MULTIDIMENSIONAL EUCLIDEAN SURFACES, SET BY SEVERAL SCALAR FUNCTIONS

Dolgarev Artur Ivanovich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), delivar@yandex.ru

514

10.21685/2072-3040-2016-2-4

Background. Nowadays, the theory of multidimensional Euclidean surfaces is actively developing. There have been investigated hypersurfaces, described by one explicit scalar function. There has been started a research of surfaces, set by several scalar functions. The aim of this work is to describe surfaces, set by several scalar functions.
Materials and methods. The author considered surfaces that are an intersection of several cylindrical surfaces.
Results. The author drew out tangential planes of cylindrical of surfaces and their intersections, obtained coordinates of vectors of cylindrical surface normals and their intersections and introduced expressions of coefficients of curvature forms of cylindrical surfaces through coefficients of their metric forms. For given coefficients of metric forms of surfaces, the author found cylindrical surfaces that are an intersection the set ones as the intersection of cylindrical surfaces.
Conclusions. Every surface of a multidimensional Euclidean space, differring from the hypersurface and the cylindrical surface, is an intersection of cylindrical surfaces and is defined with the accuracy up to the position in space in metric forms of cylindrical surfaces.

multidimensional Euclidean space, surface, cylindrical surface, metric form of surface, surface determinability.

1. Kobayasi Sh., Nomidzu K. Osnovy differentsial'noy geometrii. II. [Foundations of differential geometry. II]. Moscow: Nauka, 1981, 416 p.
2. Ivanov A. O., Tuzhilin A. A. Lektsii po klassicheskoy differentsial'noy geometrii [Classical differential geometry lectures]. Moscow: Novaya universitetskaya biblioteka, 2009, 233 p.
3. Ivanova-Karatopraklieva I., Markov P. E., Sabitov I. Kh. Fundamental'naya i prikladnaya matematika [Fundamental and applied mathemtics]. 2006, vol. 12, no. 1, pp. 3–56.
4. Dolgarev A. I. EDUCATIO. 2014, no. 3, part 6, pp. 58–61.
5. Dolgarev A. I. Moderni vymozenosti vedy – 2014: Materialy X Miedzynarodowej naukowi- praktycznej konferencji, dil 34. Matematyka. Fizyka [Modern advances of science – 2014: Proceedings of X International scientific and practical conference, Mathematica. Physics]. Praga: Education and Skience.s.r.o.,2014,pp.30–40.
6. Dolgarev I. A., Dolgarev A. I. Aktual'nye voprosy razvitiya innovatsionnoy deyatel'nosti v novom tysyacheletii: XV Mezhdunar. nauch.-prakt. konf. (22–23 maya 2015 g., Novosibirsk, Rossiya) [Topical problems of innovative activity development in the new millenium: XV International scientific and practical conference (22–23 May 2015, Novosibirsk, Russia)]. Iss. 4 (15). Novosibirsk: Mezhdunar. nezavis. institut Matematiki i Sistem «MiS», 2015, pp. 49–58.
7. Dolgarev A. I. Dny vedy – 2014: Materialy X Miedzynarodowej naukowi-praktycznej konferencji, Dil 31. Matematyka [Days of science – 2014: Proceedings of X International scientific and practical conference, Mathematics]. Praha: Education and Skience. s.r.o., 2014, pp. 72–78.
8. Dolgarev A. I. Sovremennye nauchnye issledovaniya: innovatsii i opyt. Ezhemesyachnyy zhurnal mezhotraslevogo instituta «Nauka i obrazovanie» [Modern scientific research: innovations and experience. Monthly journal of the intersectoral institute “Science and education”]. 2015, no. 1 (8), pp. 3–8.
9. Dolgarev A. I. Dni vediy – 2013: Materiali IX mezinarodni ve-decko-praktika conference, Dil 32. Matematika. Vystavba a archtektura [Days of science – 2013: Proceedings of IX International scientific and practical conference, Mathematics. Construction and architecture]. Praga: Education and Skience.s.r.o.,2013,pp.55–60.
10. Torp Dzh. Nachal'nye glavy differentsial'noy geometrii [First chapters of differential geometry]. Volgograd: Platon, 1998, 360 p.