## Gauss's Theorem for a Function in the Plane

If
$\mathbf{F}=F_1 \mathbf{i} + F_2 \mathbf{j}$
is a vector field then
$div \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}$

Gauss's Theorem says
$\oint_C \mathbf{F} \cdot \mathbf{n} \: ds = \int \int_B div \mathbf{F} \: dx \: dy$

If
$f$
is a function such that
$grad f = \mathbf{\nabla} f = \mathbf{F} = F_1 \mathbf{i} + F_2 \mathbf{j}$
then
$div \mathbf{F} = \frac{\partial^2 f}{\partial x^2}+ \frac{\partial^2 f}{\partial y^2}= \nabla^2 f$

Hence
$\oint_C (\mathbf{\nabla} f) \cdot \mathbf{n} \: ds = \int \int_B \nabla^2 f \: dx \: dy$
.
If
$f$
is harmonic so that
$\nabla^2 f=0$
then
$\oint_C (\mathbf{\nabla} f) \cdot \mathbf{n} \: ds =0$