Course detail

# Mathematics 2

FEKT-BKC-MA2Acad. year: 2022/2023

Functions of many variables, gradient. Ordinary differential equations, basic terms, exact methods, examples of use. Differential calculus in the complex domain, derivative, Caucy-Riemann conditions, holomorphic functions. Integral calculus in the complex domain, Cauchy theorem, Cauchy formula, Laurent series, singular points, residue theorem. Laplace transform, convolution, applications. Fourier transform, relation to the Laplace transform, practical usage. Z transform, discrete systems, difference equations.

Language of instruction

Number of ECTS credits

Mode of study

Guarantor

Department

Learning outcomes of the course unit

- to compute the partial derivatives of the functions of more variables and use the formulas for the gradient and the tangential plane;

- to distinguish between the separable and linear differential equations and also to solve them;

- to solve linear differential equations of higher order with a special right hand side;

- to figure out from the Cauchy Riemann conditions, if the complex function is holomorfic or not, and to derive the holomorfic funcions;

- to compute, using the definition, the integral from the complex function through a curve, to apply the Cauchy theorem for the integral of the holomorfic funcion;

- to establish the poles and to calculate a residue at asimple and at a pole of higher order, to apply the residue theorem for the integral of the meromorfic funcion;

- to solve differential equations by the Laplace transform;

- to find the real Fourier series of an odd, even and a general function, expand a function to sine or cosine series;

- to solve difference equations by the Z- transform.

Prerequisites

Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Course curriculum

2. Ordinary differential equations, basic terms.

3. Solutions of linear differential equations of first order.

4. Homogenius linear differential equations of higher order.

5. Solutions of nonhomogenious linear differential equations with constant coefficients.

6. Differential calculus in the complex domain, derivative.

7. Caucy-Riemann conditions, holomorphic functions.

8. Integral calculus in the complex domain, Cauchy theorem, Cauchy formula.

9. Laurent series, singular points.

10. Residue theorem.

11. Laplace transform, convolution, Heaviside theorems, applications.

12. Fourier transform, relation to the Laplace transform, practical usage.

13. Z transform, discrete systems, difference equations.

Work placements

Aims

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

Recommended optional programme components

Prerequisites and corequisites

Basic literature

SVOBODA, Z., VÍTOVEC, J., Matematika 2, FEKT VUT v Brně 2015 (CS)

Recommended literature

Elearning

**eLearning:**currently opened course

Classification of course in study plans

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

2. Ordinary differential equations of order 1 (separable equation, linear equation, variation of a constant).

3. Linear differential equation of order n with constant coefficients.

4. Function of complex variable - transform of a complex plane. Basic transcendental functions.

5. Differential calculus in the complex domain, Caychy-Riemann conditions, holomorphic funkction.

6. Integral calculus in the complex domain, the Cauchy theorem, the Cauchy formula.

7. Laurent series, singular points and their classification, residues and residue theorem.

8. Direct Laplace transform, convolution, grammar of the transform.

9. Inverse Laplace transform, pulses, electric circuits.

10. Fourier series (trigonometric and exponential forms, basic properties).

11. Direct and inverse Fourier transforms, relation to the Laplace transform, the pulse nad spectrum widths.

12. Direct and inverse Z transforms.

13. Difference eqautions solved using Z transform.

Computer-assisted exercise

Teacher / Lecturer

Syllabus

Project

Teacher / Lecturer

Syllabus

Elearning

**eLearning:**currently opened course