Green's Theorem and the Flux Out of a Surface

Suppose we have a force field  
\[\mathbf{F} =(f_1 , f_2 )\]
  acting on a surface  
 in the  
  plane bounded by a curve  
. We can parametrize the curve  
  in terms of the distance  
  along it, then a normal is  
\[\mathbf{n} = ( \frac{dy}{ds} , - \frac{dx}{ds})\]

\[\oint_C \mathbf{F} \cdot \mathbf{n} ds = \oint_C f_1 \: dy -f_2 \: dx= \oint_C f_1 \: dy + (-f_2 ) \: dx\]

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  
\[P=-f_2 , \: Q=f_1\]
\[\oint_C -f_2 \:dx + f_1 \: dy = \int \int_B ( \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} ) dx \: dy\]

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  
  out of the surface.

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