Vector Field Characteristics Related to Aminov’s Divergent Representations and Conservation Laws

A number of new formulas are obtained for the vector field characteristics used in vector field geometry, vector analysis, and differential geometry: curvature vector, associated vector field, degree of nonholonomy, and Laplacian. Nonclassical characteristics such as the vector potential of the curvature vector field of a vector field and the sum of three curvature vectors of vector lines of the Frenet unit vector fields of a family of curves are also studied. It is shown that all the listed quantities are related to Aminov’s divergent representations for the Gaussian curvature or for the total curvature of the second kind. The obtained formulas can be considered as properties of the family of curves. Some formulas have divergence form, which makes it possible to derive differential conservation laws for the family of curves as well as for the eikonal equation and the Euler equations of hydrodynamics.