Curl
- 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 [Fx, Fy, Fz]:
- <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 was described with a vector field, and the lanes had different speed limits, the curl on the borders between lanes would be non-zero.
See also
