Extensition of the Rolle and Darboux theorems to the functions of two variables
https://doi.org/10.29235/1561-2430-2018-54-3-290-299
Abstract
As well known, the classical Rolle and Darboux theorems for a function of one variable establish the existence of a critical point in the behavior of a function at the ends of a closed interval. The question arises of the possibility of extension of the Rolle and Darboux theorems to functions of two variables. More precisely is the existence of a critical point in Ω̅ determined by the behavior of the function f on the boundary of the ∂Ω domain Ω. As shown by A. I. Perov, such extension can be obtained with the help of the concept of rotation. In this article, we establish deeper relations between the Rolle and Darboux theorems and the rotation of a vector field on the boundary ∂Ω. We also present some new formulas for calculating the rotation of a vector field on the boundary, on the basis of which statements about the existence of critical points are formulated.
About the Authors
P. P. Zabreiko
Belarusian State University
Belarus
Belarus
Petr P. Zabreiko – Dr. Sc. (Physics and Mathematics), Professor of the Department of Functional Analysis and Analytical Economics of the Faculty of Physics and the Faculty of Mathematics.
4, Nezavisimosti Ave., 220030, Minsk.
A. V. Krivko-Krasko
Belarusian State University
Belarus
Belarus
Aleksey V. Krivko-Krasko – Ph. D. (Physics and Mathematics), Assistant Professor, Assistant Professor of the Department of Innovation Management of the School of Business.
5, Moskovskaya Str., 220007, Minsk.
References
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