Mathematical Analysis A 3

Relations:

It is not possible to register for the course 01ANA3 and for the course 01ANB3 in the same semester.

It is not possible to register for the course 01ANA3 if the student is concurrently registered for or has already completed the course 01ANB3 (mutually exclusive courses).

In order to register for the course 01ANA3, the student must have successfully completed the course 01MAN2 in a previous semester.

In order to register for the course 01ANA3, the student must have successfully completed the course 01LAL2 in a previous semester.

It is not possible to register for the course 01ANA3 if the student is concurrently registered for or has previously completed the course 01ANB3 (mutually exclusive courses).

In order to register for the course 01ANA4, the student must have received credit for the course 01ANA3 in a previous semester.

Course guarantor:

František Štampach

Lecturer:

František Štampach

Tutor:

Radek Fučík, František Štampach

Supervisor:

Department of Mathematics

Synopsis:

Function sequences and series, introduction to topology and metric spaces, differential calculus of functions of several variables.

Requirements:

To register for the course: 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: 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. Function sequences and series: pointwise and uniform convergence, interchange conditions for limits, derivatives and

integrals, power and Taylor series, trigonometric and Fourier series.

2. Topology and metric spaces: basic notions, continuity, compact, connected and complete spaces.

3. Differential calculus of function of several variables: derivative, directional and partial derivatives, derivatives of higher order, mean-value theorems,

extrema of functions of several variables.

Syllabus of tutorials:

0. Repetition: Convergence of the Riemann integral.

1. Uniform convergence of function sequences and series.

2. Fourier series.

3. Topology.

4. Multivariable calculus, continuity and differentiability.

5. Extrema of functions of several variables.

Study Objective:

Study materials:

Literature:

[1] Chp. 1-3. 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.

[7] M. Moskowitz, F. Paliogiannis: Functions of several real variables, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.

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)