1 
Course Title

Differential geometry and topology 
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Year of study, speciality 
2, speciality “Mathematics”, direction of the specialty: “research and production activities” 
3 
Semester of study 
4 
4 
Credit points 
4 
5 
Name of the lecturer, scientific degree, occupation 
Associate Professor Vitaly V. Balashchenko, PhD in Mathematics, Department of Geometry, Topology and Methods of Teaching Mathematics 
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Goals of studying 
Studying the fundamental concepts of the theory of curves, surfaces and related invariants (the curvatures of various types) as well as the study of their basic properties and relationships with specific objects in courses of analytic geometry, algebra, mathematical analysis, theoretical mechanics, differential equations 
7 
Prerequisites 
Analytic geometry, algebra, mathematical analysis, differential equations 
8 
Contents 
Curves in the 3dimensional Euclidean space. Curvature vector and the curvature. The Frenet basis and frame. The Frenet formulas. Torsion of space curves. Fundamental theorem of curve theory (existence and uniqueness). Surfaces in the 3dimensional Euclidean space. The first and the second fundamental forms of surfaces. Normal curvature. Principal directions and principal curvatures. The Euler formula. The Gaussian curvature and the mean curvature. The Gauss theorem (Theorema Egregium). Geodesics on surfaces. 
9 
Recommended literature 
1. Differential Geometry (edited by A.S. Fedenko). – Minsk. Publishing house of BSU, 1982 (in Russian). 2. A Collection of Problems in Differential Geometry (edited by A.S. Fedenko). – Nauka, Moscow, 1979 (in Russian). 3. Mishchenko A.S., Fomenko A.T. A Short Course in Differential Geometry and Topology. – Cambridge Scientific Publishers, 2009. 
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Teaching methodology 
Lectures, solving problems, discussions, consultations 
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Language of instruction 
Russian 
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Requirements for study during the semester 
– individual and collective tasks; – short tests. The final evaluation is made taking into account: 30% – assessment of current academic performance, 70% – result during the exam 
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Examination methodology 
Oral examination 