curl

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

Intuitively, the curl of a vector field measures the extent to which avector field differs from being the gradient of a scalar field. Thename ”curl” comes from the fact that vector fields at a point with anon-zero curl can be seen as somehow ”swirling around” said point. Amathematically precise formulation of this notion can be obtained inthe form of the definition of curl as limit of an integral about aclosed circuit.

Let $F$ be a vector field in ${ℝ}^3$.

Pick an orthonormal basis $\{\overrightarrow{e_1},\overrightarrow{e_2},\overrightarrow{e_3}\}$and write$\overrightarrow{F}=F^1\overrightarrow{e_1}+F^2\overrightarrow{e_2}+F^3\overrightarrow{e_3}$. Thenthe curl of $F$, notated $\mathrm{curl}\overrightarrow{F}$ or$\mathrm{rot}\overrightarrow{F}$ or $\overrightarrow{\nabla }\times \overrightarrow{F}$, is givenas follows:

$\mathrm{curl}\overrightarrow{F}$$=$$\left[{\displaystyle \frac{\partial F^3}{\partial q^2}}-{\displaystyle \frac{\partial F^2}{\partial q^3}}\right]\overrightarrow{e_1}+\left[{\displaystyle \frac{\partial F^1}{\partial q^3}}-{\displaystyle \frac{\partial F^3}{\partial q^1}}\right]\overrightarrow{e_2}+$

$\left[{\displaystyle \frac{\partial F^2}{\partial q^1}}-{\displaystyle \frac{\partial F^1}{\partial q^2}}\right]\overrightarrow{e_3}$

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

$$\mathrm{curl}\overrightarrow{F}(p)=\underset{V\to 0}{lim}\frac{1}{V}\int {\int }_S\overrightarrow{n}\times \overrightarrow{F}𝑑S$$

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

Curl is easily computed in anarbitrary orthogonal coordinate system by using the appropriatescale factors. That is

$\mathrm{curl}\overrightarrow{F}$$=$${\displaystyle \frac{1}{h_3{h}_2}}\left[{\displaystyle \frac{\partial }{\partial q^2}}\left(h_3{F}^3\right)-{\displaystyle \frac{\partial }{\partial q^3}}\left(h_2{F}^2\right)\right]\overrightarrow{e_1}+{\displaystyle \frac{1}{h_3{h}_1}}\left[{\displaystyle \frac{\partial }{\partial q^3}}\left(h_1{F}^1\right)-{\displaystyle \frac{\partial }{\partial q^1}}\left(h_3{F}^3\right)\right]\overrightarrow{e_2}+$

${\displaystyle \frac{1}{h_1{h}_2}}\left[{\displaystyle \frac{\partial }{\partial q^1}}\left(h_2{F}^2\right)-{\displaystyle \frac{\partial }{\partial q^2}}\left(h_1{F}^1\right)\right]\overrightarrow{e_3}$

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

$$h_i=\left(\frac{d}{d{x}_i}\right)\left(\frac{d}{d{x}_i}\right)\ni i\in \{1,2,3\}.$$

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

$${(\mathrm{curl}\overrightarrow{F})}^i={ϵ}^{ijk}{\nabla }_j{F}_k$$

$$\mathrm{curl}\overrightarrow{F}=*d(F_{1}d{x}^1+F_{2}d{x}^2+F_{3}d{x}^3)$$

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