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
Department
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
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.
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
Syllabus
- Mapping theory, limit and continuity of functions of more variables
- Vector analysis
- Derivative of a composed mapping
- Local, constrained and absolute extrema, Lagrange method.
- Integral calculus of functions of more variables
- Calculation of n-dimensional integrals using successive integration
- Transformation of double integrals, applications
- Transformation of triple integrals, applications
- Improper integral of functions of more variables
- Line integral in a scalar field, applications
- Line integral in a vector field, applications
- Surface integral in a scalar field, applications
- Surface integral in a vector field, integral theorems
Fundamentals seminar
12 hod., compulsory
Teacher / Lecturer
Syllabus
- Computation of limits of functions of more variables
- Computation of characteristics of scalar and vector fields
- Computation of a derivative of a composed mapping
- Computation of local, constrained and absolute extrema, Lagrange method.
- Construction of integral of functions of more variables
- Computation of n-dimensional integrals using successive integration
- Transformation of double integrals, evaluations and applications
- Transformation of triple integrals, evaluations and applications
- Improper integral of functions of more variables, evaluations
- Line integral in a scalar field, evaluations and applications
- Line integral in a vector field, evaluations and applications
- Surface integral in a scalar field, evaluations and applications
- Surface integral in a vector field, integral theorems, applications
Computer-assisted exercise
14 hod., compulsory
Teacher / Lecturer
Syllabus
- Computation of limits of functions of more variables
- Computation of characteristics of scalar and vector fields
- Computation of a derivative of a composed mapping
- Computation of local, constrained and absolute extrema, Lagrange method.
- Construction of integral of functions of more variables
- Computation of n-dimensional integrals using successive integration
- Transformation of double integrals, evaluations and applications
- Transformation of triple integrals, evaluations and applications
- Improper integral of functions of more variables, evaluations
- Line integral in a scalar field, evaluations and applications
- Line integral in a vector field, evaluations and applications
- Surface integral in a scalar field, evaluations and applications
- Surface integral in a vector field, integral theorems, applications