Course detail

Line and Surface Integral

FSI-S4AAcad. year: 2005/2006

The course provides an introduction to the theory and calculation of line and surface integrals, improper integrals and integrals depending on a parameter.

Language of instruction

Czech

Number of ECTS credits

0

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

The theory of line and surface integrals represents a basic tool for all parts of physics and technical disciplines involving the notion of the field. The theory of improper integrals is connected with the theory of surface integrals and moreover, it is a part of the mathematical apparatus of the probability theory.

Prerequisites

Differential and integral calculus of functions in a single and more variables.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on the following conditions: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge.
Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving of examples. The exam is written (possibly followed by an oral part). The final grade reflects especially the result in the written part of the exam. However, the examiner can also take account of the results in the check tests completed in seminars.
Grading scheme is as follows: excellent (90-100 points), very good
(80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points). The grading in points may be modified provided that the above given ratios remain unchanged.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to explain the theory of line and surface integrals, improper integrals and integrals depending on a parameter.
Another goal of the course is for students to master
calculation of the above mentioned types of integrals.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Fichtengolc, G.M.: Kurs differencialnogo i integralnogo isčislenija, tom II, III, , 0
Kufner, A - John,O. - Fučík, S. : Function Spaces, , 0
Škrášek, J. - Tichý, Z.: Základy aplikované matematiky II, , 0

Recommended reading

Ženíšek A.: Křivkový a plošný integrál, , 0
Ženíšek A.: Vybrané kapitoly z matematické analýzy, , 0

Classification of course in study plans

  • Programme B3901-3 Bachelor's

    branch B3910-00 , 2. year of study, summer semester, elective (voluntary)

Type of course unit

 

Lecture

13 hours, optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Curves in a plane and in a space. Line integral of the 1st and 2nd kind in a plane.
2. Green theorem in an elementary form.
3. Green theorem in a general form.
4. The independence of the line integral on the integral way.
5. The surfaces. The surface integral of the 1st kind on the strongly regular surface.
6. The surface integral of the 1st kind on the regular surface.
7. Vector flow. The surface integral of the 2nd kind.
8. Ostrogradsky theorem; its elementary and general form.
9. Line integrals in a space and Stokes theorem. Scalar and vector field.
10. Integrals depending on a parameter.
11. Improper integrals of functions in a single variable.
12. Substitution theorem and integration by parts for improper integrals.
13. The improper integral of a nonnegative function in two and three variables.

Exercise

26 hours, compulsory

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Calculation of integrals (revision).
2. Calculation of line integrals of the 1st kind.
3. Calculation of line integrals of the 2nd kind.
4. Green theorem at examples.
5. The independence of the line integral on the integral way at examples.
6. The surfaces and their properties.
7. Calculation of surface integrals of the 1st kind.
8. Calculation of surface integrals of the 2nd kind.
9. Ostrogradsky theorem at examples.
10. Calculation of line integrals in a space and Stokes theorem at examples.
11. Integrals depending on a parameter.
12. Calculation of improper integrals of functions in a single variable.
13. Calculation of improper integrals of functions in more variables.