Green;s Second Identity fro Ructions That Differentiate Twice to a Multiple of the Function

Theorem
If  
\[\phi (x,y,z), \: \psi (x,y,z) \]
  be harmonic functions with continuous first and second partial derivatives on a region  
\[R\]
  containing a region  
\[D\]
  with surface  
\[S\]
. If  
\[\phi (x,y,z), \: \psi (x,y,z) \]
  satisfy the equations  
\[\nabla^2 \phi = f \phi , \: \nabla^2 \psi = f \psi ,\]
  where  
\[f=f(x,y,z)\]
  in  
\[D\]

Then  
\[\int \int_S \phi (\frac{\partial \psi}{\partial n} - \frac{\partial \psi}{\partial n} ) dS =0 \]

Proof
Green's Second Idenity can be written  
\[\int \int_S ( \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} ) dS = \int \int \int_D (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV \]

Then
\[\int \int_S ( \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} ) dS = \int \int \int_D (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV = \int \int \int_D (\phi f \psi - \psi f \phi ) dV =0 \]