= Solution
Differentiate the position <vector>, using the angular factor from the <scale factors of orthogonal coordinates>:
$$
\boxed{d\mathbf r=\mathbf e_\rho\,d\rho+\rho\mathbf e_\phi\,d\phi+\mathbf e_z\,dz.}
$$
The <basis> is an <orthonormal basis>, so all cross terms in its squared length vanish. The Euclidean <line element> is therefore
$$
\boxed{|d\mathbf r|^2=d\rho^2+\rho^2d\phi^2+dz^2.}
$$
In particular, an angular change $d\phi$ at fixed radius traverses a physical distance $\rho\,|d\phi|$.
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