Course detail

# Mathematics 2

Partial differentiation of real functions of several variables, limits, continuity, partial derivatives. Ordinary differential equations and systems, basic concepts, examples of use. Difference equations, basic concepts. Differential calculus in the complex domain, derivative, holomorphic functions. Integral calculus in the complex domain, Cauchy theorem, Cauchy formula, Laurent series, singular points, residue theorem. Laplace transform, convolution of functions. Fourier transform, relation to the Laplace transform, practical usage. Z transform, discrete systems, difference equations. Continuous-time signals, spectra of signals. Systems and their mathematical models. Solving of input-output equation using the Laplace transform. Impulse and frequency characteristic.

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. They obtain knowledges and skills for solving of ordinary differential equations and in the second part necessary knowledges concerning Laplace, Fourier and Z transforms. Further students obtain insight into fundamental concepts of the signal and systems theory, as deterministic signals, signals with continuous time, discrete signals and of mathematical model of a system with continuous time, as well as input-output description using mathematical apliance presented in previous parts of study text.

Prerequisites

The subject knowledge on the secondary school level and AMA1 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 of course AMA1. Knowledges of infinite numerical and power series and some basic criteria of their convergence is also required.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Chvalina, J., Křehlík, Š., Svoboda, Z., Vítovec, J.: Matematika 2, FEKT VUT v Brně, 2014, s. 1-219. (CS)
Kolářová, E.: Matematika 2, Sbírka úloh, FEKT VUT v Brně, 2009, s. 1-83. (CS)
Melkes, F., Řezáč, M.: Matematika 2, FEKT VUT v Brně, 2007, s. 1-168. (CS)
Chvalina, J., Svoboda, Z., Novák, M.: Matematika 2, FEKT VUT v Brně, 2007, s. 1-148. (CS)
Pírko, Z., Veit, J.: Laplaceova transformace, Základy teorie a užití v elektrotechnice, SNTL Praha, 1970, s. 1-248. (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. In particular it consists in lectures, seminars devoted to solvin of concrete tasks, individual consultations and the study of recommended literature.

Assesment methods and criteria linked to learning outcomes

Requirements for completion of a course are specified by a regulation issued by the lecturer responsible for the course and updated for every.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Fuctions of several variables, mappings (limit, continuity). Partial derivatives, gradient.
2. Ordinary differential equations and systems of differential equations. Basic concepts and foundations of qualitative theory (existence and uniqueness of solutions of ODE´s, stability). Linear differential equations of the n-th order with constant coefficients, stability of solutions.
3. Difference equations. Basic concepts and foundations of qualitative theory (existence and uniqueness of solutions of DE´s). Linear difference equations.
4. Functions of a complex variable, derivative of complex functions. Integral calculus in complex domain, Cauchy theorem, Cauchy formula.
5.Laurent series, singular points and their classification, residuum and residua-theorem.
6. Mathematical methods for description of signals. Distribution, harmonic functions, periodical functions and Fourier series.
7. Direct and inverse Fourier transformation. Grammar of transform. Applications.
8.Direct and inverse Laplace transformation. Connection with the Fourier transform. Grammar of transform.
9. Applications of the Laplace transform to solving of differential equations and their systems.
10. Direct and inverse Z-transformation. Using the Z-trasformations for solving of difference equations.
11. Signals and their classifications. Continuous-time signals, periodical and harmonic signal, aperiodical signals, spectra of signals.
12. Sytems - concept and cassification. Mathematical model of a continuous-time system and solving of the input-output equation by Laplace transform.Impulse and frequency characteristic.
13. Connections between systems - serial, parallel connection of systems, feedback. Stability of systems.

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. Further to present explanation of fundamental concepts of the signal and systems theory, as deterministic signals, signals with continuous time, discrete signals and of mathematical model of a system with continuous time, as well as input-output description using mathematical apliance presented in previous parts of study text.

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

During semester two individual projects will be worked out and elaborated two evaluated writing tests with complete sum of 30 points.

Classification of course in study plans

• Programme BTBIO-A Bachelor's

branch A-BTB , 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

Fundamentals seminar

6 hours, compulsory

Teacher / Lecturer

Computer-assisted exercise

16 hours, compulsory

Teacher / Lecturer

The other activities

4 hours, compulsory

Teacher / Lecturer

eLearning