Kurver og Flader

Mat 5 2016
Aalborg Universitet
Institut for matematiske fag.

(from Wikipedia)

Literature

Course schedule

New lecture plans and some slides will be added a few days before the activities take place; click on No.!
No. Date Type Topics Literature Slides etc.
1   6/9 2   Curves: Regular parametrizations, tangents, speed, arc length, unit-speed parametrization   [AP], ch. 1 (apart 1.4)    
2   8/9 1   Curvature and torsion. Frenet-Serret formulas.   [AP], ch. 2.1-3 (apart Thm. 2.2.6, 2.3.6)    
3 13/9 3   Fundamental Theorem for Curves   [AP], Thm. 2.2.6, 2.3.6; [FR], ch. 1.3.5, 1.4.3    
4 15/9 4   Training Session: Fundamental Theorem for Curves   as for 3     Curves
5 20/9 2   Introduction to surfaces, including level set surfaces   [AP] 4.1, 4.2, 5.1     Horias Noter -uddrag.
6 22/9 1   (Re-)parametrizations. Transition functions. Smooth maps   [AP] 4.2, 4.3, [FR], 2.4, 2.5     Slides6
7 27/9 3   Smooth maps between surfaces and their derivatives; (local) orientations   [AP] 4.3 - 4.5    
8 29/9 4   Training Session: Smooth maps between surfaces and their derivatives   [AP] 4.3 - 4.4    
9   4/10 2   First fundamental form. Isometries   [AP] 6.1 - 6.2, 6.3 (p.133), 6.4 (p. 139 - 142)     Slides9
10   6/10 1   Gauss- and Weingarten map, second fundamental form, normal curvature   [AP] 7.1 - 7.2     "movies"
11   11/10 3   Curvatures: normal, geodesic, Gaussian, mean, and principal   [AP] 7.3, 8.1 - 8.2  
12   13/10 4   Training session: Curvature notions on surfaces   [AP] 8.1 - 8.2    
13 25/10 2   Geodesic curves on a surface   [AP] 9.1 - 9.2    
14 27/10 1   Geodesics are intrinsic. Theorema egregium.   [AP] 9.2.7 - 8. 10.1 - 10.2.1    
15 1/11 3   Codazzi-Mainardi and Gauss equations   [AP], 10.2    
16 3/11 4   Training session: Theorema Egregium   [AP], 10.2    
17  10/11 2   Local Gauss-Bonnet theorem   [AP] 13.1-2    
18   6/12 1   Global Gauss-Bonnet theorem   [AP] 8.6, 13.2-4    
19   8/12 3   Genus and Euler characteristic of compact surfaces   [AP] 13.4    
20 16/1   Spørgetime. 10-12. Lokale G4-102. Send spørgsmål til Lisbeth senest dagen før.       --

Lectures and Assignments

The teacher will be involved in different levels on various days. This is reflected in the "type" in the list schedule. The different types should be interpreted as follows:
  1. 2 hours of lectures and 2 hours of exercises with help from the instructor
  2. 2 hours of lectures and 2 hours exercises; lecturer intervenes only at the end to answer questions
  3. 1 hour of lectures and 3 hours of reading and exercises; help from the lecturer available for around 1 hour
  4. No lectures. Reading, training and exercise assignments. The lecturer will not be around. Send questions to be answered next time by email.

Eksperimenter

Maple Examples
The geometric lab.
Flotte Applets lavet af Thomas Banchoff og Steven Lovett
Visual Geometry

Links:

Famous Curves St Andrews University i Skotland.
2d-curves Mere end 600 plane kurver og historiske facts.
3d-xplormath. Appletsmed kurver og flader.
The Geometry Center
Mathematics Archives - Topics in Mathematics - Geometry
Java En abesaddel. Thomas Banchoff.. Man kan variere funktioner etc.
JavaView - diverse flader, kurver etc.
Flatland. En novelle om at være et fladt dyr, bo i en flade (!) og udforske den fra et "indre" perspektiv.
Ikke-Euklidisk geometri Blandt andet en applet, hvori man kan udforske Poincare- og øvre halvplansmodellerne for hyperbolsk geometri.

"MOVIES"

Lisbeth Fajstrup