Curl

Given a 3-dimensional vector field F(r), the curl (also known as rotation) of F(r) is the differential vector operator nabla (symbol &nabla;)  applied to F. The application of &nabla; is in the form of a cross product:

\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r})\; \stackrel{\mathrm{def}}{=} \; \mathbf{e}_x \left(\frac{\partial F_y}{\partial z} - \frac{\partial F_z}{\partial y} \right) +\mathbf{e}_y \left(\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z}\right) +\mathbf{e}_z \left(\frac{\partial F_x}{\partial y} - \frac{\partial F_y}{\partial x}\right) , $$ where ex, ey, and ez are unit vectors along the axes of a Cartesian coordinate system of axes.

As any cross product the curl may be written in a few alternative ways.

As a determinant (evaluate along the first row):

\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) = \begin{vmatrix} \mathbf{e}_x & \mathbf{e}_y & \mathbf{e}_z \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y}& \frac{\partial }{\partial z} \\ F_x & F_y & F_z \end{vmatrix} $$ As a vector-matrix-vector product

\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) = \left(\mathbf{e}_x, \; \mathbf{e}_y,\; \mathbf{e}_z\right)\; \begin{pmatrix} 0& \frac{\partial }{\partial z} & -\frac{\partial }{\partial y} \\ -\frac{\partial }{\partial z}& 0& \frac{\partial }{\partial x} \\ \frac{\partial }{\partial y}& -\frac{\partial }{\partial x} &0 \\ \end{pmatrix} \begin{pmatrix} F_x \\ F_y \\ F_z \end{pmatrix} $$ In terms of the antisymmetric Levi-Civita symbol

\Big(\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) \Big)_\alpha =\sum_{\beta,\gamma=x,y,z} \epsilon_{\alpha\beta\gamma} \frac{\partial F_\gamma}{\partial \beta}, \qquad\alpha=x,y,z, $$ (the component of the curl along the Cartesian &alpha;-axis).