Definition:Curl Operator

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Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$.

The curl of $\mathbf f$ is defined as:

\(\ds \curl \mathbf f\) \(:=\) \(\ds \nabla \times \mathbf f\) where $\nabla$ denotes the del operator
\(\ds \) \(=\) \(\ds \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } \times \paren {f_x \mathbf i + f_y \mathbf j + f_z \mathbf k}\) Definition of Del Operator
\(\ds \) \(=\) \(\ds \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ f_x & f_y & f_z \end{vmatrix}\) Definition of Vector Cross Product
\(\ds \) \(=\) \(\ds \paren {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } \mathbf k\)

Thus the curl is a vector in $\R^3$.

Also see

  • Results about curl can be found here.

Historical Note

During the course of development of vector analysis, various notations for the curl operator were introduced, as follows:

Symbol Used by
$\nabla \times$ or $\curl$ Josiah Willard Gibbs and Edwin Bidwell Wilson
$\curl$ Oliver Heaviside
Max Abraham
$\operatorname {rot}$ Vladimir Sergeyevitch Ignatowski
Hendrik Antoon Lorentz
Cesare Burali-Forti and Roberto Marcolongo