Transformation between coordinate system

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How do you transform a curvy coordinate system to that in euclidean space? An example will be greatly appreciated.
 

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How do you transform a curvy coordinate system to that in euclidean space? An example will be greatly appreciated.
I presume that by "curvy coordinate system" you mean "a coordinate system that is not Cartesian"?

The space is going to be Euclidean or not no matter which coordinates you use; I posted an example for a two-dimensional Euclidean plane in which the "curvy" coordinates are polar coordinates in your other thread.
 

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