The divergence and curl in arbitrary basis

TL;DR

This paper develops a coordinate-free approach to divergence and curl operators using differential geometry, illustrating the method with various coordinate systems and providing computational tools for specific cases.

Contribution

It introduces a coordinate-free formulation of divergence and curl in differential geometry and applies it to multiple coordinate systems with computational implementations.

Findings

Graphical representation of eleven coordinate systems where Laplace is separable

Development of basis and connection for cylindrical and paraboloidal coordinates

Provision of code for spherical orthonormal basis calculations

Abstract

In this work, the divergence and curl operators are obtained using the coordinate free non rigid basis formulation of differential geometry. Although the authors have attempted to keep the presentation self-contained as much as possible, some previous exposure to the language of differential geometry may be helpful. In this sense the work is aimed to late undergraduate or beginners graduate students interested in mathematical physics. To illustrate the development, we graphically present the eleven coordinate systems in which the Laplace operator is separable. We detail the development of the basis and the connection for the cylindrical and paraboloidal coordinate systems. We also present in [1] codes both in Maxima and Maple for the spherical orthonormal basis, which serves as a working model for calculations in other situations of interest. Also in [1] the codes to obtain the coordinate surfaces are given.