Course detail

# Selected parts from mathematics II.

The aim of this course is to introduce the basics of calculation of improper multiple integral and basics of solving of linear differential equations using delta function and weighted function.
In the field of improper multiple integral, main attention is paid to calculations of improper multiple integrals on unbounded regions and from unbounded functions.
In the field of linear differential equations, the following topics are covered: Eliminative solution method, method of eigenvalues and eigenvectors, method of variation of constants, method of undetermined coefficients, stability of solutions.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Learning outcomes of the course unit

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.

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.

Planned learning activities and teaching methods

Teaching methods include lectures and demonstration practises . Course is taking advantage of exercise bank and Maple exercises on server UMAT. Students have to write a single project/assignment during the course.

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 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)).

Course curriculum

1) Impulse function and delta function, basic properties
2) Derivative and integral of the delata function
3) Solving differential equations of the n-th order using weighted functions
4) Systems of differential equations and their properties
5) Eliminative solution method
6) Method of eigenvalues and eigenvectors
7) Method of variation of constants and method of undetermined coefficients
8) Differential transformation solution method of ordinary differential equations
9) Differential transformation solution method of functional differential equations
10) Diference operator, summation, classification od difference equations
11) Shift operator, solution methods of difference equations with variable coefficients
12) Solution methods of linear difference equations with constant coefficients
13) Solution methods of systems of difference equations.

Work placements

Not applicable.

Aims

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.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended literature

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)

Elearning

Classification of course in study plans

• Programme BPC-AMT Bachelor's 0 year of study, summer semester, elective

• Programme BPC-AUD Bachelor's

specialization AUDB-TECH , 0 year of study, summer semester, elective
specialization AUDB-ZVUK , 0 year of study, summer semester, elective

• Programme BPC-ECT Bachelor's 0 year of study, summer semester, elective
• Programme BPC-IBE Bachelor's 0 year of study, summer semester, elective
• Programme BPC-MET Bachelor's 0 year of study, summer semester, elective
• Programme BPC-SEE Bachelor's 0 year of study, summer semester, elective
• Programme BPC-TLI Bachelor's 0 year of study, summer semester, elective
• Programme BKC-EKT Bachelor's 0 year of study, summer semester, elective
• Programme BKC-MET Bachelor's 0 year of study, summer semester, elective
• Programme BKC-SEE Bachelor's 0 year of study, summer semester, elective
• Programme BKC-TLI Bachelor's 0 year of study, summer semester, elective

• Programme IT-BC-3 Bachelor's

branch BIT , 2 year of study, summer semester, elective

• Programme BIT Bachelor's 2 year of study, summer semester, elective

#### Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

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

Fundamentals seminar

12 hod., compulsory

Teacher / Lecturer

Computer-assisted exercise

14 hod., compulsory

Teacher / Lecturer

Elearning