On the Moisil-Theodoresco operator in orthogonal curvilinear coordinates

TL;DR

This paper explores the Moisil-Theodoresco operator in orthogonal curvilinear coordinates, providing new insights and defining a quaternionic Laplace operator that unifies scalar and vector Laplacians.

Contribution

It extends the understanding of the Moisil-Theodoresco operator to orthogonal curvilinear coordinates and introduces a quaternionic Laplace operator.

Findings

Clarifies the action of the Moisil-Theodoresco operator in curvilinear coordinates

Defines a quaternionic Laplace operator that encompasses scalar and vector Laplacians

Provides theoretical foundations for further applications in quaternionic analysis

Abstract

It is generally well understood the legitimate action of the Moisil-Theo\-do\-res\-co ope\-ra\-tor, over a quaternionic valued function defined on $\mathbb{R}^3$ (sum of a scalar and a vector field) in Cartesian coordinates, but it does not so in any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subject. Moreover, we introduce a notion of quaternionic Laplace operator acting on a quaternionic valued function from which one can recover both scalar and vector Laplacians in vector analysis context.