About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

S. O. Gladkov

doi:10.2478/amns.2021.2.00002

Abstract

Abstract A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho ei is found the expressions of vector projections (ΔA) i and (rot A) i , where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA) i and (rot A) i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA) i and (rot A) i as an example are made in cases of spherical and cylindrical coordinate systems.

Highlights

Our great interest in calculating the result of the action of the Laplace operator and the curl on a contravariant vector function arose mainly after reading the monographs [1, 2]
There, the results of covariant differentiation of a contravariant vector in spherical and cylindrical coordinate systems are presented in a purely formal way, with no references to the primary sources where they could be considered in detail
The question is how to calculate the results of the action of the Laplace operator and curl from a vector function in arbitrary coordinates; for example, in parabolic or elliptic ones

Summary

Introduction

Our great interest in calculating the result of the action of the Laplace operator and the curl on a contravariant vector function arose mainly after reading the monographs [1, 2]. There, the results of covariant differentiation of a contravariant vector in spherical and cylindrical coordinate systems are presented in a purely formal way, with no references to the primary sources where they could be considered in detail This raises a quite natural question, the answer to which we have not found in the monographs mentioned above. The question is how to calculate the results of the action of the Laplace operator and curl from a vector function in arbitrary (but orthogonal) coordinates; for example, in parabolic or elliptic ones Perhaps, such a question, according to the authors of classical works [1, 2], is so simple that they did not even consider it necessary to indicate a link to the source where the answer could be found. To show examples that prove the correctness of the common approach described below, we demonstrate the corresponding calculations in the cases of spherical and cylindrical coordinate systems with results that automatically coincide with the commonly accepted ones

Laplace operator

Vector function curl

Conclusion