Rewriting Green's Second Idenity

Green's Second Theorem states

\[\int \int \int_V (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV = \int \int_S(\phi \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \phi ) \cdot \mathbf{n} dS \]

The slope of a function  
\[f\]
  in a direction  
\[\mathbf{v}\]
  where  
\[\mathbf{v}\]
  is a unit vector is  
\[(\mathbf{\nabla} f) \cdot \mathbf{v}= \frac{\partial f}{\partial \mathbf{v}}\]
.
We can write
\[(\phi \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \phi ) \cdot \mathbf{n} =\phi (\mathbf{\nabla} \psi ) \cdot \mathbf{n} - ( \psi \mathbf{\nabla} \phi ) \cdot \mathbf{n} = \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \]

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