- systems of sets, Caratheodory theorem,
- d-dimensional Lebesgue measure,
- measurable functions, integral w.r.t. a measure, convergence theorems,
- L^{p} spaces,
- product measures, Fubini's theorem,
- transformation theorem,
- theorem of Radon-Nikodym.
Module MAT-12-2-M-3
Measure Theory and Differential Equations (M, 9.0 LP)
Module Identification
Module Number | Module Name | CP (Effort) |
---|---|---|
MAT-12-2-M-3 | Measure Theory and Differential Equations | 9.0 CP (270 h) |
Basedata
CP, Effort | 9.0 CP = 270 h |
---|---|
Position of the semester | 1 Sem. in SuSe |
Level | [3] Bachelor (Core) |
Language | [DE] German |
Module Manager | |
Lecturers | |
Area of study | [MAT-GRU] Mathematics (B.Sc. year 1 and 2) |
Reference course of study | [MAT-82.276-SG] B.Sc. Business Mathematics |
Livecycle-State | [NORM] Active |
Courses
Type/SWS | Course Number | Title | Choice in Module-Part | Presence-Time / Self-Study | SL | SL is required for exa. | PL | CP | Sem. | |
---|---|---|---|---|---|---|---|---|---|---|
2V+1U | MAT-12-28-K-3 | Measure and Integration Theory
| P | 42 h | 93 h |
U-Schein
| - | PL1 | 4.5 | SuSe |
2V+1U | MAT-12-25-K-3 | Introduction to Ordinary Differential Equations
| P | 42 h | 93 h |
U-Schein
| - | PL1 | 4.5 | SuSe |
- About [MAT-12-28-K-3]: Title: "Measure and Integration Theory"; Presence-Time: 42 h; Self-Study: 93 h
- About [MAT-12-28-K-3]: The study achievement "[U-Schein] proof of successful participation in the exercise classes (ungraded)" must be obtained.
- About [MAT-12-25-K-3]: Title: "Introduction to Ordinary Differential Equations"; Presence-Time: 42 h; Self-Study: 93 h
- About [MAT-12-25-K-3]: The study achievement "[U-Schein] proof of successful participation in the exercise classes (ungraded)" must be obtained.
Examination achievement PL1
- Form of examination: oral examination (20-30 Min.)
- Examination Frequency: each semester
- Examination number: 82026 ("Module Exam Measure Theory and Differential Equations")
Evaluation of grades
The grade of the module examination is also the module grade.
Contents
- first-order differential equations: autonomous first-order differential equations, variation of constants, explicitly solvable cases, initial value problems,
- existence and uniqueness: functional-analytical foundations, Banach fixed-point theorem, Picard-Lindelöf theorem, the continuability of solutions, Peano's existence theorem,
- qualitative behaviour: Gronwall's lemma, continuous dependency on data, upper and lower functions,
- linear differential equations: homogeneous linear systems, matrix exponential function, variation of constants, nth-order differential equations,
- stability: dynamical systems, phase space, Hamiltonian systems, asymptotic behaviour, stability theory according to Lyapunov.
Competencies / intended learning achievements
They know and understand the basic concepts, constructions, results and methods of proof of measure and integration theory. They are able to comprehend the proofs and independently prove or disprove statements. They are able to correctly handle measure-theoretical constructions as well as the Lebesgue integral in a general measure-theoretical context and with regard to its convergence properties. This knowledge is a prerequisite for all advanced courses in the fields of stochastics and functional analysis.
In addition, the students know and understand the basic concepts, statements and methods of the theory of ordinary differential equations, and they master their analytical treatment. By combining results from analysis and linear algebra, they are able to investigate advanced problems and to work on smaller application problems from science and technology using mathematical methods.
In the exercise classes they have acquired a confident, precise and independent handling of the terms, propositions and methods from the lectures.
Literature
- J. Elstrodt: Maß- und Integrationstheorie,
- H. Bauer: Maß- und Integrationstheorie.
- V.I. Arnold: Gewöhnliche Differentialgleichungen,
- L. Grüne, O. Junge: Gewöhnliche Differentialgleichungen,
- H. Heuser: Gewöhnliche Differentialgleichungen,
- J.W. Prüss, M. Wilke: Gewöhnliche Differentialgleichungen und dynamische Systeme,
- W. Walter: Gewöhnliche Differentialgleichungen,
- G. Teschl: Ordinary Differential Equations and Dynamic Systems.
Registration
Requirements for attendance of the module (informal)
Modules:
Requirements for attendance of the module (formal)
References to Module / Module Number [MAT-12-2-M-3]
Course of Study | Section | Choice/Obligation |
---|---|---|
[MAT-82.276-SG] B.Sc. Business Mathematics | [Core Modules (non specialised)] General Mathematics | [P] Compulsory |