2008-16.md

course	course_year	question_number	tags	title	year
Geometry	IB	16	IB 2008 Geometry	3.I.2G	2008

A smooth surface in $\mathbb{R}^{3}$ has parametrization

$$\sigma(u, v)=\left(u-\frac{u^{3}}{3}+u v^{2}, v-\frac{v^{3}}{3}+u^{2} v, u^{2}-v^{2}\right) .$$

Show that a unit normal vector at the point $\sigma(u, v)$ is

$$\left(\frac{-2 u}{1+u^{2}+v^{2}}, \frac{2 v}{1+u^{2}+v^{2}}, \frac{1-u^{2}-v^{2}}{1+u^{2}+v^{2}}\right)$$

and that the curvature is $\frac{-4}{\left(1+u^{2}+v^{2}\right)^{4}}$.