Course detail

Mathematics 2

FEKT-BPA-MA2Acad. year: 2024/2025

Functions of many variables, gradient. Ordinary differential equations, basic notions, some basic methods of solution.

Differential calculus in the complex domain, derivative of complex functions, Cauchy-Riemann conditions, holomorphic functions.

Integral calculus in the complex domain, Cauchy theorem, Cauchy formula, Laurent series, singular points, residue theorem.

Laplace transform and its applications. Fourier series and Fourier transform. Z transform, discrete systems, difference equations.

Language of instruction

English

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Michal Novák, Ph.D.

Department

Department of Mathematics (UMAT)

Offered to foreign students

Of all faculties

Entry knowledge

Knowledge of secondary school mathematics and subject matter of BPA-MA1. Students should be able to determine domains of definition of elementary functions of one real variable, understand the notion limit of functions of one real variable and a notion of number series. The skill to differentiate functions of one real variable and find primitive functions to elementary functions is needed. Students should be familiar with the concept of definite integral.

Rules for evaluation and completion of the course

Two tests (13 points each) and 1 project (4 points). In order to get the credit at least 15 points are needed in total. The exam is awarded maximum 70 points. In order to successfully pass the subject, the credit and at least 50 points in total are needed.

Conditions for the tests and the exam:

No calculators or other electronic devices are allowed.
You may bring 1 A4 sheet of paper with Laplace and 1 A4 sheet of paper with Z-tranforms formulas (in both cases prints from the teaching texts only).

Aims

To introduce students to functions of more variables, elementary method of solving ordinary differential equations, functions of complex variable and Laplace, Fourier and Z-transforms and Fourier series.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

GARNER, L.E.: Calculus and Analytical Geometry. Brigham Young University, Dellen publishing Company, San Francisco,1988, ISBN 0-02-340590-2. (EN)
SVOBODA, Z., VÍTOVEC, J., Matematics 2, FEKT VUT v Brně 2015 (EN)

Type of course unit

Lecture

39 hod., optionally

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

1. Functions of more variables (limit, continuity). Partial derivatives, gradient.
2. First order ordinary differential equations (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. Elementary functions in the complex domain.
5. Differential calculus in the complex domain, Caychy-Riemann conditions, holomorphic function.
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. Laplace transform.
9. Inverse Laplace transform, applications.
10. Fourier series (trigonometric and exponential forms, basic properties).
11. Direct and inverse Fourier transforms.
12. Direct and inverse Z transforms.
13. Difference eqautions solved using Z transform.

Fundamentals seminar

6 hod., compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

See lectures.

Exercise in computer lab

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

See lectures.

Project

6 hod., compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

See lectures.

Elearning

eLearning: currently opened course

VUT

Faculties

University Institutes

Parts

Mathematics 2

Type of course unit