\begin{definition}[Definition:Curl Operator/Cartesian 3-Space]
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:
{{begin-eqn}}
{{eqn | l = \curl \mathbf f
      | o = :=
      | r = \nabla \times \mathbf f
      | c = where $\nabla$ denotes the del operator
}}
{{eqn | r = \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}
      | c = {{Defof|Del Operator}}
}}
{{eqn | r = \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}
      | c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = \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
      | c = 
}}
{{end-eqn}}
Thus the '''curl''' is a vector in $\R^3$.
\end{definition}