## Tuesday, 2 February 2016

### 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