= Solution
For a regular parametrization $\mathbf r(u,v)$, the <first fundamental form> is the induced <Riemannian metric>
$$
I=d\mathbf r\cdot d\mathbf r=E\,du^2+2F\,du\,dv+G\,dv^2,
\quad E=\mathbf r_u\cdot\mathbf r_u,\quad F=\mathbf r_u\cdot\mathbf r_v,
\quad G=\mathbf r_v\cdot\mathbf r_v.
$$
It measures squared lengths of <tangent vectors> and therefore lengths of <curves>. In the plane put $x=\rho\cos\theta$, $y=\rho\sin\theta$. The derivatives are orthogonal, of lengths one and $\rho$, so
$$
\boxed{dx^2+dy^2=d\rho^2+\rho^2d\theta^2.}
$$
These <polar coordinates> are regular for $\rho>0$.
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