Course detail

# Mathematics 3

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.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Entry knowledge

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.

Rules for evaluation and completion of the course

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

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Jirásek, F., Čipera, S., Vacek, M., Sbírka řešených příkladů z matematiky II, SNTL Praha 1986.

(CS)

Eliáš, J., Horváth, J., Kajan, J., Śulka, R., Zbierka úloh z vzššej matamatiky 3 a 4, Alfa Bratislava 1979.

(SK)

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

(CS)

Classification of course in study plans

• Programme B-P-C-SI (N) Bachelor's

branch VS , 2. year of study, winter semester, compulsory

• Programme B-K-C-SI (N) Bachelor's

branch VS , 2. year of study, winter semester, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

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 hours, compulsory

Teacher / Lecturer

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.