Selected parts from mathematics II.

• 26 hrs lectures
• 26 hrs exercises

The aim of this course is to introduce the basics of improper multiple integrals, systems of differential equations including of investigations 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 improper multiple integral on unbounded regions and from unbounded functions.
- apply a weighted function and a delta function to solving of linear differential equations.
- select an optimal solution method for given differential equation.
- investigate a stability of solutions of systems of differential equations.

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.

1) Základní vlastnosti vícerozměrných integrálů.
2) Nevlastní vícerozměrný integrál
3) Impulzní funkce a delta funkce, základní vlastnosti.
4) Derivace a integrál delta funkce
5) Jednotková funkce a její vztah s delta funkcí, váhová funkce.
6) Řešení diferenciálních rovnic n-tého řádu užitím váhových funkcí
7) Vztah Diracovy funkce a váhové funkce
8) Systémy diferenciálních rovnice a jejich vlastnosti.
9) Eliminační metoda řešení.
10) Metoda vlastních čísel a vlastních vektorů.
11) Variace konstant a metoda neurčitých koeficientů
12) Diferenciální transformační metoda pro obyčejné diferenciální rovnice
13) Diferenciální transformační metoda pro diferenciální rovnice se zpožděným argumentem

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 improper multiple integral (10 points), three from application of a weighted function and a delta function (3 X 10 points) and three from analytical solution method of differential equations (3 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.