On Some Formulas for Families of Curves and Surfaces and Aminov’s Divergent Representations

Abstract

A unit vector field τ in the Euclidean space E³ is considered. Let P be the vector field from the first Aminov’s divergent representation K = div{(R · τ)P} for the total curvature of the second kind K of the field τ. For the field P, an invariant representation of the form P = −rotR* is obtained, where the field R* is expressed in terms of the Frenet basis (τ, ν, β) and the first curvature k and the second curvature κ of the streamlines L_τ of the field τ. Formulas relating the quantities K (or P), κ, τ, ν, and β are derived. Three-dimensional analogs of the conservation law div S_p* = 0 (which is valid for a family of plane curves L_τ) are obtained, where S_p* is the sum of the curvature vectors of the plane curves L_τ and their orthogonal curves L_ν. It is shown that if the field τ is holonomic: 1) the vector field S(τ) from the second Aminov’s divergent representation K = −1/2 div S(τ) can be interpreted as the sum of three curvature vectors of three curves related to surfaces S_τ with the normal τ; 2) the non-holonomicity values of the fields of the principal directions l₁ and l₂ are equal.