curl

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

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 ${ℝ}^3$.

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

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

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

Curl is easily computed in an arbitrary orthogonal coordinate system by using the appropriate scale 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 with tensor 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