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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025
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Welcome page > Faculty of Information Technology > Department of Applied Mathematics > Mathematical Analysis 2

Mathematical Analysis 2

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Code Completion Credits Range Language
BI-MA2.21 Z,ZK 6 3P+2C Czech
Course guarantor:
Tomáš Kalvoda
Lecturer:
Tomáš Kalvoda, Ivo Petr
Tutor:
Ondřej Bouchala, Pavel Hrabák, Tomáš Kalvoda, Pavel Paták, Ivo Petr, Jan Starý, Jana Vacková, Jan Valdman, Jaroslav Zhouf
Supervisor:
Department of Applied Mathematics
Synopsis:

The course completes the theme of analysis of real functions of a real variable initiated in BI-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.

Requirements:

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

Syllabus of lectures:

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. Multivariate functions, partial derivative, gradient, and Hessian matrix.

13. Various types of definiteness of matrices and methods of its determination.

14. The analytical method for finding local extrema of multivariate functions.

15. Principle of numerical descent methods for localization of local extrema of multivariate functions.

16. Riemann’s integral of multivariate function, Fubini’s theorem.

17. Substitution in Riemann’s integral of multivariate function.

Syllabus of tutorials:

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.

Study Objective:
Study materials:

The course is equipped with a dedicated textbook. Additionaly one can consult the following publications.

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

Note:
Further information:
https://courses.fit.cvut.cz/BI-MA2/
Time-table for winter semester 2024/2025:
06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
roomT9:155
Kalvoda T.
13:30–16:00
(lecture parallel1)
Dejvice roomT9:347
Bouchala O.
16:15–17:45
(parallel nr.2)
Dejvice roomT9:347
Bouchala O.
18:00–19:30
(parallel nr.3)
Dejvice
Tue
roomT9:346
Valdman J.
07:30–09:00
(parallel nr.4)
Dejvice
NBFIT schoolroomroomTH:A-942
Zhouf J.
09:15–10:45
(parallel nr.1)
Thákurova 7 (budova FSv) roomTH:A-942
Zhouf J.
11:00–12:30
(parallel nr.7)
Thákurova 7 (budova FSv) roomT9:346
Valdman J.
12:45–14:15
(parallel nr.9)
Dejvice
NBFIT schoolroom
roomTH:A-1242
Hrabák P.
07:30–09:00
(parallel nr.5)
Thákurova 7 (budova FSv) roomT9:346
Valdman J.
09:15–10:45
(parallel nr.6)
Dejvice
NBFIT schoolroomroomT9:346
Valdman J.
11:00–12:30
(parallel nr.8)
Dejvice
NBFIT schoolroom
roomTK:BS
Petr I.
09:15–10:45
(lecture parallel2)
Dejvice
Wed
roomTH:A-1242
Vacková J.
07:30–09:00
(parallel nr.10)
Thákurova 7 (budova FSv) roomTH:A-1242
Vacková J.
09:15–10:45
(parallel nr.11)
Thákurova 7 (budova FSv) roomTK:BS
Petr I.
12:45–14:15
ODD WEEK
(lecture parallel2)
Dejvice
roomTH:A-942
Zhouf J.
09:15–10:45
(parallel nr.12)
Thákurova 7 (budova FSv) roomTH:A-1442
Bouchala O.
11:00–12:30
(parallel nr.13)
Thákurova 7 (budova FSv) roomTH:A-1247
Vacková J.
12:45–14:15
(parallel nr.15)
Thákurova 7 (budova FSv)
roomTH:A-942
Zhouf J.
11:00–12:30
(parallel nr.14)
Thákurova 7 (budova FSv) roomTH:A-1442
Bouchala O.
12:45–14:15
(parallel nr.16)
Thákurova 7 (budova FSv)
Thu
roomTH:A-1442
Starý J.
09:15–10:45
(parallel nr.18)
Thákurova 7 (budova FSv) roomTH:A-1442
Starý J.
11:00–12:30
(parallel nr.19)
Thákurova 7 (budova FSv)
Fri
roomT9:347
Starý J.
09:15–10:45
(parallel nr.20)
Dejvice roomT9:347
Starý J.
11:00–12:30
(parallel nr.21)
Dejvice roomT9:347
Starý J.
12:45–14:15
(parallel nr.22)
Dejvice roomT9:347
Starý J.
14:30–16:00
(parallel nr.23)
Dejvice
Time-table for summer semester 2024/2025:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2024-11-21
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