First fundamental form of a surface of revolution
= First fundamental form of a surface of revolution
For
$$
X(u,v)=(f(v)\cos u,f(v)\sin u,g(v)),
$$
the coordinate tangent vectors are orthogonal and the <first fundamental form> is
$$
I=f(v)^2\,du^2+\bigl(f'(v)^2+g'(v)^2\bigr)\,dv^2.
$$