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.
First Green identity
If φ is twice continuously differentiable, and ψ is once continuously differentiable, on some region U, then:
- <math>\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<math>
Second Green identity
If φ and ψ are both twice continuously differentiable on U, then:
- <math> \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 <math>
Third Green identity
If ψ is twice continuously differentiable on U
- <math> \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<math>