The aim of the subject is to deepen and reinforce the knowledge acquired from high school mathematics.

Students will learn new procedures and mathematical methods, enhance their logical thinking to be able to understand and apply the knowledge gained in their specializations. Students should also learn how to analyze a problem, choose the appropriate procedure, select the right calculation method, and evaluate the result, which should lead to critical thinking.

The subject is divided into tree topics: the first one focusing on the fundamentals of linear algebra; the second one dealing with the differential calculus of a single variable, and the third one being dedicated to the basics of integral calculus.

Lectures

1. Basics of matrix calculus, elementary transformations of a matrix, rank of a matrix.
2. Determinants (cross rule, Sarrus' rule, Laplace expansion), rules for calculation with determinants.
3. Vector calculus (operations with vectors, dot, cross, and mixed products of vectors). Real linear space, linear combination and independent bases and dimension of a linear space.
4. Solutions to systems of linear algebraic equations by Gauss elimination method, Frobenius theorem.
5. Inverse to a matrix, matrix equations. Eigenvalues and eigenvectors of a matrix.
6. Real function of one real variable and its basic properties, explicit and parametric definition of a function. Composite function and inverse to a function. Some elementary functions (inverse trigonometric functions). 
7. Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real and complex numbers. Rational functions and their decomposition into partial fractions.
8. Limit of a function, continuous functions, basic theorems.
9.  Derivative of a function, its geometric and physical applications, rules of differentiation. 
10.  Differential of a function. Higher-order derivatives, higher-order differentials. Taylor polynomial and Taylor's theorem.
11. L'Hospital's rule, asymptotes of the graph of a function. Sketching the graph of a function.
12. Anti-derivative, indefinite integral and its properties. Integration by parts and substitution methods in calculating integrals.
13. Integration of selected functions (rational, trigonometric, irrational).

Seminars

1. High school repetition.
2. Basic operations with matrices. Elementary transformations of a matrix, rank of a matrix.
3. Determinants (cross rule, Sarrus' rule, Laplace expansion), rules for calculation with determinants.
4. Vector calculus (operations with vectors, dot, cross, and mixed products of vectors).
5. Solutions to systems of linear algebraic equations by Gauss elimination method.
6. Inverse to a matrix, matrix equations. Eigenvalues and eigenvectors of a matrix.
7. Test 1. Some elementary functions (inverse trigonometric functions). Composite function and inverse to a function.
8. Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real and complex numbers.
9. Rational functions and their decomposition into partial fractions. Limit of a function, continuous functions. Derivative of a function, its geometric and physical applications, rules of differentiation.
10. Differential of a function. Higher-order derivatives, higher-order differentials. Taylor polynomial.
11. Test 2. L'Hospital's rule, asymptotes of the graph of a function. Sketching the graph of a function.
12. Anti-derivative, indefinite integral and their properties. Integration by parts and substitution methods in calculating integrals.
13. Integration of selected functions (rational, trigonometric, irrational).

Learning outcomes

• professional knowledge

Introduction to matrix calculations and its application in solving systems of linear equations. Understanding the basic concepts of differential and integral calculus of functions of one variable and the geometric interpretation of certain concepts. Being acquainted with the use of vector calculus.

• professional skills

The students will be trained in differentiating and integrating function and they will learn how to solve the problem of function behaviour. They will manage matrix calculation, elementary transformations, and calculation of determinants and inverse matrices, being also able to solve systems of linear algebraic equations (using Gaussian elimination and applying the inverse matrix).

• general competence

The student will be able to continue further studies that require the knowledge of this subject.