On a group-theoretical approach to the curl operator

TL;DR

This paper introduces a group-theoretical matrix framework for the curl operator, extending its properties to complex domains and enabling new analytical insights into tensor calculus.

Contribution

It develops a novel matrix representation of the curl operator using group theory, applicable to tensors of any rank, and extends its analysis to complex numbers.

Findings

Constructed a self-adjoint matrix form of the curl operator

Extended the curl operator to complex plane

Preserved key properties of the traditional curl operator

Abstract

We utilize group-theoretical methods to develop a matrix representation of differential operators that act on tensors of any rank. In particular, we concentrate on the matrix formulation of the curl operator. A self-adjoint matrix of the curl operator is constructed and its action is extended to a complex plane. This scheme allows us to obtain properties, similar to those of the traditional curl operator.