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 ∇) applied to F. The application of ∇ 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 e_x, e_y, and e_z 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 α-axis).