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