# 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

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

Non-orthogonal systems are more easily handled with tensor analysis or exterior calculus.

 $(\operatorname{curl}\vec{F})^{i}=\epsilon^{ijk}\nabla_{j}F_{k}$
 $\operatorname{curl}\vec{F}=*d(F_{1}dx^{1}+F_{2}dx^{2}+F_{3}dx^{3})$
 Title curl Canonical name Curl Date of creation 2013-03-22 12:47:39 Last modified on 2013-03-22 12:47:39 Owner rspuzio (6075) Last modified by rspuzio (6075) Numerical id 17 Author rspuzio (6075) Entry type Definition Classification msc 53-01 Synonym rotor Related topic IrrotationalField Related topic FirstOrderOperatorsInRiemannianGeometry Related topic AlternateCharacterizationOfCurl Related topic ExampleOfLaminarField Defines curl of a vector field