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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025

Mathematical Analysis A 4

The course is not on the list Without time-table
Code Completion Credits Range Language
01ANA4 Z,ZK 9 4P+4C Czech
Vztahy:
It is a condition for enrolment in course 01ANA4 that the student must have received credit for the course 01ANA3 in a previous semester
Garant předmětu:
Lecturer:
Tutor:
Supervisor:
Department of Mathematics
Synopsis:

Inverse and implicit functions, constrained extrema, measure and integration theory, contour and surface integrals.

Requirements:

01ANA3, mathematical analysis and linear algebra on the level of the first year courses at FNSPE.

Syllabus of lectures:

1. Inverse and implicit function theorem.

2. Constrained extrema.

3. Measure theory: measure, outer measure, premeasure, Carathéodory construction, Borel measures, Lebesgue--Stieltjes measures.

4. Integration theory: measurable mappings, abstract Lebesgue integral w.r.t. a measure, monotone convergence, Lebesgue dominated convergence theorem, interchanging conditions, product measure, Tonelli--Fubini theorem, Lebesgue measure in the Euclidean space, change of variables.

5. Lebesgue spaces, Hölder and Minkowski inequality, completeness.

6. Curve and parametrized surface, contour and surface integrals of the first and second kind, Green theorem, Gauss theorem, Stokes theorem.

Syllabus of tutorials:

0. Quadratic functions, quadrics.

1. Implicit functions.

2. Constrained extrema.

3. Integration of multivariable functions.

4. Parametric integrals, Gamma and Beta functions.

5. Contour and surface integration.

Study Objective:
Study materials:

Literature:

[1] Chp. 3-6. in Czech lecture note „F. Štampach: Matematická analýza A3 a A4“ available at http://stampach.xyz/fjfi_stud.html

Further recommended literature:

[2] W. Rudin: Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987.

[3] B. P. Demidovich: Problems in mathematical analysis. Translated from the Russian by G. Yankovsky. Third printing. Mir Publishers, Moscow, 1973.

[4] G. B. Folland: Real Analysis: Modern Techniques and Their Applications, 2nd edition, A Willey-Interscience Publication, 1999.

[5] G. B. Folland: Advanced calculus, Pearson, 2001.

Note:
Further information:
stampach.xyz
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2024-05-28
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