calculus and analysis – How to use dimensions to analyze physical equations in polar coordinates

I already know that the geometric equation of an elastic body in polar coordinates is as follows:

begin{aligned} &varepsilon_{rho}=frac{partial u_{rho}}{partial rho} quad \ &varepsilon_{theta}=frac{u_{rho}}{rho}+frac{1}{rho} frac{partial u_{theta}}{partial theta} quad\ &\ &gamma_{rho theta}=-frac{u_{theta}}{rho}+frac{partial u_{theta}}{partial rho}+frac{1}{rho} frac{partial u_{rho}}{partial theta} quad end{aligned}

I find that in $$gamma_{rho theta}$$ there is a coefficient $$frac{1}{rho}$$ in front of $$frac{partial u_{rho}}{partial theta}$$ and not in front of $$frac{partial u_{theta}}{partial rho}$$.

I want to use MMA to analyze the dimensions of items, such as $$frac{partial u_{theta}}{partial rho}$$, $$varepsilon_{rho}$$, $$frac{partial u_{rho}}{partial theta}$$.