mean curvature at surface point

Let $P$ be a point on the surface  $F(x,y,z)=0$  where the function $F$ is twice continuously differentiable on a neighbourhood of $P$. Then the normal curvature ${\varkappa }_{\theta }$ at $P$ is, by Euler’s theorem, via the principal curvatures ${\varkappa }_1$ and ${\varkappa }_2$ as

${\varkappa }_{\theta }={\varkappa }_1{\cos }^2\theta +{\varkappa }_2{\sin }^2\theta ,$(1)

where $\theta$ is the angle between (http://planetmath.org/AngleBetweenTwoPlanes) the normal section plane corresponding ${\varkappa }_1$ and the normal section plane corresponding ${\varkappa }_{\theta }$. When we apply (1) by taking instead $\theta$ the angle $\theta +\frac{\pi }{2}$, we may write

$${\varkappa }_{\theta +\frac{\pi }{2}}={\varkappa }_1{\sin }^2\theta +{\varkappa }_2{\cos }^2\theta .$$

Adding this equation to (1) then yields

$$\frac{{\varkappa }_{\theta }+{\varkappa }_{\theta +\frac{\pi }{2}}}{2}=\frac{{\varkappa }_1+{\varkappa }_2}{2}.$$

The contents of this result is the

Theorem. The arithmetic mean of the curvatures (http://planetmath.org/CurvaturePlaneCurve) of two perpendicular normal sections has a value, which is equal to the arithmetic mean of the principal curvatures. This mean is called the mean curvature at the point in question.