Theorem
If
\[\phi\]
is a harmonic function, on a volume \[V\]
with surface \[S\]
so that \[\nabla^2 \phi =0\]
then Green's First Theorem, \[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S (\phi \nabla \psi ) \cdot \mathbf{n} dS \]
becomes \[\int \int_S \frac{\partial \phi}{\partial n} dS=0\]
.Proof
Put
\[\phi =1\]
in Green's First Theorem to get \[\int \int \int_V ( \nabla^2 \psi) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S \frac{\partial \psi}{\partial n} dS \]
Since
\[\psi\]
is harmonic \[\nabla^2 \psi =0\]
so\[\int \int_S \frac{\partial \phi}{\partial n} dS=0\]
If \[\psi = \phi \]
the Green's Firt Theorem becomes\[\int \int_V (\mathbf{\nabla \phi}) \cdot (\mathbf{\nabla \phi}) dV = \int \int_S \phi \frac{\partial \phi}{\partial n} dS \]