Mathematics 2

• 39 hrs lectures
• 22 hrs exercises
• 4 hrs projects

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.

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.

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.

• Trench, W. F.: Student Solutions Manual for Elementary Differential Equations, Trinity University, 2002, s. 1-283.
• Brown, J. W., Churchill, R. W.: Complex Variables and Applications, McGraw-Hill in New York, 2009, s. 1-468.
• Debnath, L., Bhatta, D.: Integral Transforms and their Applications, Chapman & Hall/CRC New York, 2007, s. 1-668.

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.

Individual topics in accordance with the lecture.

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).

Teaching methods include lectures, numerical exercises and computer-aided exercises.

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.