## Green's Second Identity for Harmonic Functions

Theorem
Let
$\phi (x,y,z), \: \psi (x,y,z)$
be harmonic functions with continuous first and second partial derivatives on a region
$R$
containing a region
$D$
with surface
$S$
.

Then
$\int \int_S \phi (\frac{\partial \psi}{\partial n} - \frac{\partial \psi}{\partial n} ) dS =0$

Proof
Green's Second Theorem can be written
$\int \int \int_V (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV = \int \int_S ( \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} ) dS$

Since
$\phi , \: \psi$
are both harmonic on
$R$
,
$\nabla^2 \phi = \nabla^2 \psi =0$

Hence
$\int \int_S \phi (\frac{\partial \psi}{\partial n} - \frac{\partial \psi}{\partial n} ) dS =0$