Working with surface integrals, one often needs the surface normal. For a sphere we have: $\vec n = (\cos \phi \cos \theta, \sin \phi \cos \theta, \sin \theta)$. Now I would like to find the equivalent for the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.
I understand that $\vec n \propto (a \cos \phi \cos \theta, b \sin \phi \cos \theta, c \sin \theta)$, but how do I normalise it to get a nice unit normal?
The normalized vector of a vector $r=(x,y,z)$ is
$$\hat{r} = \frac{(x,y,z)}{\sqrt{x^2+y^2+z^2}}$$
That is vector divided by its norm.