Course detail

Mathematics 3

FAST-BAA003Acad. year: 2022/2023

Double and triple integrals. Their calculation, transformation, physical and geometric interpretation.
Curvilinear integral in a scalar field, its calculation and application. Divergence and rotation of a vector field. Curvilinear integral in a vector field, its calculation and application. Independence of a curvilinear integral on the integration path. Green`s theorem.
Existence and uniqueness of solutions to first order differential equations. n-th order homogeneous linear differential equations with constant coefficients. Solutions to non-homogeneous linear differential equations with special-type right-hand sides. Variation-of-constants method.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. Ing. Vladislav Kozák, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Offered to foreign students

Of all faculties

Learning outcomes of the course unit

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

The students should be versed in the basic notions of the theory of functions of one and several variables (derivative, partial derivative, limit, continuous functions, graphs of functions). They should be able to calculate integrals of function of one variable, know their basic applications.

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 integral, basic properties and calculation.
2. Transformations and applications of double integral.
3. Definition of triple integral, basic properties and calculation.
4. Transformations and applications of triple integral.
5. Notion of a curve. 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. Green`s theorem and its application.
8. Independence of a curvilinear integral on the integration path.
9. Basics of ordinary differential equations.
10. First order differential equations - separable, linear, exact equations.
11. N-th order homogeneous linear differential equations with constant coefficients.
12. Solutions to non-homogeneous linear differential equations.
13. Variation-of-constants method. Applications in technology.

Work placements

Not applicable.

Aims

Students should learn the basics about double and tripple integrals and their applications, they should know how to calculate such integrals using the Fubini theorems and standard transformations, get familiar with line integrals both in a scalar and vector field and their applications, calculate simple line integrals.
They should learn the basic facts on selected first-order differential equations, on existence and uniqueness of solutions, be able to find analytical solutions to separated, linear, 1st-order homogeneous, and exact differential equations, calculate the solution of a non-homogeneous linear nth-order differential equation with special right-hand sides as well as using the general method of the variation of constants, understand the structure of solutions to non-homogeneous nth-order linear differential equations.

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

Eliáš, J., Horváth, J., Kajan, J., Śulka, R., Zbierka úloh z vzššej matamatiky 3 a 4, Alfa Bratislava 1979. (SK)
Jirásek, F., Čipera, S., Vacek, M.,Sbírka řešených příkladů z matematiky II, SNTL Praha 1986. (CS)

Recommended reading

Škrášek, J., Tichý Z., Základy aplikované matematiky II, Praha SNTL 1986. (CS)

Classification of course in study plans

  • Programme BPC-SI Bachelor's

    specialization VS , 2 year of study, winter semester, compulsory

  • Programme BPC-EVB Bachelor's 2 year of study, winter semester, compulsory
  • Programme BPA-SI Bachelor's 2 year of study, winter semester, compulsory
  • Programme BKC-SI Bachelor's 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Karel Mikulášek, Ph.D.
RNDr. Hana Šafářová
doc. Ing. Vladislav Kozák, CSc.
RNDr. Mgr. Lucie Zrůstová, Ph.D.

Syllabus

1. Definition of double integral, basic properties and calculation. 2. Transformations and applications of double integral. 3. Definition of triple integral, basic properties and calculation. 4. Transformations and applications of triple integral. 5. Notion of a curve. 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. Green`s theorem and its application. 8. Independence of a curvilinear integral on the integration path. 9. Basics of ordinary differential equations. 10. First order differential equations - separable, linear, exact equations. 11. N-th order homogeneous linear differential equations with constant coefficients. 12. Solutions to non-homogeneous linear differential equations. 13. Variation-of-constants method. Applications in technology.

Exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Jana Slaběňáková
Ing. Pavel Špaček, Dr.
RNDr. Radko Odehnal
RNDr. Oto Přibyl
doc. Ing. Vladislav Kozák, CSc.
RNDr. Mgr. Lucie Zrůstová, Ph.D.
Mgr. Šárka Pechancová, Ph.D.
Mgr. Blanka Morávková, Ph.D.
Mgr. Eva Jansová, Ph.D.
RNDr. Rudolf Schwarz, CSc.
Mgr. Matouš Trnka

Syllabus

1. Quadrics and integration revision. 2. Double integral calculation. 3. Double integral transformations. 4. Double integral applications. 5. Triple integral calculation. 6. Transformations and applications of triple integral. 7. Curvilinear integral in a scalar field and its applications. 8. Curvilinear integral in a vector field and its applications. 9. Green`s theorem. Independence of a curvilinear integral on the integration path. Potential. 10. First order differential equations - separable, linear. 11. Exact equation. N-th order homogeneous linear differential equations with constant coefficients. 12. Solutions to non-homogeneous linear differential equations with special-type right-hand sides. 13. Variation-of-constants method. Seminar evaluation.