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ALTERNATIVE CALCULATION OF COVARIANT DERIVATIVES WITH AN APPLICATION TO THE PROBLEMS OF MECHANICS, PHYSICS AND GEOMETRY

Вестник московского государственного областного университета. Серия: Физика-математика. 2019;(1):16-45 DOI 10.18384/2310-7251-2019-1-16-45

 

Journal Homepage

Journal Title: Вестник московского государственного областного университета. Серия: Физика-математика

ISSN: 2072-8387 (Print); 2310-7251 (Online)

Publisher: Moscow Region State University Editorial Office

Society/Institution: Moscow Region State University

LCC Subject Category: Science: Mathematics

Country of publisher: Russian Federation

Language of fulltext: Russian

Full-text formats available: PDF

 

AUTHORS


Гладков Сергей Октябринович

EDITORIAL INFORMATION

Peer review

Editorial Board

Instructions for authors

Time From Submission to Publication: 13 weeks

 

Abstract | Full Text

Based on a simple mathematical approach proposed in the paper, we demonstrate a rigorous computation of the Christoffel symbols and the Riemann tensor that obviously have a regular geometric dimension, which is extremely important in solving a huge class of purely physical problems. As examples, we consider four orthogonal coordinate systems, two of which are spherical and cylindrical, i.e. standard for describing any course of tensor analysis, and the other two are parabolic and orthogonal two-dimensional coordinate systems, for which the Christoffel symbols, the Laplace operator, and Riemann and Ricci, whose all components automatically have the correct geometric dimensions, are calculated. A number of physical applications of the described mathematical formalism are demonstrated. An example of a nonorthogonal two-dimensional coordinate system is considered, with the help of which a detailed calculation of the Christoffel symbols of both kinds is given, and an expression is found for the Laplace operator with application to the problems of elasticity theory and hydrodynamics.