Editing Curl (mathematics) (section)

== Usage ==
In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl [[operator (mathematics)|operator]] can be applied using some set of [[curvilinear coordinates]], for which simpler representations have been derived.

The notation <math>\nabla\times\mathbf{F}</math> has its origins in the similarities to the 3-dimensional [[cross product]], and it is useful as a [[mnemonic]] in [[Cartesian coordinate system|Cartesian coordinates]] if <math>\nabla</math> is taken as a vector [[differential operator]] [[del]]. Such notation involving [[operator (physics)|operators]] is common in [[physics]] and [[algebra]].

Expanded in 3-dimensional [[Cartesian coordinate system|Cartesian coordinates]] (see ''[[Del in cylindrical and spherical coordinates]]'' for [[Spherical coordinate system|spherical]] and [[Cylindrical coordinate system|cylindrical]] coordinate representations), <math>\nabla\times\mathbf{F}</math> is, for <math>\mathbf{F}</math> composed of <math>[F_x,F_y,F_z]</math> (where the subscripts indicate the components of the vector, not partial derivatives):
<math display="block">
\nabla \times \mathbf{F} =
\begin{vmatrix} \boldsymbol{\hat\imath} & \boldsymbol{\hat\jmath} & \boldsymbol{\hat k} \\[5mu]
{\dfrac{\partial}{\partial x}} & {\dfrac{\partial}{\partial y}} & {\dfrac{\partial}{\partial z}} \\[5mu]
F_x & F_y & F_z \end{vmatrix}
</math>
where {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}} are the [[unit vector]]s for the {{math|''x''}}-, {{math|''y''}}-, and {{math|''z''}}-axes, respectively. This expands as follows:<ref>{{Cite book|last=Arfken|first=George Brown |title=Mathematical methods for physicists | date=2005|publisher=Elsevier| others=Weber, Hans-Jurgen | isbn=978-0-08-047069-6 | edition=6th | location=Boston | oclc=127114279 | page = 43}}</ref>
<math display="block">
\nabla \times \mathbf{F} =
\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \boldsymbol{\hat\imath} +
\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \boldsymbol{\hat\jmath} +
\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \boldsymbol{\hat k}
</math>

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

In a general coordinate system, the curl is given by<ref name="Mathworld">{{MathWorld |title=Curl |urlname=Curl}}</ref>
<math display="block">(\nabla \times \mathbf{F} )^k = \frac{1}{\sqrt{g}} \varepsilon^{k\ell m} \nabla_\ell F_m</math>
where {{mvar|ε}} denotes the [[Levi-Civita symbol#Levi-Civita tensors|Levi-Civita tensor]], {{math|∇}} the [[covariant derivative]], <math> g</math> is the [[determinant]] of the [[metric tensor]] and the [[Einstein summation convention]] implies that repeated indices are summed over. Due to the symmetry of the [[Christoffel symbols]] participating in the covariant derivative, this expression reduces to the [[partial derivative]]:
<math display="block">(\nabla \times \mathbf{F} ) = \frac{1}{\sqrt{g}} \mathbf{R}_k\varepsilon^{k\ell m} \partial_\ell F_m</math>
where {{math|'''R'''<sub>''k''</sub>}} are the local basis vectors. Equivalently, using the [[exterior derivative]], the curl can be expressed as:
<math display="block"> \nabla \times \mathbf{F} = \left( \star \big( {\mathrm d} \mathbf{F}^\flat \big) \right)^\sharp </math>

Here {{music|flat}} and {{music|sharp}} are the [[musical isomorphism]]s, and {{math|<small>★</small>}} is the [[Hodge star operator]]. This formula shows how to calculate the curl of {{math|'''F'''}} in any coordinate system, and how to extend the curl to any [[orientation (space)|oriented]] three-dimensional [[Riemannian metric|Riemannian]] manifold. Since this depends on a choice of orientation, curl is a [[Chirality (mathematics)|chiral]] operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.