Euclidean Surface

Now set of pairs: (a, a) (a, f) (f, a) (a, h) (h, a) (b, b) (c, c) (d, d) (e, e) (e , f) (f, e) (f, f) (g, g) (h, h) (a, e) (e, a) (e, h) (h, e) (f, h) (h , f) – is an equivalence relation, we can identify it by the letter R, then we say that this is an equivalence class of R. 2. Smooth manifold. On Example of a surface. Peter Schiff has much to offer in this field. For example, this surface can not be set one-one, continuous functions in parametric form, for example, in Cartesian coordinates. Then, for example, by breaking this surface into two site if we were able to ask each section of their one-one, continuous function in Cartesian coordinates, then each part is called a map, coordinates – local, functions, linking local coordinates with the surface (ie, maps the surface in Euclidean space) – homeomorphisms. All cards are called the atlas (in the example of two), so atlas connects the manifold with a Cartesian coordinates. Section under consideration of the manifold can intersect with the region of Euclidean space (the domain of local coordinates) can not intersect. But since we have a mapping function – homeomorphisms, then the relationship between the coordinates of the neighboring regions in Euclidean space – there is a change of coordinates (in fact, if the field in the manifold intersect, the intersection region are suitable for both sets of local coordinates), and the function change of coordinates, we have one-one and continuous in both directions.