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Best to illustrate my question with an example. Take a vector expressed in some chart, then we can find the components of that vector in another chart in the standard way by using partial derivatives. However, it got my thinking why is it the case that one set of coordinates has some dependence of another? ie. why can't all the partial derivatives be zero? Geometrically speaking (I think) we view all frames as charts on our manifold, so of course any chart that is in the atlas corresponds to a frame and by compatibility the partial derivatives are non zero. But whats to stop us say, having some weird accelerated reference frame that isn't compatible with all charts and therefore not in the maximal atlas? How could we get a relation then? (I think theres an easy math answer to this that I'm missing, perhaps by playing around with the two homeomorphisms and using some sort of inverse function theorem or something). A good example would be the standard atlas for the sphere, and stereo graphic projection. As it turns out, these give rise to the same differentiable structure, but I imagine this isn't always the case?

Also...can anyone recommend any good books on GR? Ideally one suitable as reading for an undergrad course, a post grad course and then a bit more, but that's set up as more than just reading for lectures.

Thanks for your help!