\[\mathbf{F} =(f_1 , f_2 )\]
acting on a surface \[B\]
in the \[xy\]
plane bounded by a curve \[C\]
.
We can parametrize the curve \[C\]
in terms of the distance \[s\]
along it, then a normal is \[\mathbf{n} = ( \frac{dy}{ds} , - \frac{dx}{ds})\]
Then
\[\oint_C \mathbf{F} \cdot \mathbf{n} ds = \oint_C f_1 \: dy -f_2 \: dx= \oint_C f_1 \: dy + (-f_2 ) \: dx\]
According to Green's Theorem
\[\oint_C P \:dx + Q \: dy = \int \int_R ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) dx \: dy\]
.
Put \[P=-f_2 , \: Q=f_1\]
then\[\oint_C -f_2 \:dx + f_1 \: dy = \int \int_B ( \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} ) dx \: dy\]
which we can write
\[\oint \mathbf{F} \times d \mathbf{r} = \int \int div \cdot \mathbf{F} \: dx \: dy \]
The hand hand side is the flux of
\[\mathbf{F}\]
out of the surface.