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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2021/2022

Mathematical Analysis 2

The course is not on the list Without time-table
Code Completion Credits Range Language
BI-MA2.21 Z,ZK 6 3P+2C Czech
Enrollement in the course requires an successful completion of the following courses:
Lecturer:
Tutor:
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.

This course can be enrolled only after successful completion of the course BI-MA1, which can be replaced by the course BI-ZMA in the case of repetitive students.

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://course.fit.cvut.cz/BI-MA2/
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2021-10-25
For updated information see http://bilakniha.cvut.cz/en/predmet6539506.html