Course details

# Selected Parts from Mathematics 2

IVP2 FEEC BUT BPC-VPM Acad. year 2019/2020 Summer semester 5 credits

The aim of this course is to introduce the basics of calculation of local, constrained and absolute extrema of functions of several variables, double and triple integrals, line and surface integrals in a scalar-valued field and a vector-valued field including their physical applications.
In the field of multiple integrals, the main attention is paid to calculations of multiple integrals on elementary regions and utilization of polar, cylindrical and spherical coordinates, calculations of a potential of vector-valued field and application of integral theorems.

Guarantor

Deputy Guarantor

Rebenda Josef, Mgr., Ph.D. (CEITEC BUT)

Language of instruction

Czech

Completion

Examination (written)

Time span

26 hrs lectures, 12 hrs exercises, 14 hrs pc labs

Assessment points

70 exam, 30 half-term test

Department

Lecturer

Instructor

Subject specific learning outcomes and competences

Students completing this course should be able to:
• calculate local, constrained and absolute extrema of functions of several variables.
• calculate multiple integrals on elementary regions.
• transform integrals into polar, cylindrical and spherical coordinates.
• calculate line and surface integrals in scalar-valued and vector-valued fields.
• apply integral theorems in the field theory.

Learning objectives

The aim of this course is to introduce the basics of theory and calculation methods of local and absolute extrema of functions of several variables, double and triple integrals, line and surface integrals including applications in technical fields.
Mastering basic calculations of multiple integrals, especially transformations of multiple integrals and calculations of line and surface integrals in scalar-valued and vector-valued fields.
of stability of solutions of differential equations and applications of selected functions
with solving of dynamical systems.

Why is the course taught

The course provides students basic orientation in solution methods of dynamical systems which are results of mathematical models of continuous and discrete processes.

Prerequisite kwnowledge and skills

The student should be able to apply the basic knowledge of analytic geometry and mathematical analysis on the secondary school level: to explain the notions of general, parametric equations of lines and surfaces and elementary functions. From the BMA1 and BMA2 courses, the basic knowledge of differential and integral calculus and solution methods of linear differential equations with constant coefficients is demanded. Especially, the student should be able to calculate derivative (including partial derivatives) and integral of elementary functions.

Fundamental literature

• ŠMARDA, Z., RUŽIČKOVÁ, I.: Selected parts from Mathematics, el. version on UMAT server.
• KRUPKOVÁ, V.: Diferenciální a integrální počet funkce více proměnných,skripta VUT Brno, VUTIUM 1999, 123 p. (in Czech)
• BRABEC, J., HRUZA, B.: Matematická analýza II, SNTL/ALFA, Praha 1986, 579 p. (in Czech)
• GARNER, L.E.: Calculus and Analytical Geometry. Brigham Young University, Dellen publishing Company, San Francisco,1988, ISBN 0-02-340590-2.

Syllabus of lectures

1. Differential calculus of functions of several variables, limit, continuity, derivative
2. Vector analysis
3. Local extrema
4. Constrained and absolute extrema
5. Multiple integral
6. Transformation of multiple integrals
7. Applications of multiple integrals
8. Line integral in a scalar-valued field
9. Line integral in a vector-valued field
10. Potential, Green's theorem
11. Surface integral in a scalar-valued field
12. Surface integral in a vector-valued field
13. Integral theorems

Progress assessment

The student's work during the semester (written tests and homework) is assessed by a maximum of 30 points.
The written examination is evaluated by a maximum of 70 points. It consists of seven tasks (one from extrema of functions of several variables (10 points), two from multiple integrals (2 X 10 points), two from line integrals (2 x 10 points) and two from surface integrals (2 x 10 points)).

Controlled instruction

Teaching methods include lectures and demonstration practical classes (computer and numerical). The course is taking advantage of exercise bank and maplets on UMAT server.
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Course inclusion in study plans

• Programme BIT, 2nd year of study, Elective
• Programme IT-BC-3, field BIT, 2nd year of study, Elective