> [!NOTE] Definiton (Curl) > Let $\underline{F}:U\subseteq \mathbb{R}^{3} \to \mathbb{R}^{3}$ be a [[Vector Field on Subset of Real n-Space|vector field]]. The *curl* of $\underline{F}$ is, denoted $\nabla \times \underline{F}$ or $\text{curl }\underline{F}$ is defined by $\text{curl }\underline{F} = \begin{array}{|} \underline{i} & \underline{j} & \underline{k} \\ \frac{ \partial }{ \partial x } & \frac{ \partial }{ \partial y } & \frac{ \partial }{ \partial z } \\ \underline{F}_{x} & \underline{F}_{y} & \underline{F}_{z} \end{array} = \begin{pmatrix} \frac{ \partial \underline{F}_{z} }{ \partial y } - \frac{ \partial \underline{F}_{y} }{ \partial z } \\ \frac{ \partial \underline{F}_{x} }{ \partial z } - \frac{ \partial \underline{F}_{z} }{ \partial x } \\ \frac{ \partial F_{y} }{ \partial x } -\frac{ \partial F_{x} }{ \partial y } \end{pmatrix}$ # Properties Magnitude of Curl at a point is proportional to rotational speed. Curl at a point is normal to plane of rotation.