Topic 1: Differential Calculus and Its Applications

1.1. Investigation of extrema (local, constrained, global) of real functions of several variables using partial derivatives.

1.2. Partial derivative, definition of directional derivative, gradient (definition and geometric interpretation), differentiability.

1.3. Limit, continuity, and differentiability of a real function of a real variable at a point and on a set.

1.4. Application of differential calculus in analyzing the behavior of a function (extrema, monotonicity, convexity, asymptotes of the function graph).

Topic 2: Integral Calculus and Its Applications

2.5. One-dimensional and two-dimensional Riemann integral, its construction, properties, and methods of computation.

Topic 3. Measure Theory and Lebesgue integral.

3.1. Definition of Lebesgue measure, konstruction of the Lebesgue integral and its relationship with the Riemann integral.

3.2. Basic properties of the Lebesgue integral, Fubini's theorem and convergence theorem for the Lebesgue integral.

Topic 4. Algebra of vectors and matrices.

4.1. System of linear equations, methods of solution, determinants and methods of their calculation, rank of matrix, regularity, inverse matrix.

4.2. Vector space and subspace, basis and dimension, linear mapping, matrix of linear mapping, similarity and canonical forms of matrices.

Topic 5. Algebraic structures and number theory.

5.1. Ring, field, group: definitions and examples, orders of elements, cosets, quotients.

5.2. Divisibility of integers, congruences, residue classes, algebraic and transcendent numbers

Topic 6. Affine spaces.

6.1. Definition of affine space and subspace, coordinates in the affine frame, expression of affine space (parametric and general form), relative position of affine subspaces

Topic 7. Euclidean spaces.

7.1. Definition and examples of scalar product, perpendicularity of vectors, orthogonal complement, distance and angle of two affine subspaces

Topic 8. Linear programming problems, solution methods and complexity.

8.1. The form of the linear programming problem and the simplex method: justification of its correctness and finiteness.

8.2. Duality in linear programming: relation between optimal solution of primal and dual problem.

Topic 9: Structural properties of planar graphs.

9.1. Eulerian and Hamiltonian graphs, graph closure, hamiltonicity of planar graphs

9.2. Planar and plane graphs, Euler's formula and its implications, characterization of planar graphs

Topic 10. Chromatic graph theory.

10.1. Vertex and edge coloring, estimates of chromatic numbers, coloring of planar graphs