Course detail

# Mathematics 2

Differential calculus of functions of several variables, limit, continuity, partial and directional derivatives, gradient, differential, tangent plane, functions given implicitly. Ordinary differential equations, existence and uniqueness of solutions, separated and linear first order equations, n-th order equations with constant coefficients. Differential calculus in complex domain, holomorphic function, derivative. Integral calculus in complex domain, curve integral, Cauchy theorem, Cauchy formula, Laurent series, singular points, residues, residual theorem. Laplace and Fourier transform, special functions, periodic functions, Fourier series. Differential equations, Z-transform. Continuous-time signals, signal spectrum. Systems and their mathematical model. Solution of input-output equation by Laplace transform. Pulse and frequency response.

Learning outcomes of the course unit

At the end of the course students should be able to know the basic concepts and corresponding context, as well as:

- be able to find and draw the domain of the function of two variables;
- compute partial derivatives of arbitrary order for any (even implicitly) function of several variables;
- find the tangent plane to the surface specified by the function of two variables;
- solve separated and linear first order differential equations;
- solve the n-th order differential equation with constant coefficients including the special right-hand side;
- decompose a complex function into a real and imaginary component and determine the functional values ​​of complex functions;
- find the second component of a complex holomorphic function and determine this function in a complex variable including its derivative;
- calculate the integral of a complex function across a curve by parameterizing the curve, Cauchy theorem or Cauchy formula;
- be able to find singular points of complex functions and calculate their residues;
- calculate the integral of a complex function by means of a residual theorem;
- solve by the Laplace transform the n-th order differential equation with constant coefficients;
- find the Fourier series of the periodic function;
- solve by means of Z-transformation n-th order differential equation with constant coefficients;
- be familiar with the basic concepts of signal and systems theory, including the corresponding mathematical models.

Prerequisites

The knowledge on the secondary school level and the course of Mathematics 1 is required. In order to master the subject matter it is necessary to be able to determine the definition fields of common functions of one variable, understand the concept of limits of one variable function, numerical sequence and its limits. Further it is necessary to know the rules for derivation of real functions of one variable, knowledge of basic methods and methods of integration (decomposition into partial fractions, integration by parts, substitution method) for indefinite and definite integral and to be able to apply them to problems in the scope of Mathematics 1. Knowledge of infinite number series and basic criteria of their convergence as well as power series and search for fields of their convergence are 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 include lectures, numerical exercises and computer-aided exercises.

Assesment methods and criteria linked to learning outcomes

Maximum 30 points per semester for two written tests. The criterium of course-unit credit is awarded on condition of having at least 10 points from semester exercises.

The condition for passing the exam is to obtain at least 50 points out of a total of 100 possible (30 can be obtained for work in the semester, 70 can be obtained at the final written exam).

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Differential calculus of real functions of several variables, limit, continuity, partial and directional derivatives, gradient, differential, tangent plane, functions given implicitly.
2. Ordinary differential equations. Basic concepts, existence and uniqueness of solution of differential equation. First order differential equations, especially separated and linear.
3. Linear differential equations of n-th order with constant coefficients including special right side.
4. Complex function, derivative of complex function, holomorphic function.
5. Integral calculus in a complex domain, parametrization of a curve. Calculation of integral by parametrization of curve and using Cauchy theorem and Cauchy formula.
6. Laurent series, singular points and their classification, the concept of residuals and integral calculations using the residual theorem.
7. Direct and backward Laplace transform. Grammar of transformation. Use of Laplace transform in solving differential equations.
8. Direct and reverse Fourier transform. Grammar of transformation. Utilization of transformation.
9. Mathematical apparatus for signal description. Distribution, special functions, periodic functions and Fourier series.
10. Direct and reverse Z-transform. Grammar of transformation. Differential equations and the use of Z-transform in solving difference equations.
11. Signals and their classification. Continuous-time signals, periodic and harmonic signals, aperiodic signals, signal spectrum.
12. Systems - introduction of the concept and classification. Mathematical model of continuous-time system and solution of input-output equation by Laplace transform. Pulse and frequency response.
13. Relations between systems - serial, parallel connection of systems, feedback. Stability of systems.

Aims

The aim of the course is to acquaint students with basic differential calculus of functions of several variables and with general methods of solving ordinary differential equations. Another point is to teach students how to use mathematical transformations (Laplace, Fourier and Z-transformation) and thus give them a guide to alternative solutions of differential and difference equations that are widely used directly in technical fields. To learning an elements of complex analyzes (especially basic methods of integration in a complex field) offers a good tool for solving specific problems in electrical engineering. The last aim is to explain basic concepts of signal theory (e. g. deterministic signal, signal with continuous time, discrete signal) and systems, further to describe the mathematical model of system with continuous time (with the input-output model) using the previous mathematical apparatus. This final part prepared students to study in the follow-up subjects, which discuss in detail the issues discussed.

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

Numeric exercises and computer-aided exercises are required. Any absence must be duly apologized and the studied subject must be completed. During the semester two written tests with a total of 30 points are written. Specification of controlled education, way of implementation is specified by guarantor's regulation updated for every academic year.

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

Exercise

6 hours, compulsory

Teacher / Lecturer

seminars in computer labs

16 hours, compulsory

Teacher / Lecturer

The other activities

4 hours, compulsory

Teacher / Lecturer

eLearning