Course detail

Mathematics III

FAST-GA05Acad. year: 2022/2023

Double and triple integral and their applications. Transformations of double and triple integrals.
Curve integrals in scalar and vector fields, basic properties ans calculation. Independence of the curve integral of the path of integration. Green`s Theorem.
Ordinary differential equations (DE) of the first order, existence and uniqueness of the solution. DE with separable variables, homogeneous, linear and exact DE. Orthogonal and isogonal trajectories, envelope of the family of curves. Linear DE of n-th order, general solution, basic properties of solutions. Linear DE with constant coefficients.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Diblík, DrSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Students will achieve the subject's main objectives. Knowledge of double and triple integrals, their calculation and application. Knowledge of curvilinear integral in a scalar and vector field, their calculation and application. Knowledge of basic facts on existence, uniqueness and analytical methods of solutions on selected first-order differential equations and nth-order linear differential equations.

Prerequisites

Basics of calculus of one- and more-functions. The basics of linear algebra as taught in the introductory courses.
Basics of integral calculus of functions of one variable and the basic interpretations.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Definition of double and triple integrals their basic properties. Calculation of double integrals.
2. Transformations of double integrals. Physical and geometric applications of double integrals.
3. Calculation and transformations of triple integrals.
4. Physical and geometric applications of triple integrals.
5. Curvilinear integral in a scalar field and its applications.
6. Vector field. Divergence and rotation of a vector field. Curvilinear integral in a vector field and its applications.
7. Independence of a curvilinear integral on the integration path.
8. Green`s theorem and its application.
9. Basics of ordinary differential equations. First order differential equations - separable, homogeneous.
10. First order differential equations - linear, exact equations. Orthogonal and isogonal trajectories.
11. Structure of the set of solutions to an n-th order linear differential equation. Linear independence of solutions, Wronskian.
12. Homogeneous linear differential equations with constant coefficients. Solutions to non-homogeneous linear differential equations.
13. Solutions to non-homogeneous linear differential equations. Variation-of-constants method.

Work placements

Not applicable.

Aims

After the course the students should get acquainted with double and triple integrals and their basic applications, calculate such integrals using the Fubini theorems and standard transformations.
They should learn the basics of line integrals in scalar and vector fields and their aplications. They should know how to calculate simple line integrals.
They should be acquainted with selected first-order differential equations (DE) focussing on problems of existence and uniqueness of their solutions, know how to find analytical solutions to separated, linear, 1st-order homogeneous, exact DE's, calculate non-homogeneous linear nth-order DE's with a special right-hand side and using the variation of constants method. They should understand the structure of solutions of nth-order non-homogeneous linear DE's with issues of orthogonal and isogonal trajectories.

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

BUDÍNSKÝ, B. - CHARVÁT, J.: Matematika I. SNTL, Praha, 1987. (CS)
BUDÍNSKÝ, B. - CHARVÁT, J.: Matematika II. SNTL, Praha, 1990. (CS)
Daněček Josef, Dlouhý Oldřich, Přibyl Oto. Matematika II, Modul 1, Dvojný a trojný integrál, Brno, VUT, FAST, Studijní opora, 2004 (CS)
Daněček Josef, Dlouhý Oldřich, Přibyl Oto. Matematika II, Modul 2, Křivkové integrály, Brno, VUT, FAST, Studijní opora, 2004 (CS)
Diblík Josef, Přibyl Oto. Obyčejné diferenciální rovnice 1, Modul 3, Brno, VUT, FAST, Studijní opora, 2004 (CS)
Diblík Josef, Přibyl Oto. Obyčejné diferenciální rovnice 2, Modul 4, Brno, VUT, FAST, Studijní opora, 2004 (CS)
STEIN, S. K.: Calculus and analytic geometry. New York, 1989. (EN)

Recommended reading

Čermáková, H. a spol.: Sbírka příkladů z matematiky II. Stavební fakulta VUT Brno, CERM, 1994. (CS)
DANĚČEK, J., DLOUHÝ, O, PŘIBYL, O: Modul 1 Dvojný a trojný integrál. CERM Brno, 2006. (CS)
DANĚČEK, J., DLOUHÝ, O, PŘIBYL, O: Modul 2 Křivkové integrály. CERM Brno, 2006. (CS)
DIBLÍK, J., PŘIBYL,O.: Obyčejné diferenciální rovnice. CERM Brno, 2004. (CS)
Kolektiv: Elektronické studijní opory. FAST VUT Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
KOUTKOVÁ, H., PRUDILOVÁ, K.: Sbírka příkladů z matematiky III, Modul BA02_M05 Dvojný, trojný a křivkový integrál. FAST VUT, 2007. (CS)
Prudilová, K. a spol.: Sbírka příkladů z matematiky III. Stavební fakulta VUT Brno, CERM, 2001. (CS)

Classification of course in study plans

  • Programme B-P-C-GK Bachelor's

    branch GI , 2 year of study, winter semester, compulsory

  • Programme B-K-C-GK Bachelor's

    branch GI , 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

1. Definition of double and triple integrals their basic properties. Calculation of double integrals. 2. Transformations of double integrals. Physical and geometric applications of double integrals. 3. Calculation and transformations of triple integrals. 4. Physical and geometric applications of triple integrals. 5. Curvilinear integral in a scalar field and its applications. 6. Vector field. Divergence and rotation of a vector field. Curvilinear integral in a vector field and its applications. 7. Independence of a curvilinear integral on the integration path. 8. Green`s theorem and its application. 9. Basics of ordinary differential equations. First order differential equations - separable, homogeneous. 10. First order differential equations - linear, exact equations. Orthogonal and isogonal trajectories. 11. Structure of the set of solutions to an n-th order linear differential equation. Linear independence of solutions, Wronskian. 12. Homogeneous linear differential equations with constant coefficients. Solutions to non-homogeneous linear differential equations. 13. Solutions to non-homogeneous linear differential equations. Variation-of-constants method.

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

1. Basic properties of double and triple integrals. Calculation of double integrals. 2. Transformations of double integrals. Physical and geometric applications of double integrals. 3. Calculation and transformations of triple integrals. 4. Physical and geometric applications of triple integrals. 5. Curvilinear integral in a scalar field and its applications. 6. Vector field. Divergence and rotation of a vector field. Curvilinear integral in a vector field and its applications. 7. Independence of a curvilinear integral on the integration path. 8. Green`s theorem and its application. 9. First order differential equations - separable, homogeneous. 10. First order differential equations - linear, exact equations. Orthogonal and isogonal trajectories. 11. Homogeneous linear differential equations with constant coefficients. 12. Solutions to non-homogeneous linear differential equations. 13. Variation-of-constants method. Seminar evaluation.