\[\mathbf{F}=F_1 \mathbf{i} + F_2 \mathbf{j}\]
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 \[\oint_C (\mathbf{\nabla} f) \cdot \mathbf{n} \: ds =0\]