| Course Unit Code | 470-4123/01 |
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| Number of ECTS Credits Allocated | 6 ECTS credits |
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| Type of Course Unit * | Optional |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | First Year |
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| Semester when the Course Unit is delivered | Winter Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | There are no prerequisites or co-requisites for this course unit |
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| Name of Lecturer(s) | Personal ID | Name |
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| KRA04 | Mgr. Bohumil Krajc, Ph.D. |
| LUK76 | doc. Ing. Dalibor Lukáš, Ph.D. |
| KRA0220 | Ing. Jan Kracík, Ph.D. |
| Summary |
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Subject contains parts of higher mathematics omitted in some bachelor programs. Mastering selected parts is essential for the successful engineering degree in Computational Mathematics.
In the field of differential calculus of of functions of several variables we concetrate on the following parts: differentiating of composite functions, Taylor polynomials, implicit function theorem, constrained extremes. Students also learn the basic principles and methods of multidimensional integration. Attention to numerical methods is focused on issues of numerical differentiation and integration of functions. Some remarks are dedicated to a function approximation, search for zero points and extremal tasks.
A substantial part of the course is the interpretation of the relevant passages in the field of ordinary differential equations and their systems. |
| Learning Outcomes of the Course Unit |
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| A successful student of the course will have the knowledge and skills of a typical graduate bachelor degree in Computational Mathematics . |
| Course Contents |
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1.Diferential of a function of several variables. Gradient method.
2.Diferential of a composite function. Transformation of variables in the differential expressions.
3.Approximation of function . Taylor's theorem . Conditions for the existence of local extremes.
4.Numerical derivative. Approximate solutions of equations.
5.Theorem about implicitly defined function. Constrained extremes.
6.Construction of integral sums, numerical integration .
7.Definition of multiple integrals. Selected applications.
8.Fubini`s theorems. Substitution in multiple integrals . Geometric interpretation of Jacobian .
9.Theorems about the existence and uniqueness of solutions of initial value problems for ordinary differential equations. Euler's method.
10.Transformation of variables in differential equations .
11.Potential and its use for solving exact equations.
12.Ordinary differential equations of higher orders. Solving linear differential equations. Boundary value problems . |
| Recommended or Required Reading |
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| Required Reading: |
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• W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Company, New York, 1964
• W. E. Boyce, R. C. DiPrima: Elementary differential equations. Wiley, New York 1992
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• B. Budinský, J. Charvát: Matematika I., II., SNTL Praha, 1990
• J. Kuben, Š. Mayerová, P. Račková, P. Šarmanová: Diferenciální počet funkcí více proměnných, http://mi21.vsb.cz, online
• P. Vodstrčil, J. Bouchala: Integrální počet funkcí více proměnných, http://mi21.vsb.cz, 2012 (online)
• B. Krajc, P. Beremlijski: Obyčejné diferenciální rovnice, http://mi21.vsb.cz, 2012 (online)
• W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Company, New York, 1964
• W. E. Boyce, R. C. DiPrima: Elementary differential equations. Wiley, New York 1992
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| Recommended Reading: |
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| • M. Braun: Differential Equations and Their Applications. Springer, Berlin 1978. |
• K. Rektorys a kol.: Přehled užité matematiky, Prometheus, 1995
• M. Braun: Differential Equations and Their Applications. Springer, Berlin 1978. |
| Planned learning activities and teaching methods |
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| Lectures, Tutorials, Project work |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 30 | 10 |
| Examination | Examination | 70 | 21 |