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