Mathematics - Selected Topics (FSI-RMA)

Academic year 2023/2024
Supervisor: prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:

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.
Learning outcomes and competences:
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 in the extent of the first two years of study.
Course contents:

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.
Teaching methods and criteria:
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:

Classified course-unit credit based on a written test
Controlled participation in lessons:

Missed lessons can be compensated via a written test.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

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

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
Literature - fundamental:
1. Kolmogorov,A.N.,Fomin,S.V.: Elements of the Theory of Functions and Functional Analysis, Graylock Press, 1957, 1961, 2002
2. Rektorys, K.: Variační metody, Academia Praha, 1999
3. Bachman,G., Laerence, N.: Functional analysis, Dover Pub., 1966,2000
Literature - recommended:
1. Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
2. Rektorys, K.: Variační metody, Academia Praha, 1999
3. Veit, J. Integrální transformace: SNTL, Praha 1979
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MET-P full-time study --- no specialisation -- GCr 5 Compulsory 2 1 W
N-PMO-P full-time study --- no specialisation -- GCr 5 Compulsory-optional 2 1 W
N-IMB-P full-time study BIO Biomechanics -- GCr 5 Compulsory-optional 2 1 W
N-IMB-P full-time study IME Engineering Mechanics -- GCr 5 Compulsory 2 1 W