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A CHARACTERIZATION OF CONCENTRIC HYPERSPHERES IN ℝn

  • Received : 2012.12.24
  • Published : 2014.03.31

Abstract

Concentric hyperspheres in the n-dimensional Euclidean space $\mathbb{R}^n$ are the level hypersurfaces of a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$. The magnitude $||{\nabla}f||$ of the gradient of such a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ 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 : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ 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 $\mathbb{R}^n$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.

Keywords

gradient;conservative vector field;central vector field;hypersurface;principal curvature;radial function

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