Given a closed region, the sum of "microscopic" circulation within the region $D$ is equal to the macroscopic circulation of the region as a whole $C$: $ \begin{gather} \int_C\boldsymbol{F}\cdot d\boldsymbol{s}=\iint_D(\nabla\times\boldsymbol{F})\cdot\hat{\boldsymbol{k}}\,dA\\ (\nabla\times\boldsymbol{F})\cdot\hat{\boldsymbol{k}}=\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\\ \int_C\boldsymbol{F}\cdot d\boldsymbol{s}=\iint_D\Big(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\Big)\,dA \end{gather} $