Course detail

# Mathematics II-B

FSI-BMAcad. year: 2017/2018

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.

Department

Learning outcomes of the course unit

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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Thomas G.B. - Finney R.L.: Calculus and Analytic Geometry, 7th edition (EN)

Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL Praha, 1988) (CS)

Karásek J.: Matematika II (skriptum VUT) (CS)

Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016. (CS)

Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition) (EN)

Planned learning activities and teaching methods

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

Language of instruction

Czech

Work placements

Not applicable.

Aims

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.

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

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.

Classification of course in study plans

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

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

22 hours, compulsory

Teacher / Lecturer

Syllabus

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

4 hours, compulsory

Teacher / Lecturer

Syllabus

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.