Uusi opinto-opas (sisältäen myös opetusohjelmat) lukuvuodelle 2018-2019 sijaitsee osoitteessa https://opas.peppi.utu.fi . Tältä sivustolta löytyvät enää vanhat opinto-oppaat ja opetusohjelmat.

The new study guide (incl. teaching schedules) for academic year 2018-2019 can be found at https://studyguide.utu.fi. This site contains only previous years' guides.

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Archived Curricula Guide 2014–2016
Curricula Guide is archieved. Please refer to current Curricula Guides
SMAT5308 Calculus of Variations 5 ECTS
Organised by
Applied Mathematics
Person in charge
Department of Mathematics and Statistics, Yury Nikulin
Planned organizing times
Period(s) I II III IV
2014–2015 X
Preceding studies
Recommended:

General description

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.

Learning outcomes

The course provides basis for thorough understanding of the problems, methods and techniques of the calculus of variations and prepares students for the study of modern optimal control theory. The treatment is limited to extensive coverage of single integral problems in one and more unknown functions.

Contents

1 EXTREME VALUES OF FUNCTIONALS
2 THE THEORY OF THE FIRST VARIATION
3 THEORY OF FIELDS AND SUFFICIENT CONDITIONS FOR A STRONG RELATIVE EXTREMUM
4 THE HOMOGENEOUS PROBLEM
5 THE PROBLEM OF LAGRANGE AND THE ISOPERIMETRIC PROBLEM
6. THE MINIMUM PRINCIPLE OF PONTRYAGIN

Teaching methods

Teaching method Contact Online
Lectures 28 h 0 h
Exercises 14 h 0 h

Teaching language

English

Modes of study

Option 1
Available for:
  • Degree Programme Students
  • Other Students
  • Doctoral Students
  • Exchange Students
Exercises Exercise(s)
  • In English
Written exam
  • In English

Evaluation

Numeric 0-5.

Study materials

Introduction to the Calculus of Variations by Hans Sagan. Additionally available lecture notes (in Finnish)

 

Belongs to following study modules

Department of Mathematics and Statistics
2014–2015
Teaching
Archived Teaching Schedule. Please refer to current Teaching Shedule.
Department of Mathematics and Statistics
DP in Mathematics and Statistics
DP in Mathematics
Finnish Study Modules
DP in Statistics
Statistics