Mathematical Analysis A 4

Relations:

It is not possible to register for the course 01ANA4 and for the course 01ANB4 in the same semester.

It is not possible to register for the course 01ANA4 if the student is concurrently registered for or has already completed the course 01ANB4 (mutually exclusive courses).

It is a condition for enrolment in course 01ANA4 that the student must have received credit for the course 01ANA3 in a previous semester

It is not possible to register for the course 01ANA4 if the student is concurrently registered for or has previously completed the course 01ANB4 (mutually exclusive courses).

Course guarantor:

František Štampach

Lecturer:

František Štampach

Tutor:

František Štampach

Supervisor:

Department of Mathematics

Synopsis:

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

Requirements:

To register for the course: A credit from 01ANA3. A sufficient knowledge of Mathematical Analysis and Linear Algebra as taught in the first year of the FNSPE is required.

To obtain credit (zápočet): No more than 4 absences and active participation in class.

To pass the final exam: Passing the exam from 01ANA3. A minimum of 2 out of 3 points must be obtained from the written part. The student must demonstrate a sufficient understanding of a selected chapter of the covered material during an oral examination (definitions, theorems, proofs).

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.

[6] H. Amann, J. Escher: Analysis I-III, Birkhäuser, 1998, 1999, 2001.

Note:

Further information:

http://stampach.xyz/

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství - Matematická fyzika (PS)
• Matematické inženýrství - Matematická informatika (PS)
• Matematické inženýrství - Matematické modelování (PS)