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Proof That Harmonic Function Equal on a Closed Surface are Equal on the Enclosed Volume

Theorem
If  
\[\phi , \psi\]
  are harmonic functions on a region  
\[V\]
  with surface  
\[S\]
  and  
\[\phi = \psi\]
  on  
\[S\]
  then  
\[\phi = psi\]
  on  
\[V\]
. Proof
If  
\[\phi , \psi\]
  are harmonic on  
\[V\]
  then  
\[\nabla^2 \phi = \nabla^2 \psi =0 \rightarrow \nabla^2 (\phi - \psi ) =0\]
  on  
\[V\]
.
Hence  
\[\phi - \psi\]
  is harmonic on  
\[V\]
  and if  
\[\phi = \psi \]
  on  
\[S\]
  then  
\[\phi - \psi =0\]
  on  
\[S\]
.
Apply thisTheorem to the function  
\[\phi - \psi\]
  and the theorem is proved.

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