The case of the curious curl

Some important, though rarely discussed, aspects of the geometry of the curl of a vector field are investigated. In particular, the direction of the vector ∇ × A with respect to A itself is considered. Ordinarily, the vector A and its curl are neither perpendicular nor parallel. For vector fields with real components, it is shown in general that ∇ × A is orthogonal to A only on a specific surface, while ∇ × A and A are parallel (or antiparallel) only at points lying on a specific curve. Specific examples of vector fields wherein the field vector and its curl are everywhere orthogonal are frequently encountered in physics and engineering. Conditions under which this may occur are investigated. It is shown that when the field direction coincides everywhere with that of any of the unit vectors of a general system of orthogonal coordinates, the field vector and its curl will be everywhere orthogonal. The question of whether there exist vector fields wherein A and ∇ × A are everywhere parallel or antiparallel is also considered. It is shown that such fields can be constructed, and some physical examples are exhibited and discussed.