There were some typos (too many \(\varepsilon\)'s) in the proof of Lemma 4.1.1 on page 41. I corrected these, see the link in the left-hand side margin.

## Sunday, 28 February 2016

## Tuesday, 16 February 2016

### Problem sheet 5

A new problem sheet has now been posted. Given last week's test, it appeared to me that most of you had been studying for the test rather than on recent course material, so I decided to delay the next sheet for a few days.

## Sunday, 14 February 2016

### Lecture notes Chapter 5

The material we need for the Poincare-Bendixson Theorem is covered in chapter 10 of

*Differential equations, dynamical systems and an introduction to chaos*by M.W. Hirisch, S. Smale and R.L. Devaney. Please be aware that the entire book can be consulted digitally from the Imperial library website (where this chapter can also be downloaded; just search for the authors on the Library search and you find a link to the digitcal copy). For your convenience, I also created a temporary link in the left-hand side margin, under chapter 5.### Lectures, office hour and tuesday problem class this week

In my absence, the lectures on Tuesday, Thursday and Friday this week will be given by Dr Trevor Clarke. He will also be standing in for me during the office hour (in Husley 638) on Tuesday afternoon at 3pm, as usual. Tuesday's problem session after the lecture will be conducted by Dr Christian Pangerl (in Prof Turaev's absence) and will concern last week's test.

### Lecture notes Chapter 4

Please note that the temporary notes have been replaces with a proper version. Please disregard the preliminary ones.

## Thursday, 11 February 2016

### Model answers first progress test

Please find model answers for the first progress test in the left-hand side margin. Please note that it does not necessarily mean that if your answers are a little different, that they will be necessarily wrong.

## Wednesday, 10 February 2016

### Update Lecture Notes Chapter 3

Chapter 3 has now been updated to include the Jordan-Chevalley decomposition section. The stability section also includes now some text about Lyapunov functions. I further made some small changes to the text, but nothing significant.

### Jordan normal form notes

Several of you have asked me about more information about Jordan normal forms, like proofs or algorithms. There is no time in the lectures to really do the proofs and it is my experience that exercises in constructing nontrivial Jordan forms for examples is not a popular past-time for most of you. But for those interested I link here two manuscripts that may be of use: a relatively compact proof of the Jordan normal form theorem (Prof Sebastian van Strien's appendix to the M2AA1 lecture notes of last year), and
a more constructive and algorithmic approach to Jordan forms (2005 notes by Dr Stefan Friedl)

## Saturday, 6 February 2016

### Videos online

I posted videos of the proof of the Inverse Function Theorem, Implicit Function Theorem and higher dimensional derivatives in the left hand margin. Please note that there is an annoying typo in the last line of the proof of the Inverse Function Theorem: in the first posted version of chapter 1, at the end of the proof of the derivative,

*F*has the variable y as argument but it should of course be*x*(identically copied from the line above). As you see in the video, one finally inserts \(x=G(y)\) to get the result.## Friday, 5 February 2016

### Inner product in \(\mathbb{C}^n\)

I received a question concerning Chapter 3 of the lecture notes where an inner product is used in \(\mathbb{C}^n\) (instead of \(\mathbb{R}^n\)), when doing computations with complex eigenvectors. I thought that you had seen this before, but in case you haven't, please note that the standard inner product in (\(\mathbb{C}^n\)):
$$\langle \textbf{x},\textbf{y}\rangle=\sum_ix_i\cdot\overline{y}_i$$

## Thursday, 4 February 2016

### Videos proof of Inverse and Implicit Function Theorem, and derivatives of maps

I am in the process of producing some short videos where I go (more slowly) through the proofs of the Inverse and Implicit Function theorems (of Chapter 2), and also discuss in more detail how to deal with higher dimensional derivatives, as I received several questions about this. It is the first time that I am making such videos and I had some issues with the equipment slowing me down. I intend to have these videos up on this webpage still before the weekend.

### Jordan-Chevalley Decomposition

The Jordan-Chevalley decomposition subsection (3.3.3) was not yet compiled into Chapter 3 of the lecture notes. I have temporarily added a link to this subsection in the left hand side margin and will integrate it into the chapter properly, asap.

## Tuesday, 2 February 2016

### Small revision model answer 5 of problem sheet nr1

Some asked me how I concluded so quickly in the previously posted model answer for this question that in case c=0 the sequence was Cauchy and thus converged, and I decided to expand the explanation.

### How to prepare for the class test of 11 Feb?

The material to study for the class test is chapters 1 (Contractions) and 2 (Existence and uniquesness) from the lecture notes, and the problems sheets Nrs 1, 2 & 3 relating to these chapters.

I now summarize in some detail the main points in this material:

Chap 1:

- definition of metric space

- do not worry about the “elementary notions" in metric spaces on the bottom of p5 and top of p6.

- definition of contraction

- contraction mapping theorem (including proof)

- derivative test in R (including proof)

- do not learn example 1.3.3 by heart! (it is instructive to understand it, though)

- Theorem 1.3.6 (derivative test in higher dimensions) (no proof, as not given)

- Inverse function theorem in R (including proof)

- Inverse function theorem in R^n (not the proof)

- Implicit function theorem in R^n (including proof)

Chap 2:

- Picard iteration applied to examples

- Some examples of ODEs without existence and uniqueness of solutions

- Picard-Lindelof Theorem (including proof; but not regarding the completeness

of the function space C(J,U))

- Gronwall’s inequality and application to Theorem 2.2.3, establishing continuity of the finite-time

flow

Exercise sheets:

Nr 1: all * problems, question 6; questions 5&7 are not crucial

Nr 2: all * questions, question 7 (limited to intersections of surfaces and curves in R^3);

questions 5&6 are not crucial

Nr 3: all * questions, questions 5 & 7; questions 6 & 8 are not crucial

I now summarize in some detail the main points in this material:

Chap 1:

- definition of metric space

- do not worry about the “elementary notions" in metric spaces on the bottom of p5 and top of p6.

- definition of contraction

- contraction mapping theorem (including proof)

- derivative test in R (including proof)

- do not learn example 1.3.3 by heart! (it is instructive to understand it, though)

- Theorem 1.3.6 (derivative test in higher dimensions) (no proof, as not given)

- Inverse function theorem in R (including proof)

- Inverse function theorem in R^n (not the proof)

- Implicit function theorem in R^n (including proof)

Chap 2:

- Picard iteration applied to examples

- Some examples of ODEs without existence and uniqueness of solutions

- Picard-Lindelof Theorem (including proof; but not regarding the completeness

of the function space C(J,U))

- Gronwall’s inequality and application to Theorem 2.2.3, establishing continuity of the finite-time

flow

Exercise sheets:

Nr 1: all * problems, question 6; questions 5&7 are not crucial

Nr 2: all * questions, question 7 (limited to intersections of surfaces and curves in R^3);

questions 5&6 are not crucial

Nr 3: all * questions, questions 5 & 7; questions 6 & 8 are not crucial

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