Course detail

# Mathematics 2

Calculus of the more variable functions. Ordinary differential equations, basic terms, exact methods, systems of linear differential equations with constant coefficients, examples of differential equation use. Differential calculus in the complex domain, derivative, Caucy-Riemann conditions, holomorphic functions. Integral calculus in the complex domain, Cauchy theorem, Cauchy formula, Laurent series, singular points, residue theorem. Laplace transform, applications. Fourier series. Z transform, discrete systems, difference equations.

Learning outcomes of the course unit

The absolvent of the subject is able:

- to compute the partial derivatives of the functions of more variables and use the formulas for the gradient and the tangential plane;
- to distinguish between the separable and linear differential equations and also to solve them;
- to solve linear differential equations of higher order with a special right hand side;
- to figure out from the Cauchy Riemann conditions, if the complex function is holomorfic or not, and to derive the holomorfic funcions;
- to compute, using the definition, the integral from the complex function through a curve, to apply the Cauchy theorem for the integral of the holomorfic funcion;
- to establish the poles and to calculate a residue at asimple and at a pole of higher order, to apply the residue theorem for the integral of the meromorfic funcion;
- to solve differential equations by the Laplace transform;
- to find the real Fourier series of an odd, even and a general function, expand a function to sine or cosine series;
- to solve difference equations by the Z- transform.

Prerequisites

The student from his former studies of mathematics should be able: - to compute with the fractions, to solve the quadratic equation; - to apply the basics of the integral and differential calculus of the function of one variable; - to sum the geometric series with quocient |q|<1; - to apply the per pertes method for the definite integral.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Kolářová E: Matematika 2 - Sbírka úloh (CS)
Melkes F., Řezáč M.: Matematika 2 (CS)
Svoboda Z., Vítovec J.: Matematika 2 (CS)

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Maximum 20 points for individual assignments during the semester (at least 5 points for the course-unit credit); maximum 80 points for a written exam.
The exam is focused on verification of the student's knowledge in the issue of solving differencial equations, the complex calculus,
the Fourier series expansion, the Laplace and the Z-transform.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Calculus of the more variable functions.
2. Ordinary differential equations, basic terms.
3. Solutions of linear differential equations of first order.
4. Homogenius linear differential equations of higher order.
5. Solutions of nonhomogenious linear differential equations with constant coefficients.
6. Differential calculus in the complex domain, derivative.
7. Caucy-Riemann conditions, holomorphic functions.
8. Integral calculus in the complex domain, Cauchy theorem, Cauchy formula.
9. Laurent series, singular points.
10. Residue theorem.
11. Laplace transform, convolution, Heaviside theorems, applications.
12. Fourier transform, relation to the Laplace transform, practical usage.
13. Z transform, discrete systems, difference equations.

Aims

The student is acquainted with some methods for solving ordinary differential equations in the first part and with Laplace, Fourier and Z transforms in the second part.

Specification of controlled education, way of implementation and compensation for absences

Tutorials are not compulsory.

Classification of course in study plans

• Programme EEKR-BK Bachelor's

branch BK-AMT , 1. year of study, summer semester, 6 credits, compulsory
branch BK-EST , 1. year of study, summer semester, 6 credits, compulsory
branch BK-MET , 1. year of study, summer semester, 6 credits, compulsory
branch BK-SEE , 1. year of study, summer semester, 6 credits, compulsory
branch BK-TLI , 1. year of study, summer semester, 6 credits, compulsory

• Programme IBEP-TZ Bachelor's

branch TZ-IBP , 1. year of study, summer semester, 6 credits, compulsory

• Programme EEKR-CZV lifelong learning

branch ET-CZV , 1. year of study, summer semester, 6 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Exercise in computer lab

14 hours, optionally

Teacher / Lecturer

The other activities

12 hours, optionally

Teacher / Lecturer

eLearning