Green's identities

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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.

[edit] First Green's identity

This identity derives from divergence theorem applied to the vector field $\mathbf{F}=\psi \nabla \varphi$ : If φ is twice continuously differentiable, and ψ is once continuously differentiable, on some region U, then

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

[edit] Second Green's identity

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

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

[edit] Third Green's identity

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

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