For a smooth $(n-1)$-form $\omega$ on a compact oriented $n$-manifold $M$ with boundary, $\int_{\partial M}\omega=\int_M d\omega$.Suggest a slogan
Let \(M\) be a compact oriented smooth \(n\)-manifold with boundary \(\partial M\), carrying the induced orientation, and let \(\omega\) be a smooth \((n-1)\)-form on \(M\). Then \[ \int_{\partial M}\omega=\int_M d\omega. \] In particular, let \(\Omega\subseteq\CC\) be a bounded open set whose boundary consists of finitely many piecewise smooth closed curves, oriented positively with respect to \(\Omega\), and let \(f\in C^1(\overline{\Omega})\). For \(\omega=f(z)\dz\) one has \(d\omega=\frac{\partial f}{\partial\overline z}\,d\overline z\wedge dz\) and \(d\overline z\wedge dz=2i\,dx\wedge dy\), so \[ \int_{\partial\Omega}f(z)\dz=2i\iint_\Omega\frac{\partial f}{\partial\overline z}\dA. \]