A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient
The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f : ℝ2 → ℝ is a differentiable function with everywhere non-zero gradient then for every point x ∊ ℝ2 in the level set {x : f(x) = c} there is a neighborhood V of x such that {f = c} ∩ V is homeomorphic t...
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| Published in | Real analysis exchange Vol. 43; no. 2; pp. 387 - 392 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
East Lansing
Michigan State University Press
01.01.2018
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0147-1937 1930-1219 |
| DOI | 10.14321/realanalexch.43.2.0387 |
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| Abstract | The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f : ℝ2 → ℝ is a differentiable function with everywhere
non-zero gradient then for every point x ∊ ℝ2 in the level set {x : f(x) = c} there is a neighborhood
V of x such that {f = c} ∩ V is homeomorphic to an open interval or the union of finitely
many open segments passing through a point.
Mathematical Reviews subject classification: Primary: 26B10; Secondary: 26B05
Key words: Implicit Function Theorem, Non-vanishing Gradient, Locally homeomorphic |
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| AbstractList | The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f... is a differentiable function with everywhere non-zero gradient then for every point ... in the level set ... there is a neighborhood V of x such that {f = c} ∩ V is homeomorphic to an open interval or the union of finitely many open segments passing through a point.(ProQuest: ... denotes formulae omitted.) The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f : ℝ2 → ℝ is a differentiable function with everywhere non-zero gradient then for every point x ∊ ℝ2 in the level set {x : f(x) = c} there is a neighborhood V of x such that {f = c} ∩ V is homeomorphic to an open interval or the union of finitely many open segments passing through a point. Mathematical Reviews subject classification: Primary: 26B10; Secondary: 26B05 Key words: Implicit Function Theorem, Non-vanishing Gradient, Locally homeomorphic |
| Author | Wedrychowcz, Christopher M. Savvopoulou, Anna K. |
| Author_xml | – sequence: 1 givenname: Anna K. surname: Savvopoulou fullname: Savvopoulou, Anna K. organization: Department of Mathematical Sciences, Indiana University South Bend, South Bend, IN 46615, U.S.A. email – sequence: 2 givenname: Christopher M. surname: Wedrychowcz fullname: Wedrychowcz, Christopher M. organization: Department of Mathematics and Computer Science, Saint Mary's College, Notre Dame, IN, 46556, U.S.A. email |
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| Snippet | The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f : ℝ2 → ℝ is a differentiable function with everywhere... The purpose of this note is to give an alternate proof of a result of M.Elekes. We show that if f... is a differentiable function with everywhere non-zero... |
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| StartPage | 387 |
| SubjectTerms | Cattle Differentiable functions Diffraction Implicit functions INROADS Mathematical functions Mathematical theorems Open intervals Rectangles Studies Technology Acceptance Model |
| Title | A Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradient |
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