## MUC15 Mathematical Analysis 4

Faculty of Science
Spring 2022
Extent and Intensity
2/2/0. 4 credit(s). Type of Completion: zk (examination).
Taught in person.
Teacher(s)
prof. RNDr. Zuzana Došlá, DSc. (lecturer)
Guaranteed by
prof. RNDr. Zuzana Došlá, DSc.
Department of Mathematics and Statistics - Departments - Faculty of Science
Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science
Prerequisites
Mathematical analysis 1. Mathematical analysis 2. Mathematical analysis 3.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 7 fields of study the course is directly associated with, display
Course objectives
The aim of the course is to familiarize the student with the basic parts of metric spaces, integral calculus in more variables and systems of differential equations. After passing the course, the student will be able to solve selected types of integral calculus and systems of differential equations. Student will be able to understand and explain basic notions and techniques from metric spaces including the connections with other various parts of mathematic.
Learning outcomes
After passing the course, the student will be able:
to understand the basic metric spaces;
to understand and interpret the Banach fixed point theorem;
to know Riemann integral of functions in more variables;
to use effective techniques of integrating more variable functions;
to apply acquired pieces of knowledge for the solution of specific problems, mainly in geometry;
understand basic concepts connected to systems of differential equations;
use systems of differential equations to model real world phenomenons.
Syllabus
• Metric spaces: the notion of metric, convergence in the metrix space, Banach fixed point theorem and its application in numerical calculation and in economics.
• Integral calculus of the functions in more variables: measure in the plane and the space, double and triple integrals (Fubini theorem, transformations in polar and cylindric coordinates ), geometric application.
• Autonomous systems in plane: nullclines, stationary points and their classification, applications.
Literature
recommended literature
• KALAS, Josef and Jaromír KUBEN. Integrální počet funkcí více proměnných (Integral calculus of functions of several variables). 1st ed. Brno: Masarykova univerzita, 2009. 278 pp. ISBN 978-80-210-4975-8. info
• DOŠLÁ, Zuzana and Ondřej DOŠLÝ. Metrické prostory : teorie a příklady. 2. přeprac. vyd., Dotisk se. Brno: Masarykova univerzita, 2000. [iii], 83. ISBN 8021013281. info
• BRAUN, Martin. Differential equations and their applications : an introduction to applied mathematics. Fourth edition. New York: Springer, 1993. 578 stran. ISBN 3540978941. info
Teaching methods
Lectures on the given themes.
Assessment methods
Written and oral examination. For admission to exam student need to submit three homeworks.
Language of instruction
Czech