I encountered the concept of noncoordinate bases in shutz 'a first course in general relativity' but think I am missing something. The example is the 1-form basis for polar coordinates. Unnormalized polar coordinates are considered a coordinate basis. Unit (normalized) polar coordinates are considered a noncoordinate basis.

The only difference between the two systems is that the unit theta 1-form is r times the unnormalized theta basis 1-form. So if you know the components of a point in unnormalized polar coordinates, you can express it in unit polar coordinates by dividing the theta component by r.

So, except for a slight division problem at r=0, there IS a coordinate system for unit polar coordinates, but everything I have read insists there isnt.

When they try to prove that there is no coordinate system for unit polar coordinates, what they really seem to prove is that there is no covariant or contravariant transformation from unnormalized polar coordinates to unit polar coordinates. I dont understand how that implies that there is no
unit polar coordinate system.

Any help would be appreciated.