# Singularity in coordinate transformations

Ok. I would like to apologise if the question is somewhat 'stupid'. So, initially I didn't pay much attention to it, but we know that when the Jacobian vanishes for a particular point, we say that the point is a singularity of the transformation, but if we are using the transformation to describe a Physical phenomena, then shouldn't we have a complete description of the system at every time? (I am really not sure if this should be the case always)

Taking the simplest example itself to illustrate my point, a motion in the Cartesian coordinaate system, if is well defined, it necessarily won't be, while considering it in let's say, Polar coordinate system, since singularities are present at $$r,\theta = 0$$

If not then the OP raises a very fair argument related to the topic here, but my question from this is :

1. If let's say we solve this issue by taking the limits on both the sides (as suggested by one of the answers there). then what is the guarantee that the limits will match and give a unique answer?

2. What if I don't wish to change the coordinate system (in the question there's an answer suggesting that we can transition from Cartesian to polar and vice-versa to solve the problem of singularity) and what if choose to remain only in 1 coordinate system?

Thanks.