Circulation around the boundary equals total curl inside the region.Suggest a slogan
Let \(\Omega\subseteq\CC\) be a bounded open set whose boundary \(\partial\Omega\) consists of finitely many piecewise smooth closed curves, oriented positively with respect to \(\Omega\), and let \(f,g\in C^1(\overline{\Omega})\) be real-valued. Then \[ \int_{\partial\Omega}f\,dx+g\,dy=\iint_{\Omega}\Bigl(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\Bigr)\dA. \] Equivalently, for the vector field \(F\coloneqq(f,g)\) on \(\overline{\Omega}\), \[ \int_{\partial\Omega}F\cdot d\mathbf r=\iint_{\Omega}\curl F\dA, \] where \(\curl F\coloneqq\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\).