Course detail

Mathematics III-B

FSI-CMAcad. year: 2019/2020

The course is intended as an introduction to basic methods applied for solving of ordinary differential equations and problems of mathematical statistics.
The knowledge of the basic theory of differential equations and methods of solving is an important foundation for further study of physical and technical disciplines, especially those connected with mechanics.
Statistical methods are concentrated on descriptive statistics, random events, probability, random variables and vectors, random sample, parameters estimation and tests of hypotheses. The practicals cover problems and applications in mechanical engineering.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Students obtain necessary knowledge of ordinary differential equations and mathematical statistics, which enables them to understand and apply deterministic and stochastic models of technical phenomenon based on these methods.

Prerequisites

Foundations of differential and integral calculus.

Co-requisites

Not applicable.

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 is awarded on the following conditions: active attendance at seminars, understanding of the subject-matter. Fulfilment of all conditions of the running control of knowledge. At least half of all possible 30 points in both check tests. If a student does not fulfil this condition, the teacher can set an alternative one.


Examination (written form): practical part (2 examples from ordinary differential equations; 2 examples from mathematical statistics) with own summary of formulas; theoretical part (4 questions concerning basic terms, their properties, sense and practical use);

Examination (evaluation): The final grade reflects the result of the written part of the exam (maximum 70 points) and the results achieved in seminars (maximum 30 points); classification according to the total sum of points achieved: excellent (90 - 100 points), very good (80 - 89 points), good (70 - 79 points), satisfactory (60 - 69 points), sufficient (50 - 59 points), failed (0 - 49 points).

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim is to acquaint students with basic terms and methods of solving of ordinary differential equations and mathematical statistics. Another goal of the course is to form the student's thinking in modelling of real phenomenon and processes in engineering fields.

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

Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Hartman, P.: Ordinary Differential Equations. New York: John Wiley & Sons, 1964.
Montgomery, D. C. - Renger, G.: Probability and Statistics. New York : John Wiley & Sons, Inc.,1996.
Sprinthall, R. C.: Basic Statistical Analysis. Boston : Allyn and Bacon, 1997.

Recommended reading

Čermák,J.- Ženíšek, A.: Matematika III. Brno: FSI VUT V Akademickém nakladatelství CERM Brno, 2001
Karpíšek, Z.: Matematika IV - Statistika a pravděpodobnost. 2. vydání. Brno : FSI VUT v Akademickém nakladatelství CERM Brno, 2003.
Logan, J.D.: A First Course in Differential Equations. New York, Springer, 2006.

Classification of course in study plans

  • Programme B3S-P Bachelor's

    branch B-EPP , 2 year of study, winter semester, compulsory
    branch B-AIŘ , 2 year of study, winter semester, compulsory
    branch B-SSZ , 2 year of study, winter semester, compulsory
    branch B-STG , 2 year of study, winter semester, compulsory

  • Programme B3A-P Bachelor's

    branch B-PDS , 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

Ing. Josef Bednář, Ph.D.

Syllabus

1. ODE. Basic terms. Existence and uniqueness of solutions.
2. Analytical methods of solving of 1st order ODE.
3. Higher order ODEs. Properties of solutions and methods of solving of higher order homogeneous linear ODEs.
4. Properties of solutions and methods of solving of higher order non-homogeneous linear ODEs.
5. Systems of 1st order ODEs. Properties of solutions and methods of solving of homogeneous linear systems of 1st order ODEs.
6. Properties of solutions and methods of solving of non-homogeneous linear systems of 1st order ODEs.
7. Boundary value problem for 2nd order ODEs.
8. Descriptive statistics.
9. Random events and probability.
10. Random variable and vector, functional and numerical characteristics.
11. Basic probability distributions (Bi, H, Po, N), properties and use.
12. Random sample, parameter estimations (Bi, N).
13. Testing statistical hypotheses of parameters (Bi, N).

Computer-assisted exercise

13 hod., compulsory

Teacher / Lecturer

Ing. Pavel Hrabec, Ph.D.
Ing. Tomáš Kisela, Ph.D.
Ing. Zdeněk Hrazdíra, Ph.D.
doc. RNDr. Libor Žák, Ph.D.

Syllabus

1. Calculation of integrals - revision.
2. Analytical methods of solving of 1st order ODEs.
3. Analytical methods of solving of 1st order ODEs (continuation).
4. Higher order linear homogeneous ODEs.
5. Higher order non-homogeneous linear ODEs.
6. Systems of 1st order linear homogeneous ODEs.
7. Systems of 1st order linear non-homogeneous ODEs.
8. Descriptive statistics (univariate and bivariate sample).
9. Probability, conditioned probability, independent events.
10. Functional and numerical characteristics of random variable.
11. Probability distributions (Bi, H, Po, N).
12. Point and interval estimates of parameters N and Bi.
13. Testing hypotheses of parameters N and Bi.