curl

A vector measuring whether a field is spinning around a point, pointing along the axis of fastest rotation with length equal to the spin rate.

For $\mathbf{F}=(P,Q,R)$ , $\operatorname{curl}\mathbf{F} = \left(R_y - Q_z\right)\mathbf{i} + \left(P_z - R_x\right)\mathbf{j} + \left(Q_x - P_y\right)\mathbf{k}$ , computed as a determinant. Restricting to a field with no $z$ -dependence and $R\equiv0$ collapses this to $\operatorname{curl}\mathbf{F} = (Q_x-P_y)\mathbf{k}$ — the old 2D scalar curl, correctly understood as a vector pointing out of the plane. $\operatorname{curl}\mathbf{F}=\mathbf{0}$ at a point means the field is irrotational there.

Examples

For $\mathbf{F}(x,y,z) = (yz,\, 0,\, 0)$ : $\operatorname{curl}\mathbf{F} = (0,\, y,\, -z)$ .
For $\mathbf{F}=(-y,x,0)$ : $\operatorname{curl}\mathbf{F}=(0,0,2)$ , a constant swirl around the $z$ -axis.