Curl

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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.

Note that the result of the curl operator acting on a vector field is not really a vector, it is a pseudovector. This means that it takes on opposite values in left-handed and right-handed coordinate systems (see Cartesian coordinate system). (Conversely, the curl of a pseudovector is a vector.)

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 magnetic flux density.

Curl

Examples

See also