Course detail

Mathematical Analysis II

FSI-SA2Acad. year: 2024/2025

The course Mathematical Analysis II is directly linked to the introductory course Mathematical Analysis I. It concerns differential and integral calculus of functions in several real variables. Students will acquire a theoretical background that is necessary in solving some particular problems in mathematics as well as in technical disciplines.

Language of instruction

Czech

Number of ECTS credits

7

Mode of study

Not applicable.

Guarantor

doc. Ing. Luděk Nechvátal, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Mathematical Analysis I, Linear Algebra.

Rules for evaluation and completion of the course

Course-unit credit: active attendance at the seminars, successful passing through two written tests (i.e. receiving at least one half of all possible points from each of them).

Exam: will be oral based (possibly will have also a written part). Students are supposed to discuss three selected topics from the lessons. 


Seminars: obligatory.
Lectures: recommended.

Aims

Students should get familiar with basics of differential and integral calculus in several real variables. With such knowledge, various tasks of physical and engineering problems can be solved.
Application of several variable calculus methods in physical and technical problems.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

C. Bray: Multivariable Calculus, CreateSpace Independent Publishing Platform, 2013. (EN)
D. M. Bressoud: Second Year Calculus, Springer, 2001. (EN)
J. Stewart: Multivariable Calculus (8th ed.), Cengage Learning, 2015. (EN)
J. Škrášek, Z. Tichý: Základy aplikované matematiky I a II, SNTL Praha, 1989. (CS)
P. D. Lax, M. S. Terrel: Multivariable Calculus with Applications, Springer, 2017. (EN)
V. Jarník: Diferenciální počet II, Academia, 1984. (CS)
V. Jarník: Integrální počet II, Academia, 1984. (CS)

Recommended reading

J. Karásek: Matematika II, skripta FSI VUT, 2002. (CS)

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 1 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

52 hod., optionally

Teacher / Lecturer

doc. Ing. Luděk Nechvátal, Ph.D.

Syllabus

1. Metric spaces, convergence in a metric space;
2. Complete and compact metric spaces, mappings between metric spaces;
3. Function of several variables, limit and continuity;
4. Partial derivatives, directional derivative, gradient;
5. Total differential, Taylor polynomial;
6. Local and global extrema;
7. Implicit functions, differentiable mappings between higher dimensional spaces;
8. Constrained extrema, double integral;
9. Double integral over measurable sets, triple integral;
10. Substitution in a double and triple integral, selected applications;
11. Plane and space curves, line integrals, Green's theorem;
12. Path independence for line integrals and related notions, space surfaces;
13. Surface integrals, Gauss-Ostrogradsky's theorem and Stokes' theorem.

Exercise

33 hod., compulsory

Teacher / Lecturer

doc. Ing. Luděk Nechvátal, Ph.D.

Syllabus

Seminars are related to the lectures in the previous week.

Computer-assisted exercise

6 hod., compulsory

Teacher / Lecturer

doc. Ing. Luděk Nechvátal, Ph.D.

Syllabus

This seminar is supposed to be computer assisted.