A CHARACTERIZATION OF CONCENTRIC HYPERSPHERES IN ℝn

Title & Authors
A CHARACTERIZATION OF CONCENTRIC HYPERSPHERES IN ℝn
Kim, Dong-Soo; Kim, Young Ho;

Abstract
Concentric hyperspheres in the n-dimensional Euclidean space $\small{\mathbb{R}^n}$ are the level hypersurfaces of a radial function f : $\small{\mathbb{R}^n{\rightarrow}\mathbb{R}}$. The magnitude $\small{||{\nabla}f||}$ of the gradient of such a radial function f : $\small{\mathbb{R}^n{\rightarrow}\mathbb{R}}$ is a function of the function f. We are interested in the converse problem. As a result, we show that if the magnitude of the gradient of a function f : $\small{\mathbb{R}^n{\rightarrow}\mathbb{R}}$ with isolated critical points is a function of f itself, then f is either a radial function or a function of a linear function. That is, the level hypersurfaces are either concentric hyperspheres or parallel hyperplanes. As a corollary, we see that if the magnitude of a conservative vector field with isolated singularities on $\small{\mathbb{R}^n}$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.
Keywords
Language
English
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