I have posted proposed solutions of the exam problems.
Hope you enjoyed the exam!!
Have a nice summer!
We didn't get time to do the last section in the notes of TS. So from those the curriculum is
from the beginning of the second part "Calculus of variation" that starts on page 18 until the section called "Weierstrass' sufficency condition" on page 34.
Next weak I'll give short recapitulation of the course and do some exercises.
Here some exercises I can do:
Exam 2015 No 2, 3 and 5.
Exam 2010 No 1 and 2.
Exam 2006 No1
Exam 2009 No 1
I plan to start on the exercises about calculus of variation tomorrow, but
one wanted me to do 2013 No 2b, so I'll do that.
From the note by TS:
Exercise 1, 3, 5, 7, 8 and 9.
A slight change in the curriculum. Theorem 4.7 about Lyaponov functions is included.
I'll do the following exam problems in plenum:
2011 No. 1a and No 2.
2012 No. 2
2013 No. 1 and No. 2 a,b
2014 No. 1 and No.
We start in the second hour on this wednesday.
During the morning I have gotten ill - nauseous and dizzy - and I can not lecture to day. So the lecture is canceled.
There will be no lecture this Wednesday (that is, Wed Mar 5).
Since several of you have mid-term exams next week, we drop the lecture on Monday.
On Wednesday I'll give a short revue of what we have done so far, and do exercises (No 22 from exercises4, and start on exercises5).
Mon Mar 20: No lecture
Wed Mar 22: Revue + exercises
Good luck with mid-term exams!
Here are some exams problems that are relevant (later in the course orthers will be as well).
June 2015 No 1,2 and 3
June 2014 No 1
June 2013 No 1
June 2012 No 1
You will get the problems for the mandatory assignment on Mon 20. March.
They must be delivered before Thu 20. April
To day I did the two examples on page 91.
During the exercise section I did problem 17 on page 70.
Next monday I 'll speek about theorem 3.11 (without proof)
and continue with chapter 3.
I'll tone down the proves, and concentrate on the content of the results and examples.
Next wednesday we continue with exercises from "exercises 4".
New exercises from chapter 3 will come soon.
This week we shall prove theorem 3.10 on page 86, the Picard Lindelöf existence and uniqueness theorem. Every thing is in the book on pages 72 - 87. I'll only do the first proof.
We start with background stuff:
I'll the def of the Picard iteration, then recall the def of the Lipschitz condition.
Speak about the sup-norm anduniform convergence and the function spaces C(J,R^n). A little about metric spaces
Cauchy sequences, convergence and completeness.
Then the Fix point theorem for contracting maps (theorem 3.4).
Then, finally, we'll prove th 3.10
Today, Wed Feb 15, I finished 2.6 (the example on page 53) and recalled briefly the "phase portraits" in 2.2. On Mon Feb 20 I'll do 2.7.
Dagens forelesning er avlyst pga sykdom.
To day I almost finished 2.5.
On Mon Feb 6 I say a few words more about 2.5 and do 2.6.
May be we have time to start on 1.7.
To day I did section 2.3
On Mon Jan 30 I ll continue with 2.4, 2.5 and probably 2.6
And exercises 1 and 7 from the book. We continue with the exercises next wednesday.
To day Mon Jan 23 I did 2.1 and 2.1
On Wed Jan 25 we start on 2.3 and will probably have time to do 2.4 as well.
I ll do exercises in the second hour.
On Wed Jan 18 I basically finished chap 1, but I ll probably come back to some of the examples, like the paragraph "Mechanincal systems". We started on chap 2.
Today on Mon Jan 23 I'll continue with chap 2.2, 2.3 and may be 2.4.
On Wed Jan 25 I'll continue with chap 2; ie 2.4 and probably 2.5
We decided to use the second hour every Wed for exercises, starting on Wed Jan 25.
To day I basically I did 1.1, 1.2, 1.3 and the part of 1.4 about population dynamics.
Then I started on 1.5.
On Wed Jan 19 I ll redo what I started on in 1.5 and do the rest of 1.5
I ll go back to 1.4 and do the paragraph called "Mechanical systems" and may be a few words about the two last paragraphs "Oscillating circuits" and "fluid mixing".
The paragraphs 1.6 and 1.7 are not in curriculum.
Then if time permits, I'll start on chapter 2.
We are going to use a new book for this course:
Differential Dynamical Systems by James D. Meiss