Course details

# Selected parts from mathematics I.

BPC-VPA FEKT BPC-VPA Acad. year 2023/2024 Winter 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 inegrals, line and surface integrals in a scalar-valued field and a vector-valued field including their physical applications.
In the field of multiple integrals , main attention is paid to calculations of multiple integrals on elementary regions and utilization of polar, cylindrical and sferical coordinates, calculalations of a potential of vector-valued field and application of integral theorems.

Guarantor

Course coordinator

Language of instruction

Czech

Completion

Examination

Time span

• 26 hrs lectures
• 26 hrs exercises

Department

Lecturer

Instructor

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, especialy tranformations of multiple integrals and calculations of line and surface integrals in scalar-valued and vector-valued fields.
of a stability of solutions of differential equations and applications of selected functions
with solving of dynamical systems.
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 sferical coordinates.
- calculate line and surface integrals in scalar-valued and vector-valued fields.
- apply integral theorems in the field theory.

Prerequisite knowledge and skills

The student should be able to apply the basic knowledge of analytic geometry and mathamatical 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.

Study literature

• 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) Diferenciální počet funkcí více proměnných, limita, spojitost, derivace
2) Vektorová analýza
3) Lokální extrémy funkce více proměnných
4) Vázané a absolutní extrémy
5) Vícerozměrný integrál.
6) Transformace vícerozměrných integrálů
7) Aplikace vícerozměrných integrálů
8) Křivkový integrál ve skalární poli
9) Křivkový integrál ve vektorovém poli
10) Potenciál , Greenova věta
11) Plošný integrál ve skalárním poli
12) Plošný integrál ve vektorovém poli
13) Integrální věty

Progress assessment

The student's work during the semestr (written tests and homework) is assessed by maximum 30 points.
Written examination is evaluated by maximum 70 points. It consist 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)).
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.

Schedule

DayTypeWeeksRoomStartEndCapacityLect.grpGroupsInfo
Mon exam 2023-12-11 T8/T 0.10 09:0011:00 předtermín
Mon lecture lectures T8/T 0.10 09:0010:5060 2BIA 2BIB 3BIT xx Šmarda
Mon exam 2024-01-15 T8/T 5.22 10:3012:30 II. termín
Mon exam 2024-01-22 T8/T 5.22 10:3012:30 III.termín
Tue exercise even week T8/T 5.03 11:0012:5030 2BIA 2BIB 3BIT xx Šmarda
Tue exercise odd week T8/T 5.03 11:0012:5030 2BIA 2BIB 3BIT xx Šmarda
Thu exam 2024-01-04 T8/T 5.22 10:3012:30 I. termín

Course inclusion in study plans

• Programme BIT, 2nd year of study, Elective
• Programme BIT (in English), 2nd year of study, Elective
• Programme IT-BC-3, field BIT, 2nd year of study, Elective