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In undergraduate calculus we learn to maximise or minimise functions of one variable, finding optimal points. The Calculus of Variations is concerned with finding optimal functions and the properties of these optimisers. Famous examples include surfaces of minimal area or the shortest or quickest paths between given points. In the classical indirect method, the optimisers are found as solutions of the Euler-Lagrange differential equations. In the modern direct method, one uses abstract means to find optimisers, which often yields existence results for solutions of differential equations.


Autumn 2016 (Monday, October 3 to Friday, December 9)


  • Fri 13:05 - 13:55


Undergraduate calculus


In this 10-lecture series, elements of both the classical and the modern theory are presented. A tentative plan is as follows: 1. Introduction. One-dimensional variational problems 2. Fundamental lemma, Euler-Lagrange equations 3. Convexity and existence and regularity issues 4. Second variations and necessary conditions for optimality 5. Variational problems with constraints 6. Problems involving multiple integrals 7. Direct method: coercivity and lower semicontinuity 8. Sobolev spaces 9. Weak convergence and compactness 10. The direct method for integral functionals and existence of solutions for some nonlinear PDEs


Matthias Kurzke (main contact)
Phone 0115 9514984
Yves van Gennip
Phone 0115 8466166


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Hussien Abugirda
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Hayder Al-Saedi
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Manal Alqhtani
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Birzhan Ayanbayev
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Marco Baffetti
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Panagiota Birmpa
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Blesson Chacko
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Alex Close
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Liam Dobson
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Joe Driscoll
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Duc Lam Duong
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Megan Goode
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Alex Hiles
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Abdul Jumaat
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Blaine Keetch
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James Kulmer
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Amy Mallinson
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George Morrison
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Entesar Nasr
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Tom O'Neill
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George Papanikos
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Sam Povall
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Michael Roberts
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David Robertson
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Aleksandra Ross
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Anhad Sandhu
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Tomas Stary
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Ville Syrjanen
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Anthony Thompson
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Andrew Turner
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Luke Warren
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Paul Wileman
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Anthony Williams
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Jingsi Xu
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Daoping Zhang


Calculus of VariationsJost and Li-Jost
Calculus of Variations I: The Lagrangian FormalismGiaquinta and Hildebrandt
Introduction to the Calculus of VariationsSagan
Calculus of VariationsGelfand and Fomin


Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)


This course will be assessed by a single take home paper. The paper will become available in January and answers will have to be submitted online within 2 weeks. The paper will be organised in questions, and marks will be given for answers. Half of the marks will be needed to pass.

Calculus of Variations take-home exam

Files:Exam paper
Released: Monday 9 January 2017 (223.2 days ago)
Deadline: Sunday 22 January 2017 (209.2 days ago)


Files marked L are intended to be displayed on the main screen during lectures.

0Complete Lecture notes (1-10)L
0Handwritten Remarks L1
0Handwritten Remarks L2
0Handwritten Remarks L4
0Lecture 1L
0Lecture 2L
0Lecture 3L
0Lecture 4L
0Lecture 5L
0Lecture 6L
0Lecture 7L

Recorded Lectures

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