Differential Geometry of Surfaces

[fireball1]

That's quite a pretty exercise.

Show that the mean curvature H at a point $p in S$, with $S$ a regular
surface in the ambient space $RR^3$, is given by:

$H=1/pi int_0^pi k_n (theta) d theta$,

where $k_n (theta)$ is the normal curvature at $p$ along a direction making
an angle $theta$ with a fixed direction.

I have solved it, now it's up to you!

[fireball1]

Another pretty fact (Theorem of Beltrami-Enneper):

Prove that the absolute value of the torsion $tau$ at a point
of an asymptotic curve (a curve having at every point
an asymptotic direction, i.e., a direction in which the normal
curvature is zero), whose curvature is nowhere zero, is given by

$|tau|=sqrt(-K)$

where $K$ is the Gaussian curvature of the surface at the given point.