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A course is the basic teaching unit, it's design as a medium for a student to acquire comprehensive knowledge and skills indispensable in the given field. A course guarantor is responsible for the factual content of the course.
For each course, there is a department responsible for the course organisation. A person responsible for timetabling for a given department sets a time schedule of teaching and for each class, s/he assigns an instructor and/or an examiner.
Expected time consumption of the course is expressed by a course attribute extent of teaching. For example, extent = 2 +2 indicates two teaching hours of lectures and two teaching hours of seminar (lab) per week.
At the end of each semester, the course instructor has to evaluate the extent to which a student has acquired the expected knowledge and skills. The type of this evaluation is indicated by the attribute completion. So, a course can be completed by just an assessment ('pouze zápočet'), by a graded assessment ('klasifikovaný zápočet'), or by just an examination ('pouze zkouška') or by an assessment and examination ('zápočet a zkouška') .
The difficulty of a given course is evaluated by the amount of ECTS credits.
The course is in session (cf. teaching is going on) during a semester. Each course is offered either in the winter ('zimní') or summer ('letní') semester of an academic year. Exceptionally, a course might be offered in both semesters.
The subject matter of a course is described in various texts.

BIK-MA2.21 Mathematical Analysis 2 Extent of teaching: 21KP+4KC
Instructor: Olšák P. Completion: Z,ZK
Department: 18105 Credits: 6 Semester: Z

Annotation:
The course completes the theme of analysis of real functions of a real variable initiated in BIK-MA1 by introducing the Riemann integral. Students will learn how to integrate by parts and use the substitution method.The next part of the course is devoted to number series, and Taylor polynomials and series. We apply Taylor?s theorem to the computation of elementary functions with a prescribed accuracy. Then we study the linear recurrence equations with constant coefficients, the complexity of recursive algorithms, and its analysis using the Master theorem. Finally, we introduce the student to the theory of multivariate functions. After establishing basic concepts of partial derivative, gradient, and Hessian matrix, we study the analytical method of localization of local extrema of multivariate functions as well as the numerical descent method. We conclude the course with the integration of multivariate functions. This course can be enrolled only after successful completion of the course BIK-MA1, which can be replaced by the course BIK-ZMA in the case of repetitive students.

Lecture syllabus:
1. Primitive function and indefinite integral.
2. Integration by parts and the substitution method for the indefinite integral.
3. Riemann?s definite integral, Newton-Leibniz theorem, and generalized Riemann?s integral.
4. Integration by parts and the substitution method for the definite integral.
5. Numerical computation of the definite integral.
6. Number series, criteria of their convergence, estimates of asymptotic behaviour of their partial sums.
7. Taylor?s polynomials and series.
8. Taylor?s theorem and its application to computation of elementary functions with prescribed precision.
9. Homogeneous linear recurrence equations with constant coefficients.
10. Non-homogeneous linear recurrence equations with constant coefficients.
11. The complexity of recurrence algorithms, the Master theorem.
12. [2] Multivariate functions, partial derivative, gradient, and Hessian matrix.
14. Various types of definiteness of matrices and methods of its determination.
15. The analytical method for finding local extrema of multivariate functions.
16. Principle of numerical descent methods for localization of local extrema of multivariate functions.
17. Riemann?s integral of multivariate function, Fubini?s theorem.
18. Substitution in Riemann?s integral of multivariate function.

Seminar syllabus:
1. Indefinite integral, integration by parts and the substitution method.
2. Definite integral, Newton-Leibniz theorem, integration by parts and the substitution method.
3. Number series, criteria of their convergence
4. Estimates of asymptotic behaviour of partial sums of series.
5. Taylor?s polynomials and series.
6. Taylor?s theorem and its application.
7. Linear recurrence equations.
8. The Master theorem.
9. Multivariate functions, partial derivative, gradient, and Hessian matrix.
10. The analytical method for finding local extrema of multivariate functions.
11. Riemann?s integral of multivariate function, Fubini?s theorem.
12. Substitution in Riemann?s integral of multivariate function.

Literature:
1. Oberguggenberger M., Ostermann A. : Analysis for Computer Scientists. Springer, 2018. ISBN 978-0-85729-445-6.
2. Nagle R. K., Saff E. B., Snider A. D. : Fundamentals of Differential Equations (9th Edition). Pearson, 2017. ISBN 978-0321977069.
3. Graham R. L., Knuth D. E., Patashnik O. : Concrete Mathematics: A Foundation for Computer Science (2nd Edition). Addison-Wesley Professional, 1994. ISBN 978-0201558029.
4. Kopáček J.: Matematická analýza nejen pro fyziky I, Matfyzpress, 2016, ISBN 978-80-7378-353-5
5. Kopáček J.: Matematická analýza nejen pro fyziky II, Matfyzpress, 2015, ISBN 978-80-7378-282-5

Requirements:
Knowledge from BIE-MA1.21, BIE-DML.21, and BIE-LA1.21.

The course is also part of the following Study plans:
Study Plan Study Branch/Specialization Role Recommended semester
BIK-IB.21 Information Security 2021 (in Czech) PP 3
BIK-SPOL.21 Unspecified Branch/Specialisation of Study PP 3
BIK-PV.21 Computer Systems and Virtualization 2021 (in Czech) PP 3
BIK-PS.21 Computer Networks and Internet 2021 (in Czech) PP 3
BIK-SI.21 Software Engineering 2021 (in Czech) PP 3


Page updated 28. 3. 2024, semester: Z/2023-4, L/2019-20, L/2022-3, Z/2019-20, Z/2022-3, L/2020-1, L/2023-4, Z/2020-1, Z,L/2021-2, Send comments to the content presented here to Administrator of study plans Design and implementation: J. Novák, I. Halaška