Course: Mathematical Analysis III

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Course title Mathematical Analysis III
Course code MU/01003
Organizational form of instruction Lecture
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 5
Language of instruction Czech
Status of course Compulsory
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Course availability The course is available to visiting students
Lecturer(s)
  • AVERBUCH Vladimír, DrSc.
Course content
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).

Learning activities and teaching methods
unspecified
Recommended literature
  • K. Rektorys a spolupracovníci. Přehled užité matematiky. SNTL, Praha, 1968.
  • V. I. Averbuch, M. Málek. Matematická analýza III, IV. MÚ SU, Opava, 2003.
  • V. Jarník. Diferenciální počet I. ČSAV, Praha, 1963.
  • V. Jarník. Diferenciální počet II. ČSAV, Praha, 1963.
  • W. Rudin. Analýza v reálném a komplexním oboru. Academia, Praha, 1987.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Mathematical Institute in Opava Mathematics (2014-IVT) Mathematics courses 2 Winter
Mathematical Institute in Opava Mathematics (2014-F) Mathematics courses 2 Winter
Mathematical Institute in Opava Mathematics (2-F) Mathematics courses 2 Winter
Faculty of Philosophy and Science in Opava Astrophysics (1) Physics courses - Winter
Faculty of Philosophy and Science in Opava Astrophysics (2) Physics courses - Winter
Mathematical Institute in Opava Applied Mathematics (2014-F) Mathematics courses 2 Winter
Mathematical Institute in Opava Mathematical Analysis (1-IVT) Mathematics courses 2 Winter
Mathematical Institute in Opava Mathematics (1-IVT) Mathematics courses 2 Winter
Mathematical Institute in Opava Applied Mathematics (2014-IVT) Mathematics courses 2 Winter