Proof of Green's Second Identity

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