Course detail

Mathematics II-B

FSI-BMAcad. year: 2024/2025

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.

Language of instruction

Czech

Number of ECTS credits

7

Mode of study

Not applicable.

Guarantor

doc. Mgr. Zdeněk Opluštil, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Differential and integral calculus of functions in one variable.

Rules for evaluation and completion of the course

COURSE-UNIT CREDIT REQUIREMENTS:
There are two written tests. Condition for the course-unit credit: to obtain at least 50% 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 85 points from the written part and 15 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 usually consists of theoretical questions and a discussion related to the written exam. 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.  A supplementary simple example can be given, 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
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.

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Hartman, P.: Ordinary Differential Equations. New York: John Wiley & Sons, 1964. (EN)
Kurzweil, J.: Obyčejné diferenciální rovnice, Praha, SNTL, 1978. (CS)
Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL Praha, 1988) (CS)
Sneall D.B. - Hosack J.M.: Calculus, An Integrated Approach (EN)
Thomas G.B. - Finney R.L.: Calculus and Analytic Geometry, 7th edition (EN)

Recommended reading

Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016. (CS)
Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy (CS)
Karásek J.: Matematika II (skriptum VUT) (CS)
Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition) (EN)

Classification of course in study plans

  • Programme B-ENE-P Bachelor's 1 year of study, summer semester, compulsory
  • Programme B-PDS-P Bachelor's 1 year of study, summer semester, compulsory
  • Programme B-PRP-P Bachelor's 1 year of study, summer semester, compulsory

  • Programme B-STR-P Bachelor's

    specialization STR , 1 year of study, summer semester, compulsory

  • Programme B-VTE-P Bachelor's 1 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. Mgr. Zdeněk Opluštil, Ph.D.

Syllabus

1. Function in more variables, basic definitions, and properties. Limit of a function in more variables, continuous functions. Partial derivative.
2. A gradient of a function, derivative in a direction. First-order and higher-order differentials, tangent plane to the graph of a function in two variables.
3. Taylor polynomial and Taylor's theorem. Local extremes.
4. Method of Lagrange multipliers, absolute extremes.
5. Function defined implicitly. Definite integral more variables, definition, basic properties.
6. Fubini's theorem, calculation on elementary (normal) areas.
7. Transformation of the integrals (polar and cylindrical coordinates)
8. Transformation of the integrals (spherical coordinates). Applications of double and triple integrals.
9. Ordinary differential equations (ODE), basic terms, existence, and uniqueness of solutions, analytical methods of solving of 1st order ODE.
10. Higher-order ODEs, properties of solutions, and methods of solving higher-order linear ODEs. Systems of 1st order ODEs.
11. Properties of solutions and methods of solving linear systems of 1st-order ODEs.
12. Applications of ODEs.
13. Boundary value problem for 2nd order ODEs.

Exercise

33 hod., compulsory

Teacher / Lecturer

doc. Mgr. Zdeněk Opluštil, Ph.D.

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 hod., compulsory

Teacher / Lecturer

doc. Mgr. Zdeněk Opluštil, Ph.D.

Syllabus

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