\begin{definition}[Definition:Curl Operator/Determinant Form]
Let $R$ be a region of space embedded in Cartesian $3$ space $\R^3$.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf V$ be a vector field acting over $R$.
The '''curl''' of $\mathbf V$ at a point $A$ in $R$ is defined as:
:$\curl \mathbf V = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ V_x & V_y & V_z \end {vmatrix}$
where:
:$V_x$, $V_y$ and $V_z$ denote the magnitudes of the components at $A$ of $\mathbf V$ in the directions of the coordinate axes $x$, $y$ and $z$ respectively.
\end{definition}