# Name of Non-Unique Coordinate systems

Are there some examples (and a name) for non-unique coordinates (non-unique meaning may have multiple ways to represent the same point). Such as the one below.

• Polar coordinates have infinitely many ways to express any point. – Tartaglia's Stutter Mar 5 at 20:18
• I knew about that one, but are there any others. And is there an actual name for the class of these objects. – Numoru Mar 5 at 20:25
• I don't know of such a name but it wouldn't surprise me to know that there was one. – Tartaglia's Stutter Mar 5 at 20:30

Well, coordinate systems are meant to be injective (one-to-one), and it is not usually good that this condition is not satisfied. For example, if we take a coordinate system defined by$$\Phi\colon\space[0,\infty]\times[0,2\pi) \longrightarrow \mathbb{R}^2$$ so that $$\Phi(r,\varphi)=(r \cos(\varphi), r \sin(\varphi));$$ we see (either by mere exploration or more rigorously by setting the jacobian determinant equal to 0) that it fails to be injective at $$(x,y)=(0,0)$$. And, indeed, many $$(r,\varphi)$$ can represent this point, we only need $$r=0$$ so any $$\Phi(0,\varphi)$$ equals $$(0, 0)$$.
$$\vec{\nabla}\space\colon=\frac{\partial}{\partial x}\vec{u}_x+\frac{\partial}{\partial y}\vec{u}_y=\frac{\partial}{\partial r}\vec{u}_r+\frac{1}{r}\frac{\partial}{\partial \varphi}\vec{u}_\varphi.$$
This also happens with spherical coordinates at the origin, with cylindrical coordinates at every point in the $$z$$ axis, and with many more usual coordinate systems. Now, I don't know about any coordinate system made specifically to fail to be injective at specific points, nor do I know what that could be useful for, but I hope this helped you, OP, anyway.
EDIT: in all of those situations, the coordinate system fails to be injective at a 0 Lebesgue-measure subset of $$\mathbb{R}^n$$. If one happened to fail to be injective in a set with non-0 Lebesgue measure, it wouldn't even be possible to compute definite integrals within that region.