About the Kyiv school mathematicians contribution to the extrema of functions of many variables theory

Abstract

The subject of this article is the study of Kyiv school mathematicians’ contribution to the many variable extrema function theory. The purpose of this article is to study the work of Kyiv mathematician Professor M. Ye. Vashchenko-Zakharchenko. Task: for the first time, to carry out a detailed analysis of the results obtained in the article by M. Ye. Vashchenko-Zakharchenko «Signs of the highest and lowest value of functions». The research method is a historical and scientific analysis of the original source, which allows the scientific results obtained in the late 19th and early 20th centuries to be estimated from the viewpoint of modern mathematical analysis. The following results were obtained. Thanks to the analysis of this article, it was found out that here M. Ye. Vashchenko-Zakharchenko first used the methods of linear algebra and applied D. Sylvester’s criterion of positive (negative) definiteness of quadratic form to obtain sufficient conditions for the existence of an extremum of functions of many variables. The problem is considered in general for functions of many variables and for special cases of functions of two and three variables. Conclusions. Comparing the results obtained from the Kyiv mathematician, Professor M. Ye. Vashchenko-Zakharchenko with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis, we can conclude that they are included in these textbooks in virtually the same form. This is what determines the relevance of this topic: methods for solving problems for the extremum of functions of many variables, obtained at the turn of the 19th and 20th centuries, are used in modern optimization problems. The breadth of the scientific erudition of M. Ye. Vashchenko-Zakharchenko allowed him to immediately perceive new methods obtained by foreign scientists and immediately find their application to obtain sufficient conditions for the existence of an extremum of functions of many variables. This shows the high scientific level of the state of mathematics at Kyiv University in the second half of the 19th century. Further research should focus on the contribution of other mathematicians of the Kyiv school to the development of the theory of extrema of functions of many variables.

Keywords

• optimization
• extremum function of several variables
• sufficient conditions
• positive (negative) definiteness of the quadratic form