# 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 $\{\overrightarrow{{e}_{1}},\overrightarrow{{e}_{2}},\overrightarrow{{e}_{3}}\}$ 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

$$\mathrm{curl}\overrightarrow{F}(p)=\underset{V\to 0}{lim}\frac{1}{V}\int {\int}_{S}\overrightarrow{n}\times \overrightarrow{F}\mathit{d}S$$ |

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

$\mathrm{curl}\overrightarrow{F}$ | $=$ | $\frac{1}{{h}_{3}{h}_{2}}}\left[{\displaystyle \frac{\partial}{\partial {q}^{2}}}\left({h}_{3}{F}^{3}\right)-{\displaystyle \frac{\partial}{\partial {q}^{3}}}\left({h}_{2}{F}^{2}\right)\right]\overrightarrow{{e}_{1}}+{\displaystyle \frac{1}{{h}_{3}{h}_{1}}}\left[{\displaystyle \frac{\partial}{\partial {q}^{3}}}\left({h}_{1}{F}^{1}\right)-{\displaystyle \frac{\partial}{\partial {q}^{1}}}\left({h}_{3}{F}^{3}\right)\right]\overrightarrow{{e}_{2}}+$ | ||

$\frac{1}{{h}_{1}{h}_{2}}}\left[{\displaystyle \frac{\partial}{\partial {q}^{1}}}\left({h}_{2}{F}^{2}\right)-{\displaystyle \frac{\partial}{\partial {q}^{2}}}\left({h}_{1}{F}^{1}\right)\right]\overrightarrow{{e}_{3}$ |

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.

$${(\mathrm{curl}\overrightarrow{F})}^{i}={\u03f5}^{ijk}{\nabla}_{j}{F}_{k}$$ |

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

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 |