Changing coordinates mean changing one set of local coordinates

[Ratzinger]

For each patch, a coordinate function maps the local coordinates to the euclidean space. Collection of all patches with coordinate functions covers the whole manifold.
What is a transition function and what is a change of coordinates?
Is it that you got one set of local coordinates throughout (?) the whole manifold and use a transition functions to show that the two coordinate functions are equal in the overlap regions? And does changing coordinates mean changing one set of local coordinates to another (x to x')?

Would be great if someone could help me out here. Thank you

 

[Berislav]

What is a transition function and what is a change of coordinates?

A transition function is a mapping between coordinate charts. Each point of the manifold has it's own tangent space, and it's associated coordinates. You can change coordinates by defining a mapping between them.

 

[quetzalcoatl9]

For each patch, a coordinate function maps the local coordinates to the euclidean space.

the way that i learned it, the coordinate chart maps the points of the manifold, in that local region, to a subset of the euclidean plane R^n. there is no intrisic coordinate system on the manifold to start with.

Is it that you got one set of local coordinates throughout (?) the whole manifold and use a transition functions to show that the two

there is no way (in general) of having a coordinate system that is valid for the entire manifold - that is why we "paste" a coordinate system onto a small region around a particular point. however, as long as we are able to do the same thing for a different overlapping region, then we can repeat this process over the whole manifold. (the collection of all of these coordinate charts for the whole manifold is called the atlas.)

consider the 2-sphere for example, and make your coordinate charts by projecting various hemispheres onto a plane, from all six axial positions, and you'll see what I mean - projecting the "western" hemisphere onto an adjacent plane will be 1-to-1, but then couldn't include the "eastern" (opposite side) hemisphere, that would require yet another plane (aka coordinate chart).

hope this clears things up a bit.