Show that if $\displaystyle \sigma$(x,y) satisfiesLaplace's equation,$\displaystyle \frac{\partial \sigma}{\partial x^2}+\frac {\partial \sigma}{\partial y^2}=0 $ on a simply connected region R, then $\displaystyle \forall$closed curves C of R, we have

$\displaystyle \int_c (\sigma_y dx - \sigma_x dy) =0$