Take the first derivative; if that is always positive for all positive values of your variable then the transformation is positive monotone (i.e maintains ordering)

Since you have two variables, you need to check how the transformation affects both by checking two first order derivatives.

Take the first derivative; if that is always positive for all positive values of your variable then the transformation is positive monotone (i.e maintains ordering)

Since you have two variables, you need to check how the transformation affects both by checking two first order derivatives.

Edit: if you assume the function U is always positive you only need to check the first derivative of the transformation with respect to U