Mathematical Analysis II (FSI-SA2)

Academic year 2022/2023
Supervisor: doc. Ing. Luděk Nechvátal, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:
Application of several variable calculus methods in physical and technical problems.
Prerequisites:
Mathematical Analysis I, Linear Algebra.
Course contents:
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.
Teaching methods and criteria:
The course is lectured through lessons supported by exercises. The content of lessons is focused on a theoretical background of the subject. The exercises have a practical/computational character.
Assesment methods and criteria linked to learning outcomes:
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 have both a written part as well as an oral part, the condition for admission to the oral part is receiving at least one half of all possible points from the written part).
Controlled participation in lessons:
Seminars: obligatory.
Lectures: recommended.
Type of course unit:
    Lecture  13 × 4 hrs. optionally                  
    Exercise  11 × 3 hrs. compulsory                  
    Computer-assisted exercise  2 × 3 hrs. compulsory                  
Course curriculum:
    Lecture 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 Seminars are related to the lectures in the previous week.
    Computer-assisted exercise This seminar is supposed to be computer assisted.
Literature - fundamental:
1. V. Jarník: Diferenciální počet II, Academia, 1984.
2. V. Jarník: Integrální počet II, Academia, 1984.
3. D. M. Bressoud: Second Year Calculus, Springer, 2001.
4. J. Škrášek, Z. Tichý: Základy aplikované matematiky I a II, SNTL Praha, 1989.
5. J. Stewart: Multivariable Calculus (8th ed.), Cengage Learning, 2015.
6. C. Bray: Multivariable Calculus, CreateSpace Independent Publishing Platform, 2013.
7. P. D. Lax, M. S. Terrel: Multivariable Calculus with Applications, Springer, 2017.
Literature - recommended:
1. J. Karásek: Matematika II, skripta FSI VUT, 2002.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
CŽV full-time study CZV Bases of Mechanical Engineering -- Cr,Ex 7 Compulsory 1 1 S
B-MAI-P full-time study --- no specialisation -- Cr,Ex 7 Compulsory 1 1 S