Proof of Green's First Identity

Theorem
If  

\[\phi\]
  is differentiable and  
\[\psi\]
  is twice differentiable, both functions defined on a volume 
\[V\]
  with surface  
\[S\]
  then  
\[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S (\phi \nabla \psi ) \cdot \mathbf{n} dS \]

This is called Green's First Theorem. Proof
Apply the divergence Theorem with  
\[\mathbf{F} = \phi \mathbf{\nabla} \psi\]
  to give
\[\int \int \int_VV \mathbf{\nabla} \cdot ( \phi \mathbf{\nabla} \psi)dV = \int \int_S (\phi \mathbf{\nabla} \psi ) \cdot \mathbf{n} dS\]

Now use the identity  
\[\mathbf{\nabla} \cdot ( \phi \mathbf{\nabla} \psi)=phi \nabla^2 \psi +(\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi )\]
. This gives
\[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int \int_V \mathbf{\nabla} \cdot (\phi \nabla \psi ) dV = \int \int \int_V (\phi \nabla \psi ) \cdot \mathbf{n} dS \]