## Proof That Harmonic Function Equal on a Closed Surface are Equal on the Enclosed Volume

Theorem
If
$\phi , \psi$
are harmonic functions on a region
$V$
with surface
$S$
and
$\phi = \psi$
on
$S$
then
$\phi = psi$
on
$V$
. Proof
If
$\phi , \psi$
are harmonic on
$V$
then
$\nabla^2 \phi = \nabla^2 \psi =0 \rightarrow \nabla^2 (\phi - \psi ) =0$
on
$V$
.
Hence
$\phi - \psi$
is harmonic on
$V$
and if
$\phi = \psi$
on
$S$
then
$\phi - \psi =0$
on
$S$
.
Apply thisTheorem to the function
$\phi - \psi$
and the theorem is proved.