Green's identities

From Exampleproblems

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

This identity derives from divergence theorem applied to the vector field $\mathbf{F}=\nabla \phi$ : 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

Green's third identity derives from the second by the choice $\phi(.)={1 \over |\mathbf{x} - .|}$ and the observation $\nabla^2 \phi = - 4 \pi \delta \left( \mathbf{x} - . \right)$ in R³: If ψ is twice continuously differentiable on U

$\oint_{\partial U} \left[ {1 \over |\mathbf{x} - \mathbf{y}|} {\partial \psi \over \partial n} (\mathbf{y}) - \psi(\mathbf{y}) {\partial \over \partial n_\mathbf{y}} {1 \over |\mathbf{x} - \mathbf{y}|}\right]\, dS_\mathbf{y} - \int_U \left[ {1 \over |\mathbf{x} - \mathbf{y}|} \nabla^2 \psi(\mathbf{y})\right]\, dV_\mathbf{y} = k$