Paper 3, Section II, F

Geometry
Part IB, 2011

Suppose that η(u)=(f(u),0,g(u))\eta(u)=(f(u), 0, g(u)) is a unit speed curve in R3\mathbb{R}^{3}. Show that the corresponding surface of revolution SS obtained by rotating this curve about the zz-axis has Gaussian curvature K=(d2f/du2)/fK=-\left(d^{2} f / d u^{2}\right) / f.