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 ε_αβγ

${\Big (}{\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} ){\Big )}_{\alpha }=\sum _{\beta ,\gamma =x,y,z}\epsilon _{\alpha \beta \gamma }{\frac {\partial F_{\beta }}{\partial \gamma }},\qquad \alpha =x,y,z,$

(the component of the curl along the Cartesian α-axis).

Two important applications of the curl are (i) in Maxwell equations for electromagnetic fields and (ii) in the Helmholtz decomposition of arbitary vector fields.

From the Helmholtz decomposition follows that any curl-free vector field (also known as irrotational field) F(r), i.e., a vector field for which

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )=\mathbf {0}$

can be written as minus the gradient of a scalar potential Φ

$\mathbf {F} (\mathbf {r} )=-{\boldsymbol {\nabla }}\Phi (\mathbf {r} ).$

Orthogonal curvilinear coordinate systems

In a general 3-dimensional orthogonal curvilinear coordinate system u₁, u₂, and u₃, characterized by the scale factors h₁, h₂, and h₃, (also known as Lamé factors, the diagonal elements of the diagonal g-tensor) the curl takes the form of the following determinant (evaluate along the first row):

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )={\frac {1}{h_{1}h_{2}h_{3}}}{\begin{vmatrix}h_{1}\mathbf {e} _{1}&h_{2}\mathbf {e} _{2}&h_{3}\mathbf {e} _{3}\\{\frac {\partial }{\partial u_{1}}}&{\frac {\partial }{\partial u_{2}}}&{\frac {\partial }{\partial u_{3}}}\\h_{1}F_{1}&h_{2}F_{2}&h_{3}F_{3}\end{vmatrix}}$

For instance, in the case of spherical polar coordinates r, θ, and φ

$h_{r}=1,\qquad h_{\theta }=r,\qquad h_{\phi }=r\sin \theta$

the curl is

$\nabla \times \mathbf {F} ={\frac {1}{r^{2}\sin \theta }}{\begin{vmatrix}\mathbf {e} _{r}&r\mathbf {e} _{\theta }&r\sin \theta \mathbf {e} _{\phi }\\{\frac {\partial }{\partial r}}&{\frac {\partial }{\partial \theta }}&{\frac {\partial }{\partial \phi }}\\F_{r}&rF_{\theta }&r\sin \theta F_{\phi }\\\end{vmatrix}},$