Course detail

# Selected parts from mathematics I.

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.

Learning outcomes of the course unit

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.

Prerequisites

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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

ŠMARDA, Z., RUŽIČKOVÁ, I.: Vybrané partie z matematiky, el. texty na PC síti. (CS)
KRUPKOVÁ, V.: Diferenciální a integrální počet funkce více proměnných,skripta VUT Brno, VUTIUM 1999, 123s. (CS)
BRABEC, J., HRUZA, B.: Matematická analýza II, SNTL/ALFA, Praha 1986, 579s. (CS)
GARNER, L.E.: Calculus and Analytical Geometry. Brigham Young University, Dellen publishing Company, San Francisco,1988, ISBN 0-02-340590-2. (EN)

Planned learning activities and teaching methods

Teaching methods include lectures and demonstration practical classes (computer and numerical) . Course is taking advantage of exercise bank and maplets on UMAT server.

Assesment methods and criteria linked to learning outcomes

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)).

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

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.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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.

Classification of course in study plans

• Programme BPC-AUD Bachelor's

specialization AUDB-TECH , any year of study, winter semester, 5 credits, elective

• Programme BPC-AMT Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BPC-EKT Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BKC-EKT Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BPC-IBE Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BKC-MET Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BPC-MET Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BKC-SEE Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BPC-SEE Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BKC-TLI Bachelor's, any year of study, winter semester, 5 credits, elective
• Programme BPC-TLI Bachelor's, any year of study, winter semester, 5 credits, elective

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Fundamentals seminar

12 hours, compulsory

Teacher / Lecturer

Computer-assisted exercise

14 hours, compulsory

Teacher / Lecturer