M6800 Calculus of Variations

Faculty of Science
Spring 2026
Extent and Intensity
2/1/0. 5 credit(s). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
doc. Mgr. Peter Šepitka, Ph.D. (lecturer)
Guaranteed by
doc. Mgr. Peter Šepitka, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 16. 2. to Fri 22. 5. Wed 10:00–11:50 M3,01023
  • Timetable of Seminar Groups:
M6800/01: Mon 16. 2. to Fri 22. 5. Thu 12:00–12:50 M3,01023, P. Šepitka
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
Abstract
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.
Key topics
  • 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
Study resources and 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
  • KOT, Mark. A first course in the calculus of variations. Providence, R.I.: American Mathematical Society, 2014, x, 298. ISBN 9781470414955. info
Approaches, practices, and methods used in teaching
Lectures and exercises.
Method of verifying learning outcomes and course completion requirements

Two-hour written final exam (it is needed to reach at least 40 % of points) with oral evaluation of the exam with each student. The conditions (especially regarding the form of the exam) will be specified according to the epidemiological situation and valid restrictions.

Language of instruction
Czech
Further Comments
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2012, Spring 2014, Spring 2016, spring 2018, Spring 2020, Spring 2022, Spring 2024.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/spring2026/M6800