Mathematics - Selected Topics I (FSI-T1K)

Academic year 2018/2019
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 course by the topics necessary for study of physical engineering.
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 the mentioned subjects in physics.
Prerequisites:
Real and complex analysis
Course contents:
The course includes selected topics of functional analysis which are necessary for application in physics. It focuses on functional spaces, orthogonal systems and orthogonal transformations.
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:
Course-unit credit - based on a written test
Exam has a written and an oral part.
Controlled participation in lessons:
Missed lessons can be compensated for via a written test.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 1. Introduction
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
9. Usage of Fourier transform, convolution theorem
10.2D Fourier transform
11.Filtration in space and frequency domain, applications in physics
12. Operators and functionals
13. Variation methods
    Exercise 1. Introduction
2. Metric space
3. Fix point Banach's theorem applications
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
9. Usage of Fourier transform, convolution theorem
10. 2D Fourier transform
11. Filtration in space and frequency domain, applications in physics
12. Operators and functionals
13. Variation methods
Literature - fundamental:
1. Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
2. Lang, S. Real and Functional Analysis. Third Edition, Springer-Verlag 1993
Literature - recommended:
1. Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3A-P full-time study B-FIN Physical Engineering and Nanotechnology -- Cr,Ex 5 Compulsory 1 2 S