Vector Differential Operators in arbitrary coordinates: a general approach

TL;DR

This paper introduces a general method for calculating vector differential operators in arbitrary coordinates, simplifying vector calculus in curvilinear systems and providing a coordinate-independent understanding.

Contribution

It develops a unified approach using covariant differentiation to handle vector operators in any coordinate system, extending beyond orthogonal coordinates.

Findings

Simplifies vector calculus in non-orthogonal coordinates

Provides a coordinate-independent formulation of differential operators

Enhances understanding of vector calculus in generalized coordinates

Abstract

We present a method for calculating the results of operation of differential operators operating on components of vector in generalized coordinates not restricted to orthogonal one. For this we use the relationships between covariant, contravariant and physical components of a vector and the idea of covariant differentiation. This not only simplifies vector calculus in common curvilinear coordinates, e.g., cylindrical or spherical polar, but also provides a deeper understanding of these operators in coordinate independent form.