Mathematics II-B (FSI-BM)

Academic year 2019/2020
Supervisor: doc. Mgr. Zdeněk Opluštil, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
Differential and integral calculus of functions of several variables including problems of finding maxima and minima and calculating limits, derivatives, differentials, double and triple integrals. At seminars, the MAPLE mathematical software is used.
The course aims to acquaint the students with the theoretical basics of the above mentioned mathematical disciplines necessary for further study of engineering courses and for solving engineering problems encountered. Another goal of the course is to develop the students' logical thinking.
Learning outcomes and competences:
Students will acquire basic knowledge of mathematical disciplines listed in the course annotation and will be made familiar with their logical structure. They will learn how to solve mathematical problems encountered when dealing with engineering tasks using the knowledge and skills acquired.
Prerequisites:
Differential and integral calculus of functions in one variable.
Course contents:
The course takes the form of lectures and seminars dealing with the following topics:
Real functions of two and more variables, Partial derivatives - total differentials, Applications of partial derivatives - maxima, minima and saddle points, Lagrange multipliers, Taylor's approximation and error estimates, Double integrals, Triple integrals, Applications of multiple integrals.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
COURSE-UNIT CREDIT REQUIREMENTS: There are two written tests (each at most 12 points) within the seminars and a semestral work from the computer support (at most 1 point).
The student can obtain at most 25 points alltogether within the seminars. Condition for the course-unit credit: to obtain at least 6 points from each written test. Students, who do not fulfil conditions for the course-unit credit, can repeat the written test during first two weeks of examination time.

FORM OF EXAMINATIONS:
The exam has an obligatory written part.
In a 120-minute written test, students have to solve the following three problems:
Problem 1: In differential calculus of functions of several variables.
Problem 2: In double integral.
Problem 3: In tripple integral.
Above problems can also contain a theoretical question.


RULES FOR CLASSIFICATION
1. Results from seminars (at most 25 points)
2. Results from the written examination (at most 75 points)

Final classification:
0-49 points: F
50-59 points: E
60-69 points: D
70-79 points: C
80-89 points: B
90-100 points: A
Controlled participation in lessons:
Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. Missed seminars may be made up by the agreement with the teacher supervising the seminar.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  11 × 2 hrs. compulsory                  
    Computer-assisted exercise  2 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 1. Function in more variables, basic definitions.
2. Limit of a function in more variables, continuous function.
3. Partial derivative, gradient of a function, derivative in a direction.
4. First-order and higher-order differentials, tangent plane to the graph of a function in two variables, Taylor polynomial.
5. Relative maxima and minima.
6. Lagrange multipliers, absolute maxima and minima.
7. Functions defined implicitly.
8. Definite integral more variables, definition, basic properties.
9. Computing of the integrals using rectangular coordinates.
10. Calculation on elementary (normal) area's, Fubini's theorem.
11.The Jacobian and change of coordinates, transformation of the integrals, polar coordinates.
12.Cylindrical and spherical coordinates.
13.Applications of double and triple integrals.
    Exercise The first week: calculating improper integrals, applications of the Riemann integral. Following weeks: seminars related to the lectures given in the previous week.
    Computer-assisted exercise Seminars in a computer lab have the programme MAPLE as a computer support. Obligatory topics to go through: Plotting of the graph of a function of more variables (given by explicit, implicit or parametric equations), extrema of functions of more variables.
Literature - fundamental:
1. Thomas G.B. - Finney R.L.: Calculus and Analytic Geometry, 7th edition
4. Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL Praha, 1988)
Literature - recommended:
6. Karásek J.: Matematika II (skriptum VUT)
7. Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016.
9. Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-VTE-P full-time study --- no specialisation -- Cr,Ex 6 Compulsory 1 1 S
B3A-P full-time study B-MTI Materials Engineering -- Cr,Ex 6 Compulsory 1 1 S
B3S-P full-time study B-S1R Engineering -- Cr,Ex 6 Compulsory 1 1 S