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This is not a homework question, but a question related to how to use the formula for the metric of a curved sphere (cylindrical coordinates):

[tex]

ds^2 = dr^2 + R^2 \sin ^2 (r/R)d\theta ^2

[/tex]

The thing is - I haven't got the slightest idea of how to use it, and unfortunately, my astrophysics-book doesn't make a great effort out of explaining this. The Web hasn't got anything on this formula as well, so this is my last resort.

First, ds is the infinitesimal distance between two points (r, theta) and (r+dr, theta + d theta), right? If this is correct, then what is dr? Isn't that also the distance between the two points?

I really hope you can explain this to me.