Course detail

# Mathematics 2

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.

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 of task of the application type using Laplace, Fourier and Z transforms.

Prerequisites

The subject knowledge on the secondary school level and BMA1 course. For having a facility for subject matter is needed to can determined domains of usual functions of one real variable, to understanding of a concept ot the limit of functions of one real variable and a concept numerical sequences and their limits and to solve conrete standard tasks. Further there is needed the knowledge of rules for derivations of real functions of one variable, the knowledge of basic methods of integrations - the integration per partes, by the method of substitution at the indefinit and definit integral have a facilitu for applications on tasks with respect to extent of the teaching text BMA1. Knowledges of infinite numerical series and some basic criteria of their convergence is also required.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

KOLÁŘOVÁ, E., Matematika 2, Sbírka úloh, FEKT VUT v Brně 2009 (CS)
MELKES, F., ŘEZÁČ, M., Matematika 2, FEKT VUT v Brně 2002 (CS)
PÍRKO, Z., VEIT, J., Laplacova transformace. Základy teorie a užití v praxi. SNTL Praha 1970 (CS)
ARAMOVIČ, I. G., LUNC, G. L. a El´SGOLC, L. E., Funkcie komplexnej premennej, operátorový počet, teória stability. Alfa Bratislava 1973. (SK)
SVOBODA, Z., VÍTOVEC, J., Matematika 2, FEKT VUT v Brně 2015 (CS)
Zdeněk Svoboda, Jiří Vítovec: Matematika 2, FEKT VUT v Brně

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. They consists into lectures according the contents of a subject matter and in the solving of examples, as well as in the practising of other examples containing in the teaching materials of the topic.

Assesment methods and criteria linked to learning outcomes

During the semester, students elaborate two evaluated projects consisting in solving individual numerical problems evaluated by the teacher, one paper will be written.
Testing will be a distance or personal form, depending on the epidemiological situation.

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

To extent the student knowlidges on methods of functions of several variables and onto application of partial derivatives. Further, in the other part, to aquiant students with some elementary methods for solving the ordinary differential equations and to make possible a deeper inside into the theory of functions of a complex vairiable, the methods of which are a necessary theoretical equipment of a student of all electrotechnical disciplines. Finally, to provide students by abillity to solve usual tasks by methods of Laplace, Fourier and Z transforms.

Classification of course in study plans

• Programme BPC-AUD Bachelor's

specialization AUDB-ZVUK , 1. year of study, summer semester, 6 credits, compulsory
specialization AUDB-TECH , 1. year of study, summer semester, 6 credits, compulsory

• Programme BPC-AMT Bachelor's, 1. year of study, summer semester, 6 credits, compulsory
• Programme BPC-EKT Bachelor's, 1. year of study, summer semester, 6 credits, compulsory
• Programme BPC-IBE Bachelor's, 1. year of study, summer semester, 6 credits, compulsory
• Programme BPC-MET Bachelor's, 1. year of study, summer semester, 6 credits, compulsory
• Programme BPC-SEE Bachelor's, 1. year of study, summer semester, 6 credits, compulsory
• Programme BPC-TLI Bachelor's, 1. year of study, summer semester, 6 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Fundamentals seminar

6 hours, compulsory

Teacher / Lecturer

Computer-assisted exercise

14 hours, compulsory

Teacher / Lecturer

eLearning

eLearning: opened course