|
|
Main menu for Browse IS/STAG
Course info
KMA / MA5
:
Course description
Department/Unit / Abbreviation
|
KMA
/
MA5
|
Academic Year
|
2023/2024
|
Academic Year
|
2023/2024
|
Title
|
Measure and Integral
|
Form of course completion
|
Exam
|
Form of course completion
|
Exam
|
Accredited / Credits
|
Yes,
5
Cred.
|
Type of completion
|
Combined
|
Type of completion
|
Combined
|
Time requirements
|
Lecture
3
[Hours/Week]
Seminar
2
[Hours/Week]
|
Course credit prior to examination
|
Yes
|
Course credit prior to examination
|
Yes
|
Automatic acceptance of credit before examination
|
No
|
Included in study average
|
YES
|
Language of instruction
|
Czech
|
Occ/max
|
|
|
|
Automatic acceptance of credit before examination
|
No
|
Summer semester
|
0 / -
|
0 / -
|
0 / -
|
Included in study average
|
YES
|
Winter semester
|
3 / -
|
0 / -
|
1 / -
|
Repeated registration
|
NO
|
Repeated registration
|
NO
|
Timetable
|
Yes
|
Semester taught
|
Winter semester
|
Semester taught
|
Winter semester
|
Minimum (B + C) students
|
1
|
Optional course |
Yes
|
Optional course
|
Yes
|
Language of instruction
|
Czech
|
Internship duration
|
0
|
No. of hours of on-premise lessons |
|
Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
|
Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
|
Fundamental theoretical course |
No
|
Fundamental course |
No
|
Fundamental theoretical course |
No
|
Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
|
None
|
Preclusive courses
|
N/A
|
Prerequisite courses
|
N/A
|
Informally recommended courses
|
N/A
|
Courses depending on this Course
|
KMA/MMA, KMA/OBM
|
Histogram of students' grades over the years:
Graphic PNG
,
XLS
|
Course objectives:
|
The aim of this course is an introduction to metric spaces and their properties, to the theory of measure and integral and theory of Fourier series. The theoretical tools will be demonstrated on illustrative examples.
|
Requirements on student
|
Demonstrate knowledge of fundaments of theory of measure and integral. The ability to apply theoretical results in solving problems on the topics in the syllabus.
|
Content
|
Chapter 1. Measure and Lebesgue integral
2.1 Fundaments of measure theory
2.2 Measurable functions and integral
2.3 Integrals depending on parameters
2.4 Lebesgue integral in R and functions with bounded variation
Chapter 2. Spaces of integrable functions
2.1 Basic properties
2.2 Completeness, separability
2.3 Mappings in these spaces, continous embeddings
Chapter 3. Fourier series
3.1 Orthogonal a orthonormal systems of functions
3.2 Pointwise and uniform konvergence of Fourier series
|
Activities
|
|
Fields of study
|
|
Guarantors and lecturers
|
-
Guarantors:
Prof. Ing. Petr Girg, Ph.D. (100%),
-
Lecturer:
Doc. RNDr. Jiří Benedikt, Ph.D. (100%),
Prof. Ing. Petr Girg, Ph.D. (100%),
-
Tutorial lecturer:
Doc. RNDr. Jiří Benedikt, Ph.D. (100%),
Prof. Ing. Petr Girg, Ph.D. (100%),
|
Literature
|
-
Recommended:
Rudin, Walter. Analýza v reálném a komplexním oboru. Vyd. 2., přeprac. Praha : Academia, 2003. ISBN 80-200-1125-0.
-
Recommended:
Jarník, Vojtěch. Diferenciální počet II. Praha : Academia, 1976.
-
Recommended:
Jarník, Vojtěch. Integrální počet. II. Praha : Academia, 1976.
-
Recommended:
Nagy, Jozef; Nováková, Eva; Vacek, Milan. Lebesgueova míra a integrál. 1. vyd. Praha : SNTL, 1985.
-
Recommended:
Nagy, Jozef. Vybrané partie z moderní matematiky. Vyd 1. Praha : SNTL, 1976.
-
Recommended:
Kolmogorov, A. N.; Fomin, S.V. Základy teorie funkcí a funkcionální analýzy. Vyd. 1. Praha : SNTL, 1975.
-
On-line library catalogues
|
Time requirements
|
All forms of study
|
Activities
|
Time requirements for activity [h]
|
Contact hours
|
65
|
Preparation for an examination (30-60)
|
55
|
Preparation for comprehensive test (10-40)
|
40
|
Total
|
160
|
|
Prerequisites
|
Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
define and explain basic notions of mathematical analysis in one and/or several dimensions |
explain the definition and basic properties of Newton integral |
explain the definition and basic properties of Riemann integral |
explain basics of Fourier series. |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
calculate indefinite and/or definite integrals (of certain types) in one dimension using integrations-by-parts and/or substitution methods |
calculate multiple integrals using Fubini theorem within Riemann theory |
derive and prove the convergence of Fourier series for piecewise smooth functions |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
|
Learning outcomes
|
Knowledge - knowledge resulting from the course: |
define and explain basic notions of abstract measure theory |
define and explain basic notions of theory of the abstract Lebesgue integration |
define and explain basic notions of theory of the Lebesgue spaces |
define and explain basic notions of theory of Lebesgue integration in R |
define and explain basic notions of theory of Fourier series |
Skills - skills resulting from the course: |
work with abstract structures of measure theory |
use of limit theorems in calculating integrals |
use of the Fubini and Tonelli theorems in calculating multiple integrals |
analyze integrals depending on parameters |
Competences - competences resulting from the course: |
N/A |
|
Assessment methods
|
Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Skills demonstration during practicum |
A) Basics of abstract measure theory.
B) Basics of theory of the abstract Lebesgue integration.
C) Basic theory of the Lebesgue spaces.
D) Lebesgue integration in R.
E) Basic theory of Fourier series. |
Skills - skills achieved by taking this course are verified by the following means: |
Combined exam |
A) Work with abstract structures of measure theory.
B) Use of limit theorems in calculating integrals. Smazat
C) Use of the Fubini and Tonelli theorems in calculating multiple integrals.
D) Analysis of integrals depending on parameters. |
Competences - competence achieved by taking this course are verified by the following means: |
Combined exam |
|
Teaching methods
|
Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture supplemented with a discussion |
Interactive lecture |
Task-based study method |
Skills - the following training methods are used to achieve the required skills: |
Practicum |
Competences - the following training methods are used to achieve the required competences: |
Task-based study method |
|
|
|
|