Potential Vector Fields in $\mathbb R^3$ and $α$-Meridional Mappings of the Second Kind $(α\in \mathbb R)$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912170392944640 |
|---|---|
| author | Bryukhov, Dmitry |
| author_facet | Bryukhov, Dmitry |
| contents | This paper extends approach developed in a recent author's paper on analytic models of potential fields in inhomogeneous media. New three-dimensional analytic models of potential vector fields in some layered media are constructed. Properties of various analytic models in Cartesian and cylindrical coordinates in $\mathbb R^3$ are compared. The original properties 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 \emph{$α$-meridional mappings of the first and second kind}. The concept of \emph{$α$-Meridional functions of the first and second kind} naturally arises in this way. When $α=1$, the special concept of \emph{Radially holomorphic functions in $\mathbb R^3$}, introduced by Gürlebeck, Habetha and Sprössig in 2008, is developed in more detail. Certain key properties of the radially holomorphic functions $G$ and functions reversed with respect to $G$ are first characterized. Surprising properties of the radially holomorphic potentials represented by superposition of the radially holomorphic exponential function $e^{\breveβ x}$ $(\breveβ \in \mathbb R)$ and function reversed with respect to $e^{\breveβ x}$ are demonstrated explicitly. The basic properties of the radially holomorphic potential represented by the radially holomorphic extension of the Joukowski transformation in $\mathbb R^3$ are studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19536 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Potential Vector Fields in $\mathbb R^3$ and $α$-Meridional Mappings of the Second Kind $(α\in \mathbb R)$ Bryukhov, Dmitry Complex Variables Dynamical Systems 30G35, 30C65, 35J15, 35Q05, 37N10 This paper extends approach developed in a recent author's paper on analytic models of potential fields in inhomogeneous media. New three-dimensional analytic models of potential vector fields in some layered media are constructed. Properties of various analytic models in Cartesian and cylindrical coordinates in $\mathbb R^3$ are compared. The original properties 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 \emph{$α$-meridional mappings of the first and second kind}. The concept of \emph{$α$-Meridional functions of the first and second kind} naturally arises in this way. When $α=1$, the special concept of \emph{Radially holomorphic functions in $\mathbb R^3$}, introduced by Gürlebeck, Habetha and Sprössig in 2008, is developed in more detail. Certain key properties of the radially holomorphic functions $G$ and functions reversed with respect to $G$ are first characterized. Surprising properties of the radially holomorphic potentials represented by superposition of the radially holomorphic exponential function $e^{\breveβ x}$ $(\breveβ \in \mathbb R)$ and function reversed with respect to $e^{\breveβ x}$ are demonstrated explicitly. The basic properties of the radially holomorphic potential represented by the radially holomorphic extension of the Joukowski transformation in $\mathbb R^3$ are studied. |
| title | Potential Vector Fields in $\mathbb R^3$ and $α$-Meridional Mappings of the Second Kind $(α\in \mathbb R)$ |
| topic | Complex Variables Dynamical Systems 30G35, 30C65, 35J15, 35Q05, 37N10 |
| url | https://arxiv.org/abs/2412.19536 |