Subjects(version: 875)
Course, academic year 2016/2017

Mathematical Analysis II - NMAF052
Title: Matematická analýza II Laboratory of General Physics Education (32-KVOF) Faculty of Mathematics and Physics from 2016 to 2016 summer 10 summer s.:4/3 C+Ex [hours/week] unlimited unlimited taught Czech full-time
Guarantor: prof. RNDr. Josef Málek, CSc., DSc. Fyzika Physics > Mathematics for Physicists NMAF034 NMAG204, NMMA201, NMMA202, NMMA203, NMNM201, NNUM105
 Annotation - ---CzechEnglish
Last update: G_F (22.05.2008)
Second part of the basic course of mathematics for the students of general physics (bachelor study). Follows the course NMAF051.
 Aim of the course - ---CzechEnglish
Last update: T_KMA (13.05.2008)

Second part of hte basic course of mathematics for the students of physics (bachelor study). Follows the course MAF033, a simultaneously running course MAF041 is recommended.

 Literature - Czech
Last update: doc. Mgr. Milan Pokorný, Ph.D. (18.02.2019)
• Kopáček J.: Matematika pro fyziky I.,II.,III. Skripta MFF UK
• Kopáček J. a kol. : Příklady z matematiky pro fyziky I., II. Skripta MFF UK
• Jarník J.: Diferenciální počet I.,II
• Jarník J.: Integrální počet I
• Děmidovič V.: Sbírka úloh a cvičení z matematické analýzy (rusky)
• Videozáznamy přednášek
 Teaching methods - Czech
Last update: T_KMA (13.05.2008)

přednáška + cvičení

 Syllabus - ---CzechEnglish
Last update: T_KMA (13.05.2008)

1. Number series and power series

Convergent/oscilatory/divergent number series; convergence criteria for series with non-negative terms and general terms; absolute and relative convergence; product of series. Elementary power series, derivatives and primitives to series. Taylor series.

2. Ordinary differential equations

Solution of an ODE; Cauchy problem for the ODE's; basic existence and uniqueness theorems; scalar equations of the first order - basic methods of finding solutions; linear equations of the nth order - fundamental system, variation of the constant, special right-hand side. Connection to the system of ODEs. Wronskian, Bernoulli and Euler equations.

3. Functions of more than one variable

Metric, norm, open and closed sets, closure, interior, boundary. Convergence, completeness, compactness, separability. Banach and Hilbert spaces. Continuity and uniform continuity, Heine theorem. Continuous functions on a compact set. Contractive mapping. Banach fixed point theorem. Theorem on the solvability of ODE. Limit and continuity. Partial and directional derivatives, total differential. Grad, div and curl. Exact differential equations, integration factor. Chain rule, change of variables. Mean value theorem, Taylor series. Local and global extrema, Lagrange multipliers. Implicit functions.

4. Variational calculus.

Functional, Gateaux derivative, variation. Euler-Lagrange equations.

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