Potential Vector Fields in $\mathbb R^3$ and $α$-Meridional Mappings of the Second Kind $(α\in \mathbb R)$

Abstract

This paper extends author's approach developed in a recent paper on analytic models of potential fields in inhomogeneous isotropic media. The properties of different analytic models in Cartesian and cylindrical coordinates in $\mathbb R^3$ are compared. The specifics of the Jacobian matrix $\mathbf{J}(\vec V)$ of potential meridional fields $\vec V$ in cylindrically layered media, where $ϕ( ρ) = ρ^{-α}$ $(α\in \mathbb R)$, lead to the concept of $α$-meridional mappings of the first and second kind. As a consequence, a new mathematical concept of $α$-meridional functions of the first and second kind arises. The topological and geometric properties of sets of degenerate points of the Jacobian matrix of various potential meridional fields $\vec V$ are demonstrated explicitly. When $α=1$, we deal with the special concept of radially holomorphic functions and the radially holomorphic potentials in $\mathbb R^3$. Remarkable properties of the radially holomorphic potentials represented by a linear superposition of the radially holomorphic exponential function $e^{\breveβ x}$ $(\breveβ \in \mathbb R)$ and function $e^{\breveβ x}$ with coefficient $I$ are demonstrated explicitly. New properties of the radially holomorphic potentials represented by certain radially holomorphic fourth-order polynomials with reduced-quaternion-valued coefficients are established.