Analyse in meer variabelen, period 4

All references are to the books by Duistermaat and Kolk

1a. Week 18 (April 29): 2.1 Linear algebra

linear algebra, distinction of linear map and matrix, Euclidean norm, norm as trace.

Lemma 2.1.1: multiplicative properties of this norm.

Inversion of matrices, Cramer s rule
2.2 Differentiable mappings

total derivative, relation with directional derivative, Lemma 2.2.3.

Prop 2.2.9: differentiability component wise

criterion for differentiability (will be discussed next time): Thm 2.3.4, Def 2.3.6, Thm 2.3.7.
2.3 Directional and partial derivatives

Def 2.3.1, Prop. 2.3.2. criterion for differentiability: Thm 2.3.4, Def 2.3.6, Thm 2.3.7
2.4 Chain rule

Lemma 2.27: Hadamard s characterization of differentiability, Cor. 2.2.9 (proof next time).

Theorem 2.4.1: chain rule, and proof by using Hadamard.

1b. Week 18 (May 1):

proof of criterion for differentiability.

Cor 2.4.2: chain rule for partials, Lemma 2.4.7.
2.5 Mean value theorem

Lemma 2.5.1 and Thm. 2.5.3: Lipschitz continuity

2a. Week 19 (May 6):

Proof of Hadamard's criterion
3 Inverse and implicit function theorems
3.1 Diffeomorphisms
3.2 Inverse function theorem

the notion of diffeomorphism

inverse function theorem for homeomorphisms: Lemma 3.2.1 and Prop. 3.2.2, formulation.

review of contraction theorem, Lemma 1.7.2.

2b. Week 19 (May 8):

Prop 3.2.3: local inverse is homeomorphism

the notion of local diffeomorphism, local inverse function theorem Theorem 3.2.4.

global inverse function theorem, 3.2.8.

3.3: Application to polar coordinates.

3a. Week 20 (May 13):

3.4: description of implicit function question by parameter dependent system of equations.

idea of the linearized system

idea of implicit differentation

Thm 3.5.1: formulation and proof of implicit function theorem.

3.6.A: Application: smooth dependence of roots of a polynomial.

3b. Week 20 (May 15):

4.1 application of implicit function thm: hypersurface as local graph of a function.

the notion of submersion, and application of Implicit function theorem to get local graph representation.

4.2.1 definiton of submanifold see also extra notes, Def. 10, for preferred definition.

equivalence of the two notions of submanifold.

4.2.4 definition of immersion

4.3 immersion theorem, Thm 4.3.1, see also extra notes.

Representation of submanifold by immersion.

Linear algebra for surjective linear map.

Linear version of submersion theorem and C^k version of it.

4a. Week 21 (May 20):

proof of immersion theorem, see also extra notes.

def 4.2.4 of embedding; regular point; parametrization; local coordinates; chart.

Cor. 4.3.2, image of embedding is submanifold.

Several characterizations of submanifold: Thm. 4.7.1.

5.1. The notion of tangent space.

Relation to the various characterisations of submanifold, Thm 5.1.2.

4b. Week 21 (May 22):

the notion of tangent mapping: 5.2.

the Lagrange multiplier method: 5.4, see also the extra notes.

start with Book II, Ch 6: integration.

6.1: Rectangles and partitions

6.2: Riemann integrability

5a. Week 22 (May 27):

6.3 Jordan measurability

Theorem 6.3.5: area under a graph

Cor. 6.3.7: negligibility of a graph.

5b. Week 22 (May 29):

step functions

Thm. 6.4.2: iterated integration

Cor. 6.4.3: interchange of order

Thm 6.4.5 Successive integration

6.6 the change of variables theorem, formulation and examples.

Naturality of the change of variables formula under composition.

Idea of localizing the change of variable theorem.

6a. Week 23 (June 3):

6.7 partitions of unity

Proof of change of variables for continuous functions with compact support.

6b. Week 23 (June 5):

6.8 approximation of Riemann integrable functions

6.10 Absolute convergence of integrals

Chapter 7: densities and their integrals, see the extra lecture notes

volume of parallelepiped, formal definition of density

7a. Week 24 (June 10): Pentecost, no lecture.

7b. Week 24 (June 12):

pull-back of a density

continuous density on open set, pull-back

reformulation of substitution of variables

density on a submanifold, pull-back under embedding

integration of densities on submanifold

Euclidean density on a submanifold, several formulas for it.

8a. Week 25 (June 17): Final lecture

Partial integration and theorem of Gauss: see the extra lecture notes (and book 7.6).

definition of domain with C^k boundary (replaces book, 7.5)

outward unit normal, continuity

Theorem of Gauss (book: Thm 7.6.1), partial integration (book: Cor. 7.6.2) and divergence (book Thm 7.8.50) versions.

Proof of Gauss' theorem (Thm 7.6.1), by localisation and substitution of variables.

8b. Week 25 (June 19): no lecture, but question and rehearsal session.

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9. Week 26: exam week.

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Last change: April 12, 2019