> [!NOTE] Theorem (Stoke) > Let $\underline{F}:U \subset\mathbb{R}^{3}\to \mathbb{R}^{3}$ be a [[Vector Field on Subset of Real n-Space|vector field]]. Let $S$ be a [[Surface|surface]] with unit normal $\hat{\underline{n}}$ and boundary [[Curve|curve]] $\partial S$ oriented positively. Then the [[Net Circulation|net circulation]] through $S$ is given by $\oint_{\partial S} \underline{F} \cdot d\underline{r}=\int \int_{S} \nabla \times \underline{F} \cdot \hat{\underline{n}} \, dS $where $\nabla \times \underline{F}$ denotes the [[Curl of Vector Field on Real 3-Space|curl]] of $\underline{F}$ and $\oint \underline{F}\cdot d\underline{r}$ a [[Line Integral of Vector Field on Subset of Real n-Space|line integral]] of $\underline{F}.$ **Proof**: Follos from [[Divergence theorem]]. # Applications **Consequences**: [[Green's theorem|Green's theorem]], which asserts $\oint_{C} P \, dx + Q\, dy =\int \int_{D} \left( \frac{ \partial Q }{ \partial x } - \frac{ \partial P }{ \partial y } \right) \, dx \, dy$ where $C$ is a positively oriented curve in $\mathbb{R}^{2}$ and $D$ is the region bounded by $C,$ is a direct consequence of Stoke's.