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# curl

The *curl* (also known as *rotor*) is a first order linear
differential operator which acts on vector fields in $\mathbb{R}^{{3}}$.

Intuitively, the curl of a vector field measures the extent to which a vector field differs from being the gradient of a scalar field. The name ”curl” comes from the fact that vector fields at a point with a non-zero curl can be seen as somehow ”swirling around” said point. A mathematically precise formulation of this notion can be obtained in the form of the definition of curl as limit of an integral about a closed circuit.

Let $F$ be a vector field in $\mathbb{R}^{{3}}$.

Pick an orthonormal basis $\{\vec{e_{{1}}},\vec{e_{{2}}},\vec{e_{{3}}}\}$ and write $\vec{F}=F^{{1}}\vec{e_{{1}}}+F^{{2}}\vec{e_{{2}}}+F^{{3}}\vec{e_{{3}}}$. Then the curl of $F$, notated $\operatorname{curl}\vec{F}$ or $\operatorname{rot}\vec{F}$ or $\vec{\nabla}\times\vec{F}$, is given as follows:

$\displaystyle\operatorname{curl}\vec{F}$ | $\displaystyle=$ | $\displaystyle\left[\frac{\partial F^{{3}}}{\partial q^{{2}}}-\frac{\partial F^% {{2}}}{\partial q^{{3}}}\right]\vec{e_{{1}}}+\left[\frac{\partial F^{{1}}}{% \partial q^{{3}}}-\frac{\partial F^{{3}}}{\partial q^{{1}}}\right]\vec{e_{{2}}}+$ | ||

$\displaystyle\left[\frac{\partial F^{{2}}}{\partial q^{{1}}}-\frac{\partial F^% {{1}}}{\partial q^{{2}}}\right]\vec{e_{{3}}}$ |

By applying the chain rule, one can verify that one obtains the same answer irregardless of choice of basis, hence curl is well-defined as a function of vector fields. Another way of coming to the same conclusion is to exhibit an expression for the curl of a vector field which does not require the choice of a basis. One such expression is as follows: Let $V$ be the volume of a closed surface $S$ enclosing the point $p$. Then one has

$\operatorname{curl}\vec{F}(p)=\lim_{{V\to 0}}\frac{1}{V}\int\!\!\int_{{S}}\vec% {n}\times\vec{F}dS$ |

Where $n$ is the outward unit normal vector to $S$.

Curl is easily computed in an arbitrary orthogonal coordinate system by using the appropriate scale factors. That is

$\displaystyle\operatorname{curl}\vec{F}$ | $\displaystyle=$ | $\displaystyle\frac{1}{h_{{3}}h_{{2}}}\left[\frac{\partial}{\partial q^{{2}}}% \left(h_{{3}}F^{{3}}\right)-\frac{\partial}{\partial q^{{3}}}\left(h_{{2}}F^{{% 2}}\right)\right]\vec{e_{{1}}}+\frac{1}{h_{{3}}h_{{1}}}\left[\frac{\partial}{% \partial q^{{3}}}\left(h_{{1}}F^{{1}}\right)-\frac{\partial}{\partial q^{{1}}}% \left(h_{{3}}F^{{3}}\right)\right]\vec{e_{{2}}}+$ | ||

$\displaystyle\frac{1}{h_{{1}}h_{{2}}}\left[\frac{\partial}{\partial q^{{1}}}% \left(h_{{2}}F^{{2}}\right)-\frac{\partial}{\partial q^{{2}}}\left(h_{{1}}F^{{% 1}}\right)\right]\vec{e_{{3}}}$ |

for the arbitrary orthogonal curvilinear coordinate system $(q^{{1}},q^{{2}},q^{{3}})$ having scale factors $(h_{{1}},h_{{2}},h_{{3}})$. Note the scale factors are given by

$h_{{i}}=\left(\frac{d}{dx_{{i}}}\right)\left(\frac{d}{dx_{{i}}}\right)\;\ni\;i% \in\{1,2,3\}.$ |

$(\operatorname{curl}\vec{F})^{i}=\epsilon^{{ijk}}\nabla_{j}F_{k}$ |

$\operatorname{curl}\vec{F}=*d(F_{1}dx^{1}+F_{2}dx^{2}+F_{3}dx^{3})$ |

## Mathematics Subject Classification

53-01*no label found*

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## Comments

## {curl}

About time curl was mentioned!