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