Lobachevskii Journal of Mathematics

, Volume 39, Issue 1, pp 114–120 | Cite as

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



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.

Keywords and phrases

vector field total curvature family of curves family of surfaces conservation laws 


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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical GeophysicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State Technical UniversityNovosibirskRussia

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