Curvatures of a parametrized surface of revolution

Put $q=(f^{\prime 2}+g^{\prime 2}{)}^{1/2}$ for the profile parametrizationWith unit normal $N=(-g^{\prime }\cos u,-g^{\prime }\sin u,f^{\prime })/q$, the principal curvatures areConsequently $H=(k_1+k_2)/2$ and $K=k_1{k}_2$.