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학술연구/단체지원/교육 등 연구자 활동을 지속하도록 DBpia가 지원하고 있어요.
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연구자들이 자신의 연구와 전문성을 널리 알리고, 새로운 협력의 기회를 만들 수 있는 네트워킹 공간이에요.
초록· 키워드
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Concentric hyperspheres in the n-dimensional Euclidean spa- ce Rn are the level hypersurfaces of a radial function f : Rn → R. The magnitude ||∇f|| of the gradient of such a radial function f : Rn → 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 : Rn → 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 Rn is a function of its scalar potential, then either it is a central vector field or it has constant direction.
목차
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참고문헌 (9)
참고문헌 신청
D. Khavinson / 1995 / A note on entire solutions of the eiconal [eikonal] equation / Amer. Math. Monthly 102 (2) : 159 ~ 161
J. A. Thorpe / 1979 / Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics / Springer-Verlag
K. Nomizu / 1973 / Elie Cartan’s work on isoparametric families of hypersurfaces, Differential geometry / Proc. Sympos. Pure Math 27 : 191 ~ 200
L. A. Caffarelli / 2010 / Distance functions and almost global solutions of eikonal equations / Comm. Partial Differential Equations 35 (3) : 391 ~ 414
M. P. do Carmo / 1976 / Differential Geometry of Curves and Surfaces / Prentice-Hall
J. A. Thorpe / 1979 / Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics / Springer-Verlag
K. Nomizu / 1973 / Elie Cartan’s work on isoparametric families of hypersurfaces, Differential geometry / Proc. Sympos. Pure Math 27 : 191 ~ 200
L. A. Caffarelli / 2010 / Distance functions and almost global solutions of eikonal equations / Comm. Partial Differential Equations 35 (3) : 391 ~ 414
M. P. do Carmo / 1976 / Differential Geometry of Curves and Surfaces / Prentice-Hall
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