On the physical interpretation of some types of three-dimensional harmonic mappings

TL;DR

This paper develops a theory of 3D harmonic mappings, introduces new classes, and applies them to electrostatics, showing they conserve electric charge and can solve electrostatic problems.

Contribution

It introduces new classes of 3D harmonic mappings with physical interpretation and demonstrates their application to electrostatics problems.

Findings

Mappings conserve electric charge locally.

Mappings can be used to solve electrostatic problems.

The theory is validated with known electrostatic cases.

Abstract

The development of the theory of three-dimensional harmonic mappings is considered. The new classes of mappings that generate three-dimensional harmonic functions are introduced. The physical interpretation of these mappings is applied to electrostatics problems. It is found that these mappings locally conserve electric charge of the equipotential surfaces. To confirm the correctness of the theory it is shown that by using the proposed mappings the electric field in two known electrostatic problems can be found.