# P R O G R A M A introductory professional examination in mathematics for obtaining an educational master’s degree in specialty 111- “mathematics” (educational-professional level)

Mathematical analysis

1. Numerical sequences.

1.1. Definition of the boundary of the numerical sequence. Examples of sequences:

(A) those with a finite boundary;

(B) those having an infinite boundary;

(C) those that have no boundary (neither finite nor infinite);

(D) infinitesimal;

(E) infinitely large.

1.2. The properties of the boundaries of the sequences:

(A) the theorem on the boundary of the sum, of the product, of the fraction of two sequences;

(B) Theorem on the uniqueness of the boundary of the sequence;

(C) the theorem on the boundary transition in inequalities (for sequences);

(D) a theorem on three sequences.

2. Numerical functions.

2.1. Definition of the boundary of a numerical function at a point. Examples of functions:

(A) those with a finite boundary at some point;

(B) those having an infinite boundary at some point;

(C) those that have no boundary at some point (neither finite nor infinite);

(D) infinitesimal at some point;

(E) are infinitely large at some point.

2.2. Properties of boundary functions:

(A) the theorem on the boundary of the sum, of the product, of the fraction of two functions;

(B) uncertainty and examples of the application of some methods for their disclosure: the Lopital rule, the application of the Taylor formula.

3. Continuous functions.

3.1. The definition of a numerical function that is continuous at a point. Examples of functions:

(A) continuous at a certain point;

(B) discontinuous at some point.

3.2. Properties of Continuous Functions:

(A) Theorem on the preservation of the sign of a continuous function;

(B) the Bolzano-Cauchy theorem that a function that is continuous on a segment and accepts the values ​​of different signs at its ends at some point in the interval is zero;

(C) The Weierstrass theorem on the existence of a minimal and maximal values ​​of a function that is continuous on a segment.

4. Derivatives.

4.1. Definition of a derivative of a numerical function. Examples of functions:

(A) those having a derivative at some point;

(B) those that do not have a derivative at some point.

4.2. Theorems on finding derivatives:

(A) derivative amount, product, share of two functions;

(B) derivative of the composition of two functions;

(C) some “table” derivatives: ,,,,,,,.

4.3. The geometric meaning of the derivative. Fermat’s theorem on the equality of zero derivatives at the extremum point of a function. Lagrange’s theorem (the formula of finite increments) and its geometric content. Application of derivatives for the study of functions.

5. Integrals.

5.1. Definition of the primitive (indefinite integral) of a numerical function. Some “table” primitives:,,,,, ,,.

5.2. Some integration methods:

(A) replacement of the variable and the summary to the table integrals;

(B) integration into parts.

5.3. The (definite) Riemann integral of a numerical function. The geometric meaning of the integral.

5.4. Newton-Leibniz formula.

6. Rows.

6.1. The definition of a convergent numerical series. Examples of numeric rows:

(A) convergent;

(B) divergent.

6.2. The definition of a power series (from one variable). The field of convergence of the power series, the Cauchy-Hadamard formula for the radius of convergence.

6.3. Definition of a Taylor series of a numerical function. Makrolena series of some elementary functions:,,,,.

Linear algebra

1. Systems of linear equations.

1.1. The solution of the system of linear equations. Compatible and incompatible systems.

1.2. Finding a solution:

(A) elementary system transformations, Gauss method;

(B) the connection between solutions of a nonhomogeneous and corresponding homogeneous system, the general theorem of the solution.

2. Linear spaces.

2.1. Linear Independent and Linearly Dependent Vectors Systems:

(A) definitions and examples of linearly independent and linearly dependent vectors systems;

(B) the definition of the basis and dimensionality of the linear space.

2.2. Linear shell of a set of vectors. Finding the dimensionality of the linear shell of the given vectors.

2.3. Finding the dimensionality of the space of solutions of a homogeneous system of linear equations.

3. Matrices and determinants.

3.2. Calculating the determinant of a square matrix:

(A) decomposition in a row (column);

(B) Calculation using the Gauss method.

3.3. An inverse matrix. Finding an inverse matrix (arbitrary method).

4. Line operators.

4.1. The kernel and the image of a linear operator: definitions and examples.

4.2. The matrix of a linear operator in a given basis. Theorem on the change of the matrix of a linear operator when a basis is replaced.

4.3. Own numbers and own vectors of the linear operator:

(A) Defining your own number and own vector;

(B) the characteristic polynomial and the finding of eigenvalues ​​by means of the characteristic polynomial;

4.4. Diagonalization of a linear operator: definitions and examples of diagonal and non-diagonal operators.

5.1. Bringing the real quadratic form to the diagonal form by the Lagrange method. Signature of a quadratic form.

5.2. Quadratic forms are also specified. Sylvester Criterion.

Differential equations

1. Differential equations of the first order.

1.1. Statement of the Cauchy problem.

1.2. Solving of some types of first-order differential equations:

(A) equations with separable variables;

(B) linear inhomogeneous equations.

2. Linear differential equations of the nth order with constant coefficients.

2.1. Statement of the Cauchy problem.

2.2. Homogeneous linear differential equations of the nth order with constant coefficients:

(A) the general solution theorem;

(B) the characteristic equation and its application for finding a general solution.

2.3. Inhomogeneous linear differential equations of the nth order with constant coefficients:

(A) the general solution theorem;

(B) the principle of superposition of solutions;

(C) finding a partial solution of an inhomogeneous equation (arbitrary method).

3. Systems of linear differential equations with constant coefficients.

3.1. Statement of the Cauchy problem. Homogeneous and heterogeneous systems.

3.2. Homogeneous systems of linear differential equations with constant coefficients:

(A) the general solution theorem;

(B) finding a general solution (arbitrary method);

(C) the Lyapunov stability of the zero solution, asymptotic stability, asymptotic stability criterion.

Geometry and topology

1. Direct on the plane. Direct and planes in space.

1.1. Parametric and implicit methods for directing a plane on a plane. Mutual arrangement of straight lines on a plane.

1.2. Parametric and implicit methods of plotting space.

1.3. Parametric and implicit methods of direct problem in space. The mutual arrangement of direct and planes in space.

1.4. Find the equation of the perpendicular, which is omitted from a given point on a given line in space. Finding a common perpendicular to two commuting lines.

1.5. Finding the distance from point to line in space and distance between two straight lines in space.

2. R n as a linear space.

2.1. Vectors in R n :

A) linear operations over vectors;

B) the linear shell of the system of vectors in R n and its basis.

2.2. Linear subspaces in R n :

A) parametric and implicit methods of the problem of linear and affine subspaces;

B) finding the basis of the sum and the basis of the intersection of parametrically given subspaces.

2.3. Convex sets:

A) the definition of a convex set, examples of convex sets;

B) the convexity of the intersection of convex sets;

C) the analytic expression for a convex shell of finite number of points;

D) polyhedra in R n , the extremum theorem on a linear function on a closed convex polyhedron.

2.4. Separation of convex sets:

A) definition of the reference hyperplane;

B) the definition of a hyperplane dividing the plural;

C) conditions of strict separation of two convex sets.

3. R n as metric space.

3.1. Definition of metric space. Examples of metrics in R n .

3.2. The definition of the topological space. Metric topology in R n .

3.3. Internal, boundary, boundary points, points of contact, closure of a subset in R n .

3.4. The sequence in R n and its boundary. Fundamental sequence, the Cauchy criterion.

3.5. Compact subsets in R n . Weierstrass theorem on continuous functions given on compact subsets.

List of recommended literature

1. Ter-krikorov AM, Shabunin MI Course of Mathematical Analysis. – M.: Fizmatlit, 2003.
2. Kudryavtsev LD Short course of mathematical analysis. – M .: Nauka, 1989.
3. Pontryagin L.S. Ordinary differential equations. – Moscow: Nauka, 1974.
4. Samoilenko AM, Krivoshey SA, Perestyuk NA Differential equations. Examples and tasks. – M .: Higher school, 1989.
5. Gelfand I. M., Lectures on linear algebra. – M.: Science, 1970.
6. Ilyin VA, Pozdnyak E. G. Linear algebra. – M.: Science, 1984.
7. Pogorelov AV Analytical geometry. – M .: Nauka, 1978.
8. Borisenko O.A., Ushakova L.M. An analytical geometry. – X.: Basis, 1993.
9. Borisenko O.A. Differential geometry and topology. – X.: Basis, 1995.

Approved at the meeting of the Academic Council of the Faculty of Mathematics and Informatics, Minutes No. 1 dated 07.02.2017

Dean of the Faculty of Mathematics and Informatics G. M. Zholtkevich

Approved at the meeting of the admissions committee, protocol No. from p.

O. Anoshenko, Executive Secretary of the Admissions Committee