Course detail

# Mathematics 2

FEKT-BPC-MA2AAcad. year: 2021/2022

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

Number of ECTS credits

Mode of study

Guarantor

Department

Learning outcomes of the course unit

- 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

Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

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

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

Aims

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

Recommended optional programme components

Prerequisites and corequisites

Basic literature

Kolářová, E.: Matematika 2, Sbírka úloh, FEKT VUT v Brně, 2009, s. 1-83. (CS)

Recommended reading

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

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.

Computer-assisted exercise

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Syllabus

Project

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