Mathematik

I. Science​ content

The goal of teaching the subject is to develop in students the skills to apply topological methods and differential calculus methods in the study of geometric objects.

The task of the science is to study topological, open, and closed mappings, Tikhonov products of topological spaces, necessary and sufficient conditions for a topological space to be a metric space, and the geometry of geometric objects parameterized by differentiable functions, that is, lines and surfaces.

To achieve this goal, the tasks of introducing students to the necessary methods of studying science and forming a scientific worldview are carried out.

This knowledge is widely used in applications to the theory of cardinal invariants, continuation of continuous functions, manifolds, differential topology, and homology, which are modern branches of differential geometry and topology, as well as in the education system.

II. Main theoretical part (lectures)

II. I. The subject includes the following topics:

Topic 1. Introduction to differential geometry and topology

Basic concepts of topology. General concepts of the theory of lines and surfaces. Subject and methods of differential geometry and topology.

Topic 2. Topology in Euclidean space.

Euclidean space, Euclidean topology.

Topic 3. Metric spaces and metric topology.

Topic 4. Topological spaces.

Basic properties of open and closed sets in topological spaces. Continuity and convergence.

Topic 5. Construction of topological spaces.

Topological spaces multiplication, partial spaces.

Topic 6. Topology base. Building a topology using a base.

Topic 7. Axioms of divisibility. Urison's lemmas.

Topic 8. B pain kits.

Bounded sets in topological space and their properties .

Topic 9. Compact sets.

Compact sets in topological space and their properties.

Topic 10. Compactness in Euclidean spaces.

Compact set in Euclidean space. Compactness of a cut and a closed cube.

Topic 11. Continuous accentuations.

of continuous accentuation in topological space, examples. Theorems on continuity .

Topic 1 2. Connectedness and compactness in continuous mapping . Theorems on linearly connected sets and their intersections.

1 3 -topic. Topological reflections .​​​

Topological reflection properties , examples. Stereographic projection .

1 4th topic. Topological groups.

Topic 15. Homotopic reflections.

Topic 16. Vector function and its differential calculus.

Vector function calculus. Limit, derivative and differential of a vector function. Rules of differentiation and integration for vector functions.

Topic 17. Lines and their methods of representation

Elementary, simple and general smooth curves, methods of curve representation, parametrization methods. Simple and special points of a curve.

Topic 1 8. Curved line attempt and adhesion plane

Definition and properties of a curve. Equation of the normal plane of a curve . The equation of a plane of adhesion, its properties. The equations of the principal normal and binormal.

Topic 1 9. Arc length of a curve and the natural parameterization method

E gr i line Arc length and its calculation . Straightening curve . Natural parameter of curve .

Topic 20. Curvature and torsion of a curve

Curve of a line and its calculation . Curve of a line and its calculation . Frene formulas . Natural curve of a line equations.

Topic 21. Curvilinear coordinate system .

Curvilinear coordinate system . A linear arc in a curvilinear coordinate system​​​ length. The concept of Roman metric .

Topic 22. The concept of surface and its methods of representation .

Elementary, simple and general surface concepts. Methods of rendering surfaces.

Topic 23. Surface plane of incidence

Curves lying on a surface. Equation of the plane of incidence and normal of a surface . Basis for the plane of incidence. Incidence vector and its coordinates . Change of coordinates of the incidence vector when passing from one basis to another .

Topic 24. The first quadratic form of a surface

The first quadratic form of a surface. Calculating the length of lines lying on the surface , the angle between two curved lines .

Topic 25. The second quadratic form of a surface

The second quadratic form of a surface. Menye's formula. Normal curvature of a surface . Principal curvatures and directions . Euler's formula. Classification of surface points . Dupin index matrix .

Topic 26. Basic surface equations

Derivative formulas of Gauss and Weingarten. Christoffel symbols.

Topic 27. The connection between the first and second quadratic forms. Bonne's theorem.

Topic 28. Internal geometry of the surface

Internal geometry of surfaces . Geodesic lines . Semi-geodesic coordinate system. Parallel translation of vectors .

Topic 29. Vector fields and their integral lines

Vector fields in Euclidean space. Integral lines of vector fields . Vector fields given on surfaces and their integral lines .

Topic 30. Parallel vectors on a surface

Covariant differential of a vector field and its properties. Parallel translation of the test vectors .

III . Instructions and recommendations for practical exercises

The following topics are recommended for practical training :

1. Euclidean space and Euclidean topology.
2. Metric spaces, metric topology or .
3. Topological spaces: open and basic properties of closed sets .
4. Topological spaces : k - spaces , partial spaces , and factor spaces.
5. Topology basics . Hausdorff, regular and normal spaces.
6. Hausdorff axiom and Ur 's number lemmas.
7. Connectivity and linear connectivity .​​
8. Theorems about linearly connected sets and their properties .​​​​
9. Compact sets and Tikhonov 's theorems.
10. Compactification of topological spaces.
11. Compactness in Euclidean spaces: compactness of a cube with a cut and closure q .
12. Continuous reflections: necessary and sufficient conditions. Open and closed reflections.
13. Connectedness and compactness in continuous reflection .
14. Topological reflections: properties, examples. Stereographic projection.
15. Topological groups. Construction of the Myobius leaf.
16. Homotopic reflections. First fundamental group. Homotopic type.
17. Elementary, simple , and general linear curves .
18. Methods of representing the curve q , methods of parameterization.
19. Simple and special points of a curve .
20. Definition and properties of the curve . Equation of the normal plane of a curve .
21. The equation of a plane, its properties . The equations of the general normal and binormal.
22. Arc length of a curve and its calculation. Torsion of a curve and its calculation .Frenet formulas​ Natural equations of a curve .
23. of a curve and its calculation . Twist of a line and its calculation . Frene formulas. Natural equations of a line .
24. Curvilinear coordinate system. Arc length in a curvilinear coordinate system . Concept of Riemannian metric .
25. Elementary, simple and general concepts of surfaces. Methods of representing surfaces. Curves lying on a surface .
26. Equation of the plane of incidence and normal to a surface. Basis for the plane of incidence. Incidence vector and its coordinates.
27. The change in the coordinates of the stress vector when moving from one basis to another . The first quadratic form of the surface.
28. length of the lines lying on the surface Calculation of the angle between two e - lines . The second quadratic form of a surface.
29. . Normal curvature of a surface. Principal curvatures and directions . Euler 's formula.
30. Classification of surface points . Dupin ind i matrix . Derivative formulas of Gauss and Weingarten. Christoffel symbols.
31. The relationship between the first and second quadratic forms . Bonnet 's theorem. The interior geometry of surfaces.
32. Geodesic lines . Semi-geodesic coordinate system. Parallel translation of vectors .
33. fields in Euclidean space . Integral lines of vector fields . Vector fields given on surfaces and their integral lines .
34. Covariant differential of a vector field and its properties. Parallel translation of random vectors .

IV. Independent learning and independent work

A modern specialist is required to have a high level of training, the ability to make independent decisions, select the necessary information from a large amount of information to perform assigned tasks, and the ability to process this information.

The main goals of independent learning for students are:

• mastering new methods of acquiring knowledge, being able to independently analyze processes;
• consolidate, deepen, expand, and organize the knowledge gained in classroom sessions;
• learn to work with information and special literature;
• independent study of educational materials.

Recommended topics for independent study :

1. Special topologies: Zarissky topology, Alexandrov 2-arrows;
2. Set power: cardinal numbers;
3. Factor space and factor topology;
4. Strong and weak topology;
5. Construct examples of the separability of topological spaces;
6. Homotopy. Homotopic reflections;
7. Euler characteristics of polynomials;
8. 1st fundamental group of the circle;
9. Jordan's theorem for polygons;
10. Cantor sets ;
11. Alexandrov Square ;
12. Vector functions and operations on them. Differentiation rules for vector functions ;
13. lines . The indicatrix of a line and its equation ;
14. The family of lines is a coil. Evolute and involute;
15. Making surfaces ;
16. Surt family scroll;
17. Spherical representation of surfaces;
18. Isometric views;
19. Derivative formulas;
20. Scalar and vector fields.

It is recommended that students prepare and present abstracts on topics that are being studied independently.

V. Results of science teaching (competences to be formed)

As a result of mastering the subject, the student:

• Metric and topological open and closed sets; connected sets and spaces; compact sets and spaces; continuous mappings; linearly connected sets; topological mappings; methods of curve representation; methods of surface representation; curves lying on a surface; principal curvatures and principal directions; Dupin indicatrix; parallel translation of vectors; covariant differential of a vector field; understanding of surfaces with invariant curvature;
• The concept of a topological space base; giving examples of connected and unconnected sets; constructing an example of a continuous reflection; constructing examples of linearly connected sets; constructing examples of topological reflections; calculating the length of lines on a surface; finding the angle between curved lines on a surface; determining the normal curvature of a surface using the Menye formula; determining the principal curvatures and directions of a surface; covariant differentiation and parallel translation of vectors; acquiring the skills to use the Gauss-Bonne theorem for surfaces with invariant curvature;
• be able to mathematically model processes using mathematical symbols and simple systems, build models for a specific economic process, conduct calculations within the framework of the constructed model, and be able to apply this knowledge to the use of basic methods and guidelines for developing experimental data .

VI. Educational technologies and methods:

• Lectures;
• Interactive case studies;
• Seminars (logical thinking, quick questions and answers);
• Working in groups;
• Making presentations;

VII. Requirements for obtaining loans:

Fully master the theoretical and methodological concepts of the subject, be able to correctly reflect the results of the analysis, conduct independent observations of the processes being studied, and complete the tasks and assignments given in the forms of current and intermediate control, and submit a written work for final control.

VII. Main literature

1. Narmanov A.Ya. Differential geometry. T. Turon-Iqbal, 2016. 225 pages.
2. Narmanov A.Ya., Sharipov AS, Aslonov JO Collection of problems from the course of differential geometry and topology, T.: University, 2014.
3. Mishchenko A. S, Fomenko A. T. Short course differential geometry and topology. M.: fizmatlit, 2004. 304 p .

Additional literature

1. Armstrong M.A.​ Basic Topology. Springer, 1998.
2. Mishchenko A.S., Solovev Ya.T., Fomenko A.T. Sbornik zadach po differential geometry and topology, MGU, 2004 .
3. SIDNEY A. MORRIS. TOPOLOGY WITHOUT TEARS, 2003.
4. Alexei Sossinsky. Knots: mathematics with a twist. Harvard University Press, 2002.
5. Fedorchuk V.V., Filippov V.V. General topology. Basic construction. Moscow, Fizmatlit, 2006.

4. Fedorchuk V.V. Introduction to topology. Moscow, izd. MSU, 2012.

6. Arkhangelsky A.V., Ponomarev V.I. Osnovy Obshchey topology v zadachax i uprajneniyax. Moscow, Fizmatlit, 1974. – 462 p.
7. Engelking R. General topology. Moscow: Mir, 1986. – 752 p.
8. G. Poznyak, E. V. Shikin. Differential geometry: first acquaintance. M.: Izd-vo MGU, 1990. 384 pages.
9. Pogorelov A.V. Differential geometry. M. Nauka, 1969. 176 pages.

Information sources

1. www.ziyonet.uz
2. www.allmath.ru
3. www.exponenta.ru