On the construction of unit longitudinal-vortex vector fields with the use of smooth mappings

A solution is given for the problem of constructing a unit vector field collinear to the field of its curl. The solution is based on the use of a suitably parametrized orthogonal transformation of a unit vector field that is potential in $\mathbb R^3$. The result is stated in the theorem that contains the recipe for constructing the required field.