Definition:Curl Operator/Cartesian 3-Space
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Definition
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 known as
The curl of a vector quantity is also known in some older works as its rotation, denoted $\operatorname {rot}$.
However, curl is now practically universal, being unambiguous and compact.
Also see
- Results about the curl operator 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 |
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): curl
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): curl