Mathematics III-B (FSI-CM-K)

Academic year 2020/2021
Supervisor: prof. RNDr. Jan Čermák, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Course contents:
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.
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 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).
Controlled participation in lessons:
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.
Type of course unit:
    Guided consultation in combined form of studies  1 × 13 hrs. compulsory                  
    Guided consultation  1 × 26 hrs. optionally                  
Course curriculum:
    Guided consultation in combined form of studies 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).
    Guided consultation 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.
Literature - fundamental:
1. Hartman, P.: Ordinary Differential Equations. New York: John Wiley & Sons, 1964.
2. Sprinthall, R. C.: Basic Statistical Analysis. Boston : Allyn and Bacon, 1997.
3. Montgomery, D. C. - Renger, G.: Probability and Statistics. New York : John Wiley & Sons, Inc.,1996.
Literature - recommended:
1. Čermák,J.- Ženíšek, A.: Matematika III. Brno: FSI VUT V Akademickém nakladatelství CERM Brno, 2001
2. Karpíšek, Z.: Matematika IV - Statistika a pravděpodobnost. 2. vydání. Brno : FSI VUT v Akademickém nakladatelství CERM Brno, 2003.
3. Karpíšek, Z. – Popela, P. – Bednář, J.: Statistika a pravděpodobnost. Učební pomůcka - studijní opora pro kombinované studium. Brno : FSI VUT v Akademickém nakladatelství CERM Brno, 2002.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3S-K combined study B-STG Manufacturing Technology -- Cr,Ex 4 Compulsory 1 2 W
B3S-K combined study B-AIŘ Applied Computer Science and Control -- Cr,Ex 4 Compulsory 1 2 W
B3S-K combined study B-SSZ Machine and Equipment Construction -- Cr,Ex 4 Compulsory 1 2 W