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.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

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.

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 three 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).

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.

Work placements

Not applicable.

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 three 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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Svoboda, Z., Vítovec, J.: Matematika 2, FEKT VUT v Brně, 2014, s. 1-189. (CS)
Kolářová, E.: Matematika 2, Sbírka úloh, FEKT VUT v Brně, 2009, s. 1-83. (CS)

Not applicable.

Classification of course in study plans

• Programme BPC-BTB Bachelor's, 1. year of study, summer semester, compulsory
• Programme BIT Bachelor's, 2. year of study, summer semester, elective

• Programme IT-BC-3 Bachelor's

branch BIT , 2. year of study, summer semester, elective

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

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.

Fundamentals seminar

4 hours, compulsory

Teacher / Lecturer

Computer-assisted exercise

18 hours, compulsory

Teacher / Lecturer

Syllabus

Individual topics in accordance with the lecture.

Project

4 hours, compulsory

Teacher / Lecturer

Syllabus

Individual topics in accordance with the lecture.