Title:

Selected Parts from Mathematics 2

Code:IVP2 (FEKT BPC-VPM)
Ac.Year:2019/2020
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
BIT-2ndElective
IT-BC-3BIT2ndElective
Language of Instruction:Czech
Credits:5
Completion:examination (written)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:26120140
 ExamsTestsExercisesLaboratoriesOther
Points:7030000
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
 
Learning objectives:
  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, especially transformations of multiple integrals and calculations of line and surface integrals in scalar-valued and vector-valued fields.
of stability of solutions of differential equations and applications of selected functions
with solving of dynamical systems.
Description:
  The aim of this course is to introduce the basics of calculation of local, constrained and absolute extrema of functions of several variables, double and triple integrals, line and surface integrals in a scalar-valued field and a vector-valued field including their physical applications.
In the field of multiple integrals, the main attention is paid to calculations of multiple integrals on elementary regions and utilization of polar, cylindrical and spherical coordinates, calculations of a potential of vector-valued field and application of integral theorems.
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 local, constrained and absolute extrema of functions of several variables.
  • calculate multiple integrals on elementary regions.
  • transform integrals into polar, cylindrical and spherical coordinates.
  • calculate line and surface integrals in scalar-valued and vector-valued fields.
  • apply integral theorems in the field theory.
Why is the course taught:
  The course provides students basic orientation in solution methods of dynamical systems which are results of mathematical models of continuous and discrete processes.
Syllabus of lectures:
 
  1. Differential calculus of functions of several variables, limit, continuity, derivative
  2. Vector analysis
  3. Local extrema
  4. Constrained and absolute extrema
  5. Multiple integral
  6. Transformation of multiple integrals
  7. Applications of multiple integrals
  8. Line integral in a scalar-valued field
  9. Line integral in a vector-valued field
  10. Potential, Green's theorem
  11. Surface integral in a scalar-valued field
  12. Surface integral in a vector-valued field
  13. Integral theorems
Fundamental literature:
 
  • ©MARDA, Z., RU®IČKOVÁ, I.: Selected parts from Mathematics, el. version on UMAT server.
  • KRUPKOVÁ, V.: Diferenciální a integrální počet funkce více proměnných,skripta VUT Brno, VUTIUM 1999, 123 p. (in Czech)
  • BRABEC, J., HRUZA, B.: Matematická analýza II, SNTL/ALFA, Praha 1986, 579 p. (in Czech)
  • 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 practical classes (computer and numerical). The 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.
Progress assessment:
  The student's work during the semester (written tests and homework) is assessed by a maximum of 30 points.
The written examination is evaluated by a maximum of 70 points. It consists 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)).
 

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