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.
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.