuously differentiable on U, then:

\int_U \left( \psi \nabla^2 \phi - \phi \nabla^2 \psi\right)\, dV = \oint_{\partial U} \left( \psi {\partial \phi \over \partial n} - \phi {\partial \psi \over \partial n}\right)\, dS

Third Green identity

If ψ is twice continuously differentiable on U

\oint_{\partial U} \left[ {1 \over |\mathbf{x} - \mathbf{y}|} {\partial \psi \over \partial n} (\mathbf{y}) - \psi(\mathbf{y}) {\partial \over \partial n_\mathbf{y}} {1 \over |\mathbf{x} - \mathbf{y}|}\right]\, dS_\mathbf{y} - \int_U \left[ {1 \over |\mathbf{x} - \mathbf{y}|} \nabla^2 \psi(\mathbf{y})\right]\, dV_\mathbf{y} = k
k = 4πψ(x) if x ∈ Int U, 2πψ(x) if x ∈ ∂U and has a tangent plane at x, and 0 elsewhere.

See also: Green's identities, Continuously differentiable, George Green, Green's theorem, Interior, Vector calculus