Extend a vector field of normal vectors beyond the surface I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question.
We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where $0\leq u,v\leq1$.  We can find the normal vector at any point $(u_0,v_0)$ by
$$
\frac{\partial \vec{r}}{\partial u}\times \frac{\partial \vec{r}}{\partial v}
$$
The question is how we can extend this into a vector field all throughout $R^3$ rather than just on the surface.
The reason why is because we want to calculate the mean curvature of the surface, which is given by 
$$
-\frac{1}{2}\nabla \cdot \vec{n}
$$
where $\vec{n}$ is a normal vector and I'm not sure how to do this when our surface is given para-metrically by $\vec{r}(u,v)$.
Thank you.
 A: If you wish to calculate the mean curvature of a surface $M$ at a point $p$, you need to extend the normal vector field to an open neighborhood of $p$ in $R^3$. This can be done by using the implicit function theorem to represent $M$ as the level surface of a suitable function $f(x,y,z)$ with nonzero gradient at $p$. Once you have $f$ in a neighborhood of $p$, just take the gradient of $f$ in a possibly smaller neighborhood, normalize it to unit length, and you got your extension.
A: The easiest way to extend the unit normal vector field beyond a surface so that the extended field is gradient field, is to assume it constant along the transversal direction to the surface. 
   But for your problem-to calculate the mean curvature it is sufficient to take Guenter's tangential derivatives $\mathcal{D}_j:=\partial_j-\nu_j\partial_\nu_j$ from the normal vector $\nu=(nu_1,\nu_2,\nu_3)$ because $\mathcal{D}_j\nu_k:=\partial_j\nu_k$ but the advantage is that for the derivative $\mathcal{D}_j\nu_k$ you does not need to leave a surface.
References:\
1. R. Duduchava, D.Mitrea, M.Mitrea, Differential operators and
boundary value problems on surfaces. {\em Mathematische Nach-ric-hten} {\bf 9-10}, 2006, 996-1023.\
2. R. Duduchava, Partial differential equations on hypersurfaces,
{\em Memoirs on Differential Equations and Mathematical Physics} {\bf 48}, 2009, 19-74.\
3. R. Duduchava, D Kapanadze, Extended normal vector fields and
Weingarten map on hypersurfaces, {\em Georgian Mathematical Journal}, {\bf 15}, No. 3 (2010), 485--501. 
I can sent you a new paper concerning this problem:\
R. Duduchava, G. Tephnadze, EXTENSION OF THE UNIT NORMAL VECTOR FIELD TO A HYPERSURFACE.
