Like Green’s theorem but in a higher dimension. I.e. you can evaluate flux just by looking at the boundary.
Given an oriented piecewise-smooth surface \(S\) that is bounded by a simple, closed, piecewise-smooth, positively oriented boundary curve \(\partial S,\) and a vector field \(\bm{F}\) whose components have continuous partial derivatives on a open region in space containing \(S,\) \[ \int\limits_{\partial S} \bm{F}\cdot\mathrm{d}\bm{r} = \iint\limits_S \operatorname{curl}\bm{F} \cdot \mathrm{d}\bm{S} \,. \]