Definition:Curl Operator

Definition
Let $\R^3 \left({x, y, z}\right)$ denote the real Cartesian space of $3$ dimensions..

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

Let $\mathbf f \left({f_x, f_y, f_z}\right): \R^3 \to \R^3$ be a vector-valued function on $\R^3$.

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

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