Course detail

Mathematics - Selected Topics

FSI-RMAAcad. year: 2019/2020

The course familiarises studetns with selected topics of mathematics which are necessary for study of mechanics and related subjects. It deals with spaces of functions, orthogonal systems of functions, orthogonal transformations and numerical methods used in mechanics.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Basic knowledge of functional analysis, metric, vector, unitary spaces, Hilbert space, orthogonal systems of functions, orthogonal transforms, Fourier transform and spectral analysis, application of mentioned subjects in mechanics and physics.

Prerequisites

Mathematical analysis and linear algebra

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Course-unit credit - based on a written test
Exam has a written and (possibly) and oral part.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to extend students´knowledge acquired in the basic mathematical courses by the topics necessary for study of mechanics and related subjects.

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

Missed lessons can be compensated for via a written test.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Bachman,G., Laerence, N.: Functional analysis, Dover Pub., 1966,2000
Kolmogorov,A.N.,Fomin,S.V.: Elements of the Theory of Functions and Functional Analysis, Graylock Press, 1957, 1961, 2002
Rektorys, K.: Variační metody, Academia Praha, 1999

Recommended reading

Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
Rektorys, K.: Variační metody, Academia Praha, 1999
Veit, J. Integrální transformace: SNTL, Praha 1979

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-IMB , 1 year of study, winter semester, compulsory
    branch M-MET , 1 year of study, winter semester, compulsory
    branch M-PMO , 1 year of study, winter semester, compulsory-optional

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Mapping, binary relations, equivalence, factor set
2. Metric space
3. Contraction, fix point Banach's theorem
4. Vector space, base, dimension, Vector spaces of functions
5. Unitary space orthogonal a orthonormal spaces
6. Hilbert space, L2 and l2 space
7. Orthogonal bases, Fourier series
8. Orthogonal transforms, Fourier transform, spectral analysis
9. Usage of Fourier transform, convolution theorem, filters
10. 2D Fourier transform and its application
11. Filtration in space and frequency domain, applications in physics and mechanics
12. Operators and functionals
13. Variation methods

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Revision of selected topics
2. Metric space
3. Contraction, fix point Banach's theorem
4. Vector space, base, dimension, Vector spaces of functions
5. Unitary space orthogonal a orthonormal spaces
6. Hilbert space, L2 and l2 space
7. Orthogonal bases, Fourier series
8. Orthogonal transforms, Fourier transform, spectral analysis
9. Usage of Fourier transform, convolution theorem, filters
10. 2D Fourier transform and its application
11. Filtration in space and frequency domain, applications in physics and mechanics
12. Operators and functionals
13. Variation methods