btw (de inverse van) een kaart (U,phi) is een (proper) coördinaten patch (engelse term naast chart) in de oefeningen is de manifold het oppervlak x(D) en x-1 de kaart
check maar
http://resources.metapress.com/pdf-prev ... ze=largest
en van wiki:
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball. On the other hand, there are surfaces which cannot be embedded in three-dimensional Euclidean space without introducing singularities(dit zijn punten waarvoor oppervlak niet regulier is) or intersecting itself — these are the unorientable surfaces. (dit is de veralgemening naar inbedding in andere ruimtes)
To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide coordinates on it — except at the International Date Line and the poles, where longitude is undefined. This example illustrates that not all surfaces admits a single coordinate patch. In general, multiple coordinate patches are needed to cover a surface.
en dit geeft plots een goed inzicht in de discussie bij het begin van manifolds