Proving the uniqueness of a co-ordinate transformation

Hi,

I have a mapping defined from a curvilinear co-ordinate system (let's
call it u,v,w) to a cartesian co-ordinate system (x,y,z). The mapping
does not preserve length. How can I prove that the inverse mapping
(from x,y,z space back to u,v,w space) is unique?

Stated differently, I would like to be sure that each point in (x,y,z)
space maps to exactly one corresponding point in (u,v,w) space.

The mapping from (u,v,w) space to (x,y,z) space is defined and I can
transform vectors from one co-ordinate system to the other using the
Jacobian or inverse Jacobian as appropriate. I would now also like to
find the point in (u,v,w) space corresponding to a specific point in
(x,y,z) space.

Thanks,
Julian

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