Course detail

Mathematics - Selected Topics

FSI-RMAAcad. year: 2023/2024

The course familiarises studetns with selected topics of mathematics which are necessary for study of mechanics, mechatronics  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)

Entry knowledge

Mathematical analysis and linear algebra in the extent of the first two years of study.

Rules for evaluation and completion of the course

Classified course-unit credit based on a written test


Missed lessons can be compensated via a written test.

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.


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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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
Bachman,G., Laerence, N.: Functional analysis, Dover Pub., 1966,2000

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

eLearning

eLearning: currently opened course

Classification of course in study plans

  • Programme N-IMB-P Master's

    specialization BIO , 1. year of study, winter semester, compulsory-optional
    specialization IME , 1. year of study, winter semester, compulsory

  • Programme N-MET-P Master's, 1. year of study, winter semester, compulsory
  • Programme N-PMO-P Master's, 1. year of study, winter semester, compulsory-optional

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Revision of selected topics
2. Metric space, complete metric space
3. Contraction, fix-point Banach's theorem and its applications
4. Vector space, base, dimension, isomorphism
5. Automorphism of vector spaces, eigenvectors and eigenvalues
6. Unitary space orthogonal a orthonormal bases
7. Hilbert space, L2 and l2 spaces
8. Orthogonal bases, Fourier series
9. Complex Fourier series, discrete Fourier transform
10. Usage of Fourier transform, convolution theorem
11. L2 space for functions of more variable
12. Operators and functionals in Hilbert space
13. Applications

Exercise

26 hours, compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Revision of selected topics
2. Metric space, complete metric space
3. Contraction, fix-point Banach's theorem and its applications
4. Vector space, base, dimension, isomorphism
5. Automorphism of vector spaces, eigenvectors and eigenvalues
6. Unitary space orthogonal a orthonormal bases
7. Hilbert space, L2 and l2 spaces
8. Orthogonal bases, Fourier series
9. Complex Fourier series, discrete Fourier transform
10. Usage of Fourier transform, convolution theorem
11. L2 space for functions of more variable
12. Operators and functionals in Hilbert space
13. Applications

eLearning

eLearning: currently opened course