M2100F Mathematical Analysis II

Faculty of Science
Spring 2025
Extent and Intensity
4/2/0. 6 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
prof. Mgr. Petr Hasil, Ph.D. (lecturer)
Mgr. Stanislav Hronek (seminar tutor)
Mgr. Michael Krbek, Ph.D. (seminar tutor)
Guaranteed by
prof. Mgr. Petr Hasil, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
M1100 Mathematical Analysis I || M1101 Mathematical Analysis I || M1100F Mathematical Analysis I
The knowledge from course Mathematical Analysis I is assumed.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 7 fields of study the course is directly associated with, display
Course objectives
Second part of the basic course of the mathematical analysis. The subject of the course are the theory of metric spaces, the differential calculus of functions of several variables, and the integral calculus of functions of several variables. After passing the course, the student will be able: to define and interpret the basic notions used in the basic parts of mathematical analysis and to explain their mutual context; to formulate relevant mathematical theorems and statements and to explain methods of their proofs; to use effective techniques utilized in basic fields of analysis; to apply acquired pieces of knowledge for the solution of specific problems including problems of applicative character.
Learning outcomes
At the end of the course students will be able to:
- define and interpret the notions from the theory of metric spaces and the differential and integral calculus of functions of several variables;
- formulate relevant mathematical theorems and to explain methods of their proofs;
- analyse problems from the topics of the course;
- understand to theoretical and practical methods of the theory of metric spaces and the differential and integral calculus of functions of several variables;
- apply the methods of mathematical analysis to concrete problems.
Syllabus
  • I. Metric spaces: basic definitions, convergence, open and closed sets, continuous mappings, complete spaces, compact spaces, Banach contraction principle. II. Differential calculus of functions of several variables: limit, continuity, partial derivatives, Taylor's formula, extrema, mappings between higherdimensional spaces, implicit function theorem, constrained extrema. III. Integral calculus of functions of several variables: Jordan measure, Riemann integral, Fubini theorem, transformation theorem for multiple integrals, elements of curvelinear and surface integral, integrals depending on a parameter.
Literature
    recommended literature
  • DOŠLÁ, Zuzana and Ondřej DOŠLÝ. Metrické prostory : teorie a příklady. 4. vydání. Brno: Masarykova univerzita, 2016, viii, 90. ISBN 9788021083578. info
  • DOŠLÁ, Zuzana and Ondřej DOŠLÝ. Diferenciální počet funkcí více proměnných. 1. dotisk 3. vyd. Brno: Masarykova univerzita, 2010, 144 pp. ISBN 978-80-210-4159-2. info
  • KALAS, Josef and Jaromír KUBEN. Integrální počet funkcí více proměnných (Integral calculus of functions of several variables). 1st ed. Brno: Masarykova univerzita, 2009, 278 pp. ISBN 978-80-210-4975-8. info
    not specified
  • HASIL, Petr and Petr ZEMÁNEK. Sbírka řešených příkladů z matematické analýzy II (Collection of Solved Problems in Mathematical Analysis II). Masarykova univerzita, 2016. URL info
  • ADAMS, R. A. and Christopher ESSEX. Calculus : a complete course. 7th ed. Toronto: Pearson, 2010, xvi, 973. ISBN 9780321549280. info
  • BUCK, R. Creighton. Advanced calculus. 3d ed. Long Grove: Waveland Press, 2003, x, 622. ISBN 1577663020. info
  • RÁB, Miloš. Zobrazení a Riemannův integrál v En. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1988, 97 s. info
  • JARNÍK, Vojtěch. Integrální počet. Vyd. 2. Praha: Academia, 1976, 763 s. URL info
  • BRAND, Louis. Advanced calculus : an introduction to classical analysis. New York: John Wiley & Sons, 1955, x, 574. info
Teaching methods
Standard theoretical lectures with excercises.
Assessment methods
Adjustment for pandemic period (onsite/online teaching):
Lectures and seminars are NOT compulsory.
The exam will be probably online. Specific course according to the situation at the time.
If possible, other standard rules will be maintained.

Standard rules for regular semesters:
Lectures: 4 hours/week. Seminars (compulsory): 2 hours/week.
5 written intrasemestral tests in seminars (10% of the overall evaluations).
Final exam: Written test (55%) and oral exam (35%).
To pass: at least 5 of 10 points from intrasemestral tests, then 45% in total.
Results of the intrasemestral tests are included in the overall evaluation. All percentages are given relative to the overall total for the whole semester.
Language of instruction
Czech
Follow-Up Courses
Further Comments
The course is taught annually.
The course is taught: every week.
Listed among pre-requisites of other courses
The course is also listed under the following terms Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024.
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