Curl - Definition


Curl - Definition

This article is about curl in mathematics, see also Curl programming language and cURL, the Unix command line tool for transferring files.

In vector calculus, curl is a vector operator that shows a vector field's tendency to rotate about a point.

A vector field which has zero curl everywhere is called irrotational.

In mathematics the curl is noted by:

<math>\nabla \times F<math>

where <math>\nabla<math> is the vector differential operator del, and F is the vector field the curl is being applied to.

Expanded in Cartesian coordinates, <math>\nabla \times F<math> is, for F composed of [F_x, F_y, F_z]:

<math>\begin{pmatrix}

{\partial F_z / \partial y} - {\partial F_y / \partial z} \\ \\ {\partial F_x / \partial z} - {\partial F_z / \partial x}\\ \\ {\partial F_y / \partial x} - {\partial F_x / \partial y} \end{pmatrix}<math>

A simple way to remember the expanded form of the curl is to think of it as:

<math>\begin{pmatrix}

{\partial / \partial x} \\ \\ {\partial / \partial y} \\ \\ {\partial / \partial z} \end{pmatrix} \times F<math>

or as the determinant of the following matrix:

<math>\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\

{\partial / \partial x} & {\partial / \partial y} & {\partial / \partial z} \\

\\  F_x & F_y & F_z \end{pmatrix}<math>

where i, j, and k are the unit vectors for the x, y, and z axes, respectively.

In Einstein notation, with the Levi-Civita symbol it is written as:

<math>(\nabla \times F)_k = \epsilon_{klm} \partial_l F_m<math>

Using the exterior derivative, it is written simply as:

Note that taking the exterior derivative of a vector field does not result in another vector field, but a 2-form or bivector field, properly written as <math>P\,(dx \wedge dy) + Q\,(dy \wedge dz) + R\,(dx \wedge dz) <math>. However, since bivectors are generally considered less intuitive than ordinary vectors, the R³-dual is commonly used instead: this is a chiral operation, producing a pseudovector that takes on opposite values in left-handed and right-handed coordinate systems.

Examples

In a tornado the winds are rotating about the eye, and a vector field showing wind velocities would have a non-zero curl at the eye, and possibly elsewhere (see vorticity).
In a vector field that describes the linear velocities of each individual part of a rotating disk, the curl will have a constant value on all parts of the disk.
If a freeway were described with a vector field, and the lanes had different speed limits, the curl on the borders between lanes would be non-zero.
Faraday's law of induction, one of Maxwell's equations, can be expressed very simply using curl. It states that the curl of an electric field is equal to the opposite of the rate of change of the magnetic flux density.

Examples

See also