## Proof of Green's First Identity

Theorem
If

$\phi$
is differentiable and
$\psi$
is twice differentiable, both functions defined on a volume
$V$
with surface
$S$
then
$\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S (\phi \nabla \psi ) \cdot \mathbf{n} dS$

This is called Green's First Theorem. Proof
Apply the divergence Theorem with
$\mathbf{F} = \phi \mathbf{\nabla} \psi$
to give
$\int \int \int_VV \mathbf{\nabla} \cdot ( \phi \mathbf{\nabla} \psi)dV = \int \int_S (\phi \mathbf{\nabla} \psi ) \cdot \mathbf{n} dS$

Now use the identity
$\mathbf{\nabla} \cdot ( \phi \mathbf{\nabla} \psi)=phi \nabla^2 \psi +(\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi )$
. This gives
$\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int \int_V \mathbf{\nabla} \cdot (\phi \nabla \psi ) dV = \int \int \int_V (\phi \nabla \psi ) \cdot \mathbf{n} dS$