First fundamental form of an arc-length surface of revolution
= First fundamental form of an arc-length surface of revolution
For $X(u,v)=(Y(u)\cos v,Y(u)\sin v,Z(u))$ with $Y'^2+Z'^2=1$, the induced metric is
$$
du^2+Y(u)^2\,dv^2.
$$
= First fundamental form of an arc-length surface of revolution
For $X(u,v)=(Y(u)\cos v,Y(u)\sin v,Z(u))$ with $Y'^2+Z'^2=1$, the induced metric is
$$
du^2+Y(u)^2\,dv^2.
$$