Course detail

# Mathematics II-B

FSI-BMAcad. year: 2020/2021

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, Methods of solving ordinary differential equations

A significant part of the course is devoted to applications of the studied topics. The acquired knowledge is a prerequisite for understanding the theoretical foundations in the study of other specialized subjects.

Supervisor

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. Moreover, they improve their skills in mathematical software, which can be used to solving problems.

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)

Sneall D.B. - Hosack J.M.: Calculus, An Integrated Approach
(EN)

Hartman, P.: Ordinary Differential Equations. New York: John Wiley & Sons, 1964. (EN)

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

Kurzweil, J.: Obyčejné diferenciální rovnice, Praha, SNTL, 1978. (CS)

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

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

Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy
(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 seminar with the computer support. The student can obtain at most 24 points altogether within the seminars. Condition for the course-unit credit: to obtain at least 6 points from each written test. Students, who do not fulfill conditions for the course-unit credit, can repeat the written test during the first two weeks of examination time.

FORM OF EXAMINATIONS:

The exam has an obligatory written and oral part. The student can obtain 75 points from the written part and 25 points from the oral part (the examiner can take into account the results of the seminar).

EXAMINATION:

- The written part ranges from 90 to 120 minutes according to the difficulty of the test.

- The written part will contain at least one question (example) from each of the following topics:

1. Differential calculus of functions of several variables.

2. Multiple integrals

3. Ordinary differential equations

- The written part may also include theoretical questions from the above-mentioned themes.

- The oral part is usually realized as a discussion of the test. For each example, the student must be able to justify his calculation procedure - otherwise, the test will not be recognized and will be evaluated for zero points. An additional theoretical question can be asked, or a supplementary simple example, which the student calculates immediately.

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

The course aim is 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 of the agreement with the teacher supervising the seminar.

Classification of course in study plans

- Programme B-ENE-P Bachelor's, 1. year of study, summer semester, 7 credits, compulsory
- Programme B-STR-P Bachelor's
specialization STR , 1. year of study, summer semester, 7 credits, compulsory

- Programme B3S-P Bachelor's
branch B-PRP , 1. year of study, summer semester, 7 credits, compulsory

- Programme B-VTE-P Bachelor's, 1. year of study, summer semester, 7 credits, compulsory
- Programme B-PRP-P Bachelor's, 1. year of study, summer semester, 7 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Function in more variables, basic definitions, and properties. Limit of a function in more variables, continuous function.

2. Partial derivative, a gradient of a function, derivative in a direction.

3. First-order and higher-order differentials, tangent plane to the graph of a function in two variables, Taylor polynomial.

4. Local extremes, Method of Lagrange multipliers.

5. Absolute extremes function defined implicitly.

6. Definite integral more variables, definition, basic properties, computing of the integrals using rectangular coordinates.

7. Fubini's theorem, calculation on elementary (normal) areas.

8.Transformation of the integrals (polar, cylindrical and spherical coordinates).

9. Applications of double and triple integrals.

10. Ordinary differential equations (ODE), basic terms, existence, and uniqueness of solutions, analytical methods of solving of 1st order ODE.

11. Higher-order ODEs, properties of solutions and methods of solving of higher-order linear ODEs.

12. Systems of 1st order ODEs., properties of solutions and methods of solving of linear systems of 1st order ODEs.

13. Boundary value problem for 2nd order ODEs.

Exercise

33 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

6 hours, compulsory

Teacher / Lecturer

Syllabus

Seminars in a computer lab have suitable mathematical software as computer support. Obligatory topics correspond to the course syllabus.