Suppose I have a tensor T, they always say that T has an "admissible" change of coordinate systems as:
$\mathcal{T} : \bar{x}^i = \bar{x}^i(x^1, x^2, \dots, x^n)$
But, what would be an example of an "un-admissible" change of coordinate systems?
In other words, is there an example of a transform $\mathcal{T}$ that is un-admissible so that tensor T does not obey the transformation laws $\bar{T}^i = T_r J^i_r$?
If the transform $\mathcal{T}$ is not bijective, then, this is an un-admissible change of coordinates. (bijective means that we have a one-to-one mapping of coordinates from the "$x^i$ system" to the "$\bar{x}^i$ system")
However, many non-bijective functions are bijection on a restricted region, thus, a transform that is un-admissible maybe be admissible on a range of coordinate values. For example, restricting the angle coordinate $\theta$ of a polar coordinate system between the ranges $\pi$ and $-\pi$ will convert the transform $\mathcal{T}$ from an un-admissible change of coordinates to an admissible change of coordinates.