| week | Material to cover [with relevant book sections] |
Exercises |
(To hand in on week n+1) |
| BLOK
3 |
|||
| 6 |
Standard notation for R^n inner product, norms,
Hilbert-Schmidt norm for linear operators [§§ 1.1, 2.1]. Metric space, open set, neighborhood, continuity of a map between metric spaces [§§ 1.2, 1.3]. Differentiability of a map at a point, total and directional derivatives [§§ 2.2, 2.3]. |
Prove
that two norms on a real vector space give rise to the same collection
of open sets if and only if either norm is continuous with respect to
the metric defined by the other. Prove that this is the case for ||x||
and the norms on R^n given by ||x||′ =max{ |x_1| ,.., |x_n| }
and ||x||″= |x_1| +...+| x_n|. 2.8 |
2.10 |
| 7 |
Partial derivatives and Jacobian
matrix [§ 2.3]. Hadamard Lemma 2.2.7 and chain rule, C^1-maps [§§ 2.2, 2.3, 2.4]. Lipschitz property of a C^1-map on a convex compact set [§ 2.5]. Higher-order derivatives, C^k-maps and C^k-diffeomorphisms [§§2.7, 3.1]. |
Prove that a nonempty open
subset of R^n cannot be C^1-diffeomorphic to an open subset of R^p
unless n=p. 2.22 |
2.27 |
| 8 |
Formulation and discussion of
the Local Inverse Function Theorem 3.2.4 [§ 3.2, p.92] and the Local
Implicit Function Theorem [§ 3.5]. Contraction Mapping Principle for complete metric spaces and Lemma 1.7.2 [§ 1.7]. Part I of the Local Inverse Function Theorem (Existence of a continuous inverse) [§ 3.2, Proposition 3.2.3]. |
3.2, 3.36 |
3.31 |
| 9 |
Part II of the Local Inverse Function Theorem (Continuous
differntiability of the inverse) [§ 3.2, Lemma
3.2.1 and Propositions 3.2.2 and 3.2.9]. Applications of the Implicit Function Theorem [§§ 3.4, 3.6]. Diffeomorphism interpreted as a new coordinate system, examples (such as polar coordinates [§ 3.1]) , Implicit Function Theorem as providing a normal form. Example: a simple root of a polynomial depends differentiably on the coefficients [§ 3.6, p. 101] . |
2.21, 3.41 |
Hand-in |
| 10 |
Submanifold of
R^n, immersion, embedding, submersion [§ 4.2]. Immersion Theorem 4.3.1 (Part (i) only). The image of an embedding is a submanifold [§ 4.3]. Examples of immersions and embeddings [§ 4.4]. Submersion Theorem 4.5.2 (Parts (i) and (ii) only). Fiber of a submersion is a submanifold [§ 4.5]. Examples of submersions [§ 4.6] |
4.3, 4.8, 4.12 |
4.29 |
| 11 |
Hertentamen week | ||
| 12 |
Normal forms for
immersion [Theorem 4.3.1 Part(ii)] and submersion [Theorem 4.5.2 Parts
(iii) and (iv)]. Critical Points, Hessian [§ 2.9]. Morse Lemma [§ 4.6]. |
2.58, 2.59, 2.62 [Hint: Use
Morse Lemma] |
4.31 |
| 13 |
Equivalent definitions of a
submanifold [§ 4.7]. Tangent Space of a submanifold [§ 5.1], derivative as a linear map between tangent spaces [§ 5.]. The tangent space to a 2D submanifold in R^3 [Examples 5.3.3 and 5.3.5]. The tangent space of the orthogonal group O(n) [Example 4.6.2] at the unit matrix is the set of all anti-symmetric matrices. |
Let f be a submersion defined on a neighborhood of a point a in R^n and denote by V the fiber of f through a. Prove that the tangent space of V at a is the kernel of Df(a). 5.2, 5.10 |
5.18 |
| 14 |
Cusps of planar curves;
Example 5.3.6 (Cycloid). Example 5.3.7 (Descartes' Folium). Critical point and value of a scalar function on a submanifold, Theorem 5.4.2 on the Lagrange multipliers. Example 5.5.1. |
5.35, 5.40, 5.42 | 5.47 (only (i)-(iii)) |
| 15 |
Geen hoorcollege, wel werkcolege | |
|
| 16 |
Deeltentamen
I: Dinsdag 16-04-2013,
ANDROCLUSGEBOUW zaal C101, 13.30-16.30 Midterm Exam 16-04-2013 with Solutions. |
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