Green's Second Identity for Harmonic Functions

Theorem
Let  
\[\phi (x,y,z), \: \psi (x,y,z) \]
  be harmonic functions with continuous first and second partial derivatives on a region  
\[R\]
  containing a region  
\[D\]
  with surface  
\[S\]
.

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

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

Since  
\[\phi , \: \psi\]
  are both harmonic on  
\[R\]
,  
\[\nabla^2 \phi = \nabla^2 \psi =0\]

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

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