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 $frac{partial u_{theta}}{partial rho}$, $varepsilon_{rho}$, $frac{partial u_{rho}}{partial theta}$.