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Rewriting Green's First Identity

The slope of a function  

\[\phi\]
  in a direction  
\[\mathbf{v}\]
  where  
\[\mathbf{v}\]
  is a unit vector is  
\[\mathbf{\nabla} \cdot \mathbf{v}= \frac{\partial \phi}{\partial \mathbf{v}}\]
.
With this, we can rewrite Green's First Theorem
\[\int \int \int_V ( \phi \nabla^2 \psi + (\mathbf{\nabla} \phi ) \cdot (\mathbf{\nabla} \psi ))dV = \int \int_S (\phi \nabla \psi ) \cdot \mathbf{n} dS \]

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

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