Curl

This article is about curl in mathematics. You may want Curl programming language.

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In vector calculus, curl is a vector operator[?] that shows a vector field's tendency to rotate about a point. Common examples include:

• 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 was described with a vector field, and the lanes had different speed limits, the curl on the borders between lanes would be non-zero.

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, and is composed of [F_x, F_y, F_z].

Expanded, <math>\nabla \times F</math> is

<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.)

See also:

• Gradient
• Divergence