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

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

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