Theorem>br> If
\[\phi, \: \psi\]
are twice differentiable and both functions defined on a volume \[V\]
with surface \[S\]
then \[\int \int \int_V (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV = \int \int_S(\phi \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \phi ) \cdot \mathbf{n} dS \]
This called Green's Second Theorem, or the Symmetrical Identity.
Proof
Green's First Theorem states
\[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S (\phi \nabla \psi ) \cdot \mathbf{n} dS \]
Interchange
\[\phi, \: \psi\]
to obtain\[\int \int \int_V ( \psi \nabla^2 \phi + (\mathbf{\nabla} \psi ) \cdot (\mathbf{\nabla} \phi ))dV = \int \int_S (\psi \nabla \phi ) \cdot \mathbf{n} dS \]
Subtracting the second of these from the first gives
\[\int \int \int_V (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV = \int \int_S(\phi \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \phi ) \cdot \mathbf{n} dS \]