Green's First Idenity for Harmonic Functions

Theorem
If  

\[\phi\]
  is a harmonic function, on a volume  
\[V\]
  with surface  
\[S\]
  so that  
\[\nabla^2 \phi =0\]
  then Green's First Theorem,  
\[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S (\phi \nabla \psi ) \cdot \mathbf{n} dS \]
  becomes  
\[\int \int_S \frac{\partial \phi}{\partial n} dS=0\]
.
Proof
Put  
\[\phi =1\]
  in Green's First Theorem to get  
\[\int \int \int_V ( \nabla^2 \psi) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S \frac{\partial \psi}{\partial n} dS \]

Since  
\[\psi\]
  is harmonic  
\[\nabla^2 \psi =0\]
  so
\[\int \int_S \frac{\partial \phi}{\partial n} dS=0\]
If  
\[\psi = \phi \]
  the Green's Firt Theorem becomes
\[\int \int_V (\mathbf{\nabla \phi}) \cdot (\mathbf{\nabla \phi}) dV = \int \int_S \phi \frac{\partial \phi}{\partial n} dS \]

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