Selected parts from mathematics I.

• 39 hrs lectures

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.

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.

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.

• KRUPKOVÁ, V.: Diferenciální a integrální počet funkce více proměnných,skripta VUT Brno, VUTIUM 1999, 123s.
• BRABEC, J., HRUZA, B.: Matematická analýza II, SNTL/ALFA, Praha 1986, 579s.
• GARNER, L.E.: Calculus and Analytical Geometry. Brigham Young University, Dellen publishing Company, San Francisco,1988, ISBN 0-02-340590-2.

• ŠMARDA, Z., RUŽIČKOVÁ, I.: Vybrané partie z matematiky, el. texty na PC síti.

1.Some notions from differential calculus of a function of multi variables.
2.Multiple integrals.
3.Transformation of multiple integrals.
4.Improper multiple integrals.
5.Lines in Rn, undirected line integral.
6.Directed line integral, indenpedence on an
integrable way.
7.Surfaces in R3, undirected surface integral.
8.Orientation of a surface, directed surface
integral.
9.Integral theorems.
10.Systems of differential equations, elementary
methods of solving.
11.General methods of solving of differential
equations.
12.Solving of systems of differential equations
with selected rightside,stability of solutions.
13.Criterions of stability of solutions.

The student's work during the semestr (written tests and project 15/15) 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)).

Teaching methods include lectures and demonstration practical classes . 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.