Directional Energy Functionals Through Anholonomic Coordinates

Abstract

In this paper, a special case of directional energy functional is investigated by computing the directional energy and pseudo-angle of unit vector fields in the ordinary three-dimensional space. This approach is also extended simultaneously to define the critical points of the directional energy functionals of the velocity fields. Then, the restriction of the harmonic maps and the extrema of the directional energy functionals is considered. Finally, we compute directional harmonic and biharmonic equations of the curvature vector fields to generalize total bending or energy of vector fields.