Course detail

Selected Parts from Mathematics for Engineers

FEKT-MPC-VPAAcad. year: 2025/2026

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 inegrals, line and surface integrals in a scalar-valued field and a vector-valued field including their physical applications. In the field of multiple integrals , main attention is paid to calculations of multiple integrals on elementary regions and utilization of polar, cylindrical and sferical coordinates, calculalations of a potential of vector-valued field and application of integral theorems.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Zdeněk Šmarda, CSc.

Department

Department of Mathematics (UMAT)

Entry knowledge

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. Especially, the student should be able to calculate derivative (including partial derivatives) and integral of elementary functions.

Rules for evaluation and completion of the course

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

Aims

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 field.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Krupková, V., Diferencilní a integrállní počet funkce více proměnných, skripta VUT, 1998, 199 s. (CS)

Recommended reading

BRABEC, J., HRUZA, B.: Matematická analýza II, SNTL/ALFA, Praha 1986, 579s. (CS)

Classification of course in study plans

  • Programme MPC-KAM Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Zdeněk Šmarda, CSc.

Syllabus

  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
    

Fundamentals seminar

12 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Šmarda, CSc.

Syllabus

  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
   

Computer-assisted exercise

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Šmarda, CSc.

Syllabus

  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