Course detail

Mathematics 2

FEKT-BMA2Acad. year: 2012/2013

Functions of many variables, gradient. Ordinary differential equations, basic terms, exact methods, examples of 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, convolution, applications. Fourier transform, relation to the Laplace transform, practical usage. Z transform, discrete systems, difference equations.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Chvalina, DrSc.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

Students will be acquainted with some exact and numerical methods for differential equation solving and with the grounding of technique for formalized solution using Laplace, Fourier and Z transforms.

Prerequisites

The subject knowledge on the secondary school level and BMA1 course.

Co-requisites

Not applicable.

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

See the end of the syllable.

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. Fourier series, Fourier integral, Fourier transform, convolution, practical usage.
12. Laplace transform, relation to the Fourier transform, applications.
13. Z transform, discrete systems, difference equations.

Work placements

Not applicable.

Aims

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

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

Not applicable.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Aramanovič I.G., Lunc G.L., Elsgolc L.E.: Funkcie komlexnej premennej operátorový počet - teória stability, Alfa Bratislava 1973
Hlávka J., Klátil J., Kubík S.: Komplexní proměnná v elektrotechnice. SNTL Praha 1990.
Chvalina J., Svoboda Z., Novák M.: Matematika 2
Kolářová, E.:MATEMATIKA 2 Sbírka úloh
Melkes F., Řezáč M.: Matematika 2 (BMA2 et KMA2)
Škrášek J., Tichý Z.: Základy aplikované matematiky II. SNTL Praha 1983.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme EECC Bc. Bachelor's

    branch B-AMT , 1 year of study, summer semester, compulsory
    branch B-MET , 1 year of study, summer semester, compulsory
    branch B-TLI , 1 year of study, summer semester, compulsory
    branch B-SEE , 1 year of study, summer semester, compulsory
    branch B-EST , 1 year of study, summer semester, compulsory

  • Programme EEKR-CZV lifelong learning

    branch EE-FLE , 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. RNDr. Zdeněk Svoboda, CSc.
Mgr. Jiří Vítovec, Ph.D.
doc. RNDr. Edita Kolářová, Ph.D.
prof. RNDr. Jan Chvalina, DrSc.

Syllabus

1. Multivariable functions (limit, continuity). Partial derivatives, gradient.
2. Ordinary differential equations of order 1 (separable equation, linear equation, variation of a constant).
3. Linear differential equation of order n with constant coefficients.
4. Function of complex variable - transform of a complex plane. Basic transcendental functions.
5. Differential calculus in the complex domain, Caychy-Riemann conditions, holomorphic funkction.
6. Integral calculus in the complex domain, the Cauchy theorem, the Cauchy formula.
7. Laurent series, singular points and their classification, residues and residue theorem.
8. Direct Laplace transform, convolution, grammar of the transform.
9. Inverse Laplace transform, pulses, electric circuits.
10. Fourier series (trigonometric and exponential forms, basic properties).
11. Direct and inverse Fourier transforms, relation to the Laplace transform, the pulse nad spectrum widths.
12. Direct and inverse Z transforms.
13. Difference eqautions solved using Z transform.

Exercise in computer lab

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Edita Kolářová, Ph.D.
doc. RNDr. Michal Novák, Ph.D.
Mgr. Jiří Vítovec, Ph.D.
Mgr. Martin Tůma, Ph.D.
doc. RNDr. Zdeněk Svoboda, CSc.
Mgr. Marie Tomšová

Syllabus

Individual topics in accordance with the lecture.