Coordinate transformations in GR, worked examples?

[m4r35n357]

I have read over and over in various places about coordinate transformations, and understand the theory (really!), but can't find any worked examples of actual use of the transformation equations. Does anyone know of any web references or tutorials on the subject?
To make things a little more specific, I am ultimately interested in converting between Schwarzschild <=> Gullstrand-Painleve and Boyer-Lindquist <=> Doran coordinate systems.
I find it very puzzling that such an important topic does not seem to be applied in any depth in any of the books and other sources that I have. Perhaps just looking in the wrong places . . . .

 

[m4r35n357]

This is the best I've managed to find so far, rectangular to polar, starts at about 1:37:00 . . .
(Susskind)

 

[Mentz114]

Going from Schwarzschild to Gull-Painleve coordinates is described in this paper

http://arxiv.org/pdf/gr-qc/0411060.pdf, and in the Wiki article

http://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinatesThe only other reference I've got for Doran coords is the Visser paper you cited before.

 

[m4r35n357]

Thanks for your reply Mentz114 ;)
Ah, OK (I should have mentioned this paper, sorry), equations 5, 25 and 26 do seem to address this in part, and in fact I am using precisely those equations in my "simulation", but I am not happy with it because I cannot be sure I'm doing it correctly.
They appear (to me) to cover the transformation of two of the differentials (ie. the coordinate velocities) but not the coordinates themselves. I attempted to integrate them in Maxima (trying to get a relation between the time and azimuth coordinates) but it wouldn't cooperate, just printing out the unevaluated integrals.
Going back to the "simulation", I am simply passing all the coordinates through unchanged when I go from Boyer-Lindquist to Doran, and just altering the Ut and Uphi velocities according to those equations. I just think this is too much guesswork, hence my request for references/worked examples.
So yeah, I have now fully disclosed all my sources (apologies once more), but the few equations I know about are merely quoted and not derived.
Can anyone give me any pointers how I could derive them myself (with the aid of Maxima of course)?
Also, the original request stands, are there any worked examples using any of the metrics I mentioned in the OP?