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

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

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

If ψ is twice continuously differentiable on U

\oint_{\partial U} {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}|}\, dS_\mathbf{y} - \int_U {1 \over |\mathbf{x} - \mathbf{y}|} \nabla^2 \psi(\mathbf{y})\, 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.

Word Browser