# Definition Let $F\in\mathbb{R}^3$ be a vectorial function. Let $S$ be a flat surface with boundary curve $C$ and normal direction $n$. Denote by $|S|$ the area of the surface The $\text{curl}$ of $F$ along $n$ is defined as $ (\nabla\times F)(x)\cdot n = \lim_{S\rightarrow 0} \frac{1}{|S|}\int_{C}F\cdot dr $ # Cartesian coordinates In cartesian coordinates the $\text{curl}$ is given as $ \nabla\times F = (\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z})\hat{\mathbf{i}} + (\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x})\hat{\mathbf{j}} + (\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y})\hat{\mathbf{z}}. $ # Related Items [[Stokes Theorem]] [[Divergence]]