MU01003 Mathematical Analysis III

Mathematical Institute in Opava
Winter 2017
Extent and Intensity
4/0/0. 5 credit(s). Type of Completion: zk (examination).
Teacher(s)
Vladimír Averbuch, DrSc. (lecturer)
Guaranteed by
Vladimír Averbuch, DrSc.
Mathematical Institute in Opava
Prerequisites (in Czech)
MU01002 Mathematical Analysis II && ( MU01006 Algebra II || MU01016 Algebra II ) && MU01903 Mathematical Analysis III - Ex
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
Course objectives
The main attention of the third part of the basic course of mathematical analysis is given to normed spaces, Fréche and Gateaux derivatives, chain rule, inverse function theorem and implicite function theorem, higher derivatives, Taylor formula and conditions of extremum of functions, including Lagrange theorem on multiplicators.
Syllabus
  • 1. Normed spaces (normed speces, topology of a normed space, equivalent norms, equivalence of all the norms in finite-dimensional spaces, the natural topology of a finite-dimensional space, basic normes, product of normed spaces, compact sets in a finite-dimensional space, continuity of some basic mappings).
    2. The first derivative (Fréche derivative, Gateaux derivative, directional derivative, differential, their basic properties and relations between them, derivatives of basic mappings, Chain Rule and its corollaries, partial derivatives, continuous differentiability).
    3. Theorems on inverse function and on imlicite function (Banach spaces, contraction lemma, Theorem on inverse function, Theorem on imlicite function).
    4. Higher derivatives (definition and properties of higher derivatives, symmetry of higher derivatives, higher partial derivatives, Taylor formula, extreme problems without constrains, Fermat theorem, necessary conditions and sufficient conditions of the second order for local extremum, extreme problems with constrains, tangent vectors and normal vectors, necessary condition of local extremum in problems with constrains in terms of normal vectors, Lagrange Theorem on multiplicators).
Literature
    recommended literature
  • V. I. Averbuch, M. Málek. Matematická analýza III, IV. MÚ SU, Opava, 2003. URL info
  • W. Rudin. Analýza v reálném a komplexním oboru. Academia, Praha, 1987. info
  • K. Rektorys a spolupracovníci. Přehled užité matematiky. SNTL, Praha, 1968. info
  • V. Jarník. Diferenciální počet I. ČSAV, Praha, 1963. info
  • V. Jarník. Diferenciální počet II. ČSAV, Praha, 1963. info
Language of instruction
English
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
The course is also listed under the following terms Winter 1997, Winter 1998, Winter 1999, Winter 2000, Winter 2001, Winter 2002, Winter 2003, Winter 2004, Winter 2005, Winter 2006, Winter 2007, Winter 2008, Winter 2009, Winter 2010, Winter 2011, Winter 2012, Winter 2013, Winter 2014, Winter 2015, Winter 2016, Winter 2018, Winter 2019, Winter 2020, Winter 2021, Winter 2022.
  • Enrolment Statistics (Winter 2017, recent)
  • Permalink: https://is.slu.cz/course/sumu/winter2017/MU01003