SubjectsSubjects(version: 882)
Course, academic year 2020/2021
  
Mathematical Analysis 2 - NMAI055
Title: Matematická analýza 2
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Martin Klazar, Dr.
doc. Mgr. Robert Šámal, Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Real and Complex Analysis
Incompatibility : NMAX055
Interchangeability : NMAX055
Is incompatible with: NMAX055
Is interchangeable with: NMAX055
Annotation -
Last update: doc. RNDr. Pavel Töpfer, CSc. (26.01.2018)
The second part of the mathematical analysis course for students of computer science with the focus on the differential function of several variables. Students will learn to use partial derivatives and differentials to analyze multivariate functions (extremes, approximations). The knowledge of integrals obtained in Mathematical Analysis 1 will be deepened and extended. A comprehensive framework for the whole study will be provided by a study of metric spaces. It will be assumed that the students understand material covered by Mathematical Analysis 1.
Course completion requirements -
Last update: prof. RNDr. Aleš Pultr, DrSc. (25.09.2020)

The credit will be given for active participation in tutorials, homeworks and successful completion of tests (the exact weight of each of these criteria is determined by the

TA).

The nature of the first two requirements does not make it possible for repeated attempts for the credit.

The teacher can, however, determine alternative conditions for replacing the missing requirements.

The exam oral, possibly in distance form. Obtaining the credit is necessary before the final exam.

Literature -
Last update: doc. RNDr. Martin Klazar, Dr. (26.11.2012)

T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974 (2nd edition).

Ch. Ch. Pugh, Real Mathematical Analysis, Undergraduate Text in Mathematics, Springer, 2002.

T. Tao, Analysis I, Hindustan Book Agency, 2006.

T. Tao, Analysis II, Hindustan Book Agency, 2006.

V. A. Zorich, Mathematical Analysis I, Universitext, Springer, 2004.

V. A. Zorich, Mathematical Analysis II, Universitext, Springer, 2004.

Requirements to the exam -
Last update: Mgr. Tereza Klimošová, Ph.D. (18.02.2019)

Exam will be written. A student must obtain credit from the tutorial to take the exam. The material for the exam corresponds to the syllabus to the extent to which topics were covered during lectures and tutorials and in reading assignments. Ability to generalize and apply theoretical knowledge to solving problems will be required.

Syllabus -
Last update: doc. RNDr. Pavel Töpfer, CSc. (26.01.2018)

More details of integrals of functions of one variable: partial fractions decomposition, simple standard substitutions, fundamental theorem of calculus.

Integrals of functions of several variables: Riemann's integral on a box, Fubini's theorem, calculation by repeated integration.

Differential calculus of functions of several variables:

  • Partial derivatives, differential, C^1 functions.
  • Rules for calculation (chain rule).
  • Use: extremes on an open set, saddle points classification, implicitly defined functions, constrained extremes (Lagrange multipliers).
  • Informatively line integral.
  • Extremes of continuous function on a compact set.

Metric spaces: a framework for the whole analysis, limits, continuity, informatively topology.

 
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