structural:summarization
Usage

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

The notation ∇ × F has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra.

Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ∇ × F is, for F composed of (where the subscripts indicate the components of the vector, not partial derivatives): ∇ × F = | ı ^ ȷ ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | {\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}{\boldsymbol {\hat {\imath }}}&{\boldsymbol {\hat {\jmath }}}&{\boldsymbol {\hat {k}}}\\{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}} where i, j, and k are the unit vectors for the x -, y -, and z -axes, respectively. This expands as follows: ∇ × F = (∂ F z ∂ y − ∂ F y ∂ z) ı ^ − (∂ F z ∂ x − ∂ F x ∂ z) ȷ ^ + (∂ F y ∂ x − ∂ F x ∂ y) k ^ {\displaystyle \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_{z}}{\partial x}}-{\frac {\partial F_{x}}{\partial z}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}}

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 (∇ × F) k = 1 g ε k ℓ m ∇ ℓ F m {\displaystyle (\nabla \times \mathbf {F})^{k}={\frac {1}{\sqrt {g}}}\varepsilon ^{k\ell m}\nabla _{\ell }F_{m}} where ε denotes the Levi-Civita tensor, ∇ the covariant derivative, g {\displaystyle g} 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: (∇ × F) = 1 g R k ε k ℓ m ∂ ℓ F m {\displaystyle (\nabla \times \mathbf {F})={\frac {1}{\sqrt {g}}}\mathbf {R} _{k}\varepsilon ^{k\ell m}\partial _{\ell }F_{m}} where R are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: ∇ × F = (⋆ (d F ♭)) ♯ {\displaystyle \nabla \times \mathbf {F} =\left(\star {\big (}{\mathrm {d} }\mathbf {F} ^{\flat }{\big)}\right)^{\sharp }}

Here ♭ and ♯ are the musical isomorphisms, and ★ is the Hodge star operator. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.