Vector field characteristics related to Aminov's divergent representations and conservation laws
A.G. Megrabov1,2
1Institute of Computational Mathematics and Mathematical Geophysics Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
2Novosibirsk State Technical University, Novosibirsk, Russia
Keywords: family of curves, curvature vector of a vector field and its vector potential, associated vector field, degree of nonholonomy, Aminov's divergent representations, conservation laws
Abstract
A number of new formulas are obtained for vector fields and vector analysis used in geometry and the vector field characteristics used in differential geometry: curvature vector, associated vector field, degree of nonholonomy, and the Laplacian value. Non-classical 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 Euler's hydrodynamic equations
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