<abstract xmlns="http://www.w3.org/1999/xhtml">

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 <italic>ei</italic> is found the expressions of vector projections (Δ<bold>A</bold>)<italic>i</italic> and (<italic>rot</italic><bold>A</bold>)<italic>i</italic>, where <italic>A</italic> is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (Δ<bold>A</bold>)<italic>i</italic> and (<italic>rot</italic><bold>A</bold>)<italic>i</italic> for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (Δ<bold>A</bold>)<italic>i</italic> and (<italic>rot</italic><bold>A</bold>)<italic>i</italic> as an example are made in cases of spherical and cylindrical coordinate systems.
</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 (rotA)i, where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA)i and (rotA)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 (rotA)i as an example are made in cases of spherical and cylindrical coordinate systems.

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

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

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...

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

Published Online: May 14, 2021

Page range: 459 - 468

Received: Dec 04, 2020

Accepted: Apr 11, 2021

DOI: https://doi.org/10.2478/amns.2021.2.00002

Keywords
curvilinear basis, orthonormal basis, covariant derivative, Christoffel symbol

© 2021 S. O. Gladkov, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Published Online: May 14, 2021

Page range: 459 - 468

Received: Dec 04, 2020

Accepted: Apr 11, 2021

DOI: https://doi.org/10.2478/amns.2021.2.00002

Keywordscurvilinear basis, orthonormal basis, covariant derivative, Christoffel symbol

© 2021 S. O. Gladkov, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
curvilinear basis, orthonormal basis, covariant derivative, Christoffel symbol