On Some Formulas for Families of Curves and Surfaces and Aminov’s Divergent Representations
- Авторы: Megrabov A.1,2
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Учреждения:
- Institute of Computational Mathematics and Mathematical Geophysics
- Novosibirsk State Technical University
- Выпуск: Том 39, № 1 (2018)
- Страницы: 114-120
- Раздел: Article
- URL: https://journals.rcsi.science/1995-0802/article/view/201055
- DOI: https://doi.org/10.1134/S1995080218010201
- ID: 201055
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Аннотация
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 Lτ 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 Sp* = 0 (which is valid for a family of plane curves Lτ) are obtained, where Sp* is the sum of the curvature vectors of the plane curves Lτ and their orthogonal curves Lν. 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 Sτ with the normal τ; 2) the non-holonomicity values of the fields of the principal directions l1 and l2 are equal.
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Об авторах
A. Megrabov
Institute of Computational Mathematics and Mathematical Geophysics; Novosibirsk State Technical University
Автор, ответственный за переписку.
Email: mag@sscc.ru
Россия, pr. Akademika Lavrentieva 6, Novosibirsk, 630090; pr. Karla Marksa 20, Novosibirsk, 630073