Course detail

Mathematics 2

FEKT-BKC-MA2Acad. year: 2019/2020

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

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Edita Kolářová, Ph.D.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

Students will be acquainted with some exact methods for differential equation solving and with the grounding of technique for formalized solution of task of the application type using Laplace, Fourier and Z transforms.

Prerequisites

The subject knowledge on the secondary school level and BMA1 course. For having a facility for subject matter is needed to can determined domains of usual functions of one real variable, to understanding of a concept ot the limit of functions of one real variable and a concept numerical sequences and their limits and to solve conrete standard tasks. Further there is needed the knowledge of rules for derivations of real functions of one variable, the knowledge of basic methods of integrations - the integration per partes, by the method of substitution at the indefinit and definit integral have a facilitu for applications on tasks with respect to extent of the teaching text BMA1. Knowledges of infinite numerical series and some basic criteria of their convergence is also required.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations. They consists into lectures according the contents of a subject matter and in the solving of examples, as well as in the practising of other examples containing in the teaching materials of the topic.

Assesment methods and criteria linked to learning outcomes

Maximum 20 points for individual assignments during the semester (at least 5 points for the course-unit credit); maximum 80 points for a written exam. The exam is focused on verification of the student's knowledge in the issue of solving differencial equations, the complex calculus,
the Fourier series expansion, the Laplace and the Z-transform.

Course curriculum

1. Calculus of the more variable functions.
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

Not applicable.

Aims

To extent the student knowlidges on methods of functions of several variables and onto application of partial derivatives. Further, in the other part, to aquiant students with some elementary methods for solving the ordinary differential equations and to make possible a deeper inside into the theory of functions of a complex vairiable, the methods of which are a necessary theoretical equipment of a student of all electrotechnical disciplines. Finally, to provide students by abillity to solve usual tasks by methods of Laplace, Fourier and Z transforms.

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

The tutorials are not obligatory.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

KOLÁŘOVÁ, E., Matematika 2, Sbírka úloh, FEKT VUT v Brně 2009 (CS)
SVOBODA, Z., VÍTOVEC, J., Matematika 2, FEKT VUT v Brně 2015 (CS)

Type of course unit

Lecture

39 hod., optionally

Teacher / Lecturer

doc. RNDr. Edita Kolářová, Ph.D.

Syllabus

1. Multivariable functions (limit, continuity). Partial derivatives, gradient.
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.

Fundamentals seminar

6 hod., compulsory

Teacher / Lecturer

doc. RNDr. Edita Kolářová, Ph.D.

Computer-assisted exercise

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Edita Kolářová, Ph.D.

Syllabus

Individual topics in accordance with the lecture.

VUT

Faculties

University Institutes

Parts

Mathematics 2

Type of course unit