Selected parts from mathematics I.

• 26 hrs lectures
• 26 hrs exercises

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.

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.

• GARNER, L.E.: Calculus and Analytical Geometry. Brigham Young University, Dellen publishing Company, San Francisco,1988, ISBN 0-02-340590-2.

1. Mapping theory, limit and continuity of functions of more variables
2. Vector analysis
3. Derivative of a composed  mapping
4. Local, constrained and absolute extrema, Lagrange method.
5. Integral calculus of functions of more variables
6. Calculation of n-dimensional integrals using successive integration
7. Transformation  of double integrals, applications
8. Transformation  of triple integrals, applications
9.  Improper  integral of  functions  of more variables
10. Line integral in a scalar field, applications
11. Line integral in a vector field, applications
12. Surface integral in a scalar field, applications
13. Surface integral in a vector field, integral  theorems

1. Computation of limits of functions of more variables
2. Computation of  characteristics of  scalar and vector fields
3. Computation of a  derivative of a composed  mapping
4. Computation of local, constrained and absolute extrema, Lagrange method.
5. Construction  of  integral of functions of more variables
6. Computation  of n-dimensional integrals using successive integration
7. Transformation  of double integrals,  evaluations and applications
8. Transformation  of triple integrals, evaluations and applications
9.  Improper  integral of  functions of more variables, evaluations
10. Line integral in a scalar field, evaluations and applications
11. Line integral in a vector field, evaluations and applications
12. Surface integral in a scalar field, evaluations and applications
13. Surface integral in a vector field, integral  theorems, applications

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.