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\[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int \int_V \mathbf{\nabla} \cdot (\phi \nabla \psi ) dV = \int \int \int_V (\phi \nabla \psi ) \cdot \mathbf{n} dS \]