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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