I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have come up with the metric in polar coordinates [1 0;0 r^2]. Just for grins, I decided to use the partial derivatives and convert back to cartesian using the polar metric, expecting to come up with the exact same thing I started with, namely [1 0;0 1]. Unfortunately, that is not what happened. Shouldn't this work? Can anyone help me in where my thought process is wrong here? Note, this is not a HW question; I am a degreed engineer teaching myself relativity from a workbook.