Course Details

Mathematics 3

BAA003 course is part of 4 study plans

Bc. full-t. program BPC-EVB compulsory Winter Semester 2nd year 5 credits

Bc. full-t. program BPC-SI > spVS compulsory Winter Semester 2nd year 5 credits

Bc. full-t. program BPA-SI compulsory Winter Semester 2nd year 5 credits

Bc. combi. program BKC-SI compulsory Winter Semester 2nd year 5 credits

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.

Course Guarantor

doc. Ing. Vladislav Kozák, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Learning outcomes

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. Knowledge of 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.

Corequisites

Not required.

Planned educational activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lecture, seminar.

Forms and criteria of assessment

Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops. The result of the semester examination is given by the sum of maximum of 70 points obtained for a written test and a maximum of 30 points from the seminar.

Objective

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 instruction, the form of instruction, and the form of compensation of the absences

Vymezení kontrolované výuky a způsob jejího provádění stanoví každoročně aktualizovaná vyhláška garanta předmětu.

Lecture

2 hours/week, 13 weeks, elective

Syllabus of lectures

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.

Practice

2 hours/week, 13 weeks, compulsory

Syllabus of practice

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.