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PřF:M6800 Calculus of Variations - Course Information

## M6800 Calculus of Variations

**Faculty of Science**

Spring 2022

**Extent and Intensity**- 2/1/0. 5 credit(s). Type of Completion: zk (examination).

Taught in person. **Teacher(s)**- Mgr. Peter Šepitka, Ph.D. (lecturer)
**Guaranteed by**- prof. RNDr. Roman Šimon Hilscher, DSc.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Prerequisites**- Differential and integral calculus of functions of one and several variables, linear algebra, differential equations.
**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**- Economics (programme ESF, N-MA)
- Mathematical Analysis (programme PřF, N-MA)

**Course objectives**- Elementary course in calculus of variations. The content copies the classical differential calculus of functions in infinite dimension. Students will understand necessary and sufficient conditions for (weak) extrema in such problems and applications.
**Learning outcomes**- The student will understand the role of necessary and sufficient conditions for a weak extremum and will be able to use them for solving calculus of variations problems. Student will understand the differences between the classical calculus of functions of one (or several) variables, as well as the historical background of the calculus of variations.
**Syllabus**- Functional
- Simple variational problems
- Function spaces
- Variation of a functional
- Necessary conditions for weak extremum
- Euler's equation
- Fixed and variable endpoints
- Second variation
- Sufficient conditions for an extremum
- Relationship between weak and strong extremum

**Literature**- GEL'FAND, Izrail Moisejevič and Sergej Vasil'jevič FOMIN.
*Calculus of variations*. Edited by Richard A. Silverman. Mineola, N. Y.: Dover Publications, 2000. vii, 232 s. ISBN 0-486-41448-5. info *A history of analysis*. Edited by Hans Niels Jahnke. Providence: American Mathematical Society, 2003. ix, 422 s. ISBN 0-8218-2623-9. info- MESTERTON-GIBBONS, Mike.
*A primer on the calculus of variations and optimal control theory*. Providence, R.I.: American Mathematical Society, 2009. xiii, 252. ISBN 9780821847725. info - WEINSTOCK, Robert.
*Calculus of variations : with applications to physics and engineering*. New York: Dover Publications, 1974. x, 326. ISBN 0486630692. info - SAGAN, Hans.
*Introduction to the calculus of variations*. New York, N.Y.: Dover Publications, 1969. xvi, 449. ISBN 0486673669. info

- GEL'FAND, Izrail Moisejevič and Sergej Vasil'jevič FOMIN.
**Teaching methods**- Lectures and exercises.
**Assessment methods**- Two-hour written final exam (it is needed to reach at least 50 % of points) with oral evaluation of the exam with each student.
**Language of instruction**- Czech
**Further Comments**- The course is taught once in two years.

The course is taught: every week.

- Enrolment Statistics (Spring 2022, recent)
- Permalink: https://is.muni.cz/course/sci/spring2022/M6800