Selected Parts from Mathematics 1

Language of Instruction:Czech
Completion:examination (written)
Type of
Guarantor:©marda Zdeněk, doc. RNDr., CSc. (DMAT)
Deputy guarantor:Rebenda Josef, Mgr., Ph.D. (CEITEC)
Lecturer:©marda Zdeněk, doc. RNDr., CSc. (DMAT)
Instructor:©marda Zdeněk, doc. RNDr., CSc. (DMAT)
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEEC BUT
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Learning objectives:
  The aim of this course is to introduce the basics of improper multiple integrals, systems of differential equations including of investigations of stability of solutions of differential equations and applications of selected functions with solving of dynamical systems.
  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, the 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, a method of eigenvalues and eigenvectors, method of variation of constants, a method of undetermined coefficients, the stability of solutions.
Knowledge and skills required for the course:
  The student should be able to apply the basic knowledge of analytic geometry and mathematical 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.
Learning outcomes and competencies:
  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 the given differential equation.
  • investigate the stability of solutions of systems of differential equations.
Why is the course taught:
  The course provides students basic orientation in differential and integral calculus of functions of several variables that are necessary for the description of the behaviour of the scalar and vector-valued fields and as well for determination of characteristics of vector-valued random variables.
Syllabus of lectures:
  1. Basic properties of multiple integrals.
  2. Improper multiple integral
  3. Impulse function and delta function, basic properties
  4. Derivative and integral of the delata function
  5. Unit function and its relation with the delta function, the weighted function
  6. Solving differential equations of the n-th order using weighted functions
  7. The relation between Dirac function and weighted function
  8. Systems of differential equations and their properties
  9. Eliminative solution method
  10. Method of eigenvalues and eigenvectors
  11. Method of variation of constants and method of undetermined coefficients
  12. Differential transformation solution method of ordinary differential equations
  13. Differential transformation solution method of functional differential equations
Syllabus of computer exercises:
 Computer classes are based on demonstration practical classes with the utilization of exercise bank and maplets on UMAT server.

1. Calculation of the characteristics of the scalar and vector fields.
2. Calculation and simulation of local, constrained and absolute extrema of functions of several variables.
3. Calculation and simulation of multiple integrals in elementary regions.
4. Transformation integrals into polar, cylindrical and spherical coordinates, simulation.
5. Calculation and simulation of line integrals in scalar-valued and vector-valued fields.
6. Calculation and simulation of surface integrals in scalar-valued and vector-valued fields.
7. Application of integral theorems in the field theory, simulation.
Fundamental literature:
  • ©MARDA, Z., RU®IČKOVÁ, 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.
Controlled instruction:
  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.
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.
Progress assessment:
  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)).

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