##### Green's Identities
ems.ca.fe.g

George Green

See a short bio on wikipedia

[1793-1841] was a self-taught English mathematician whose academic life story did not follow the normal path: he was a miller! Despite having no formal education, in 1828, he published at his own expense what was to become a truly lasting contribution to science: An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, in which the results discussed here can be found. He was later admitted, at the young age of 40, as an undergraduate student in Cambridge where he remained until his early demise at age 47.

We can provide a very precise statement about uniqueness of solutions to Poisson's (or Laplace's) equation with some very basic considerations starting from the divergence theorem \[ \int_{\cal V} d\tau ~{\boldsymbol \nabla} \cdot {\bf F} = \oint_{\cal S} da ~{\bf F} \cdot {\bf n} \] Let \({\bf F} = \phi {\boldsymbol \nabla} \psi\), where \(\phi\) and \(\psi\) are scalar fields. We can then write \[ {\boldsymbol \nabla} \cdot (\phi {\boldsymbol \nabla} \psi) = \phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi \] and \[ \phi {\boldsymbol \nabla} \psi \cdot {\bf n} = \phi \frac{\partial \psi}{\partial n}. \] Substituting this in the divergence theorem gives Green's first identity

- J (1.34)

\[ \int_{\cal V} d\tau ~(\phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi) = \oint_{\cal S} da ~\phi \frac{\partial \psi}{\partial n}. \tag{Green1}\label{Green1} \] This first identity will prove crucial in the argument that follows. As an aside for now, for completeness, if we do the same thing again but with \(\phi\) and \(\psi\) interchanged, and subtract the result, we obtain another useful result known as Green's second identity or Green's theorem

- J (1.35)

\[ \int_{\cal V} d\tau (\phi {\boldsymbol \nabla}^2 \psi - \psi {\boldsymbol \nabla}^2 \phi) = \oint_{\cal S} da \left(\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right). \tag{Green2}\label{Green2} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42