Direction of unit normal defined from surface implicit function

Question

If $F$ is the implicit function of a three dimensional surface $\partial S$ , the unit normal vector function $\mathbb{\hat{n}}(x,y,z):=\nabla F(x,y,z)/\|\nabla F(x,y,z)\|$ at every point $(x,y,z)\in \partial S$ is always inward or outward ?

Answer 1

$\nabla F$ points in the direction of $F$ increasing. If that is inward or outward depends then on your definitions.

Example.

If $F(x,y,z)=x^2+y^2+z^2$ then $F=1$ represents a sphere of radius $1$ centred at the origin. $\nabla F=2(x,y,z)$ points outwards, because $F$ increases in that direction ($F=1+\varepsilon$ is a larger sphere for $\varepsilon>0$, external to $F=0$).

On the other hand, if $F(x,y,z)=-x^2-y^2-z^2$ then $F=-1$ represents the same sphere as before, but $F=-1+\varepsilon$ is a smaller sphere for $\varepsilon>0$, hence $\nabla F=-2(x,y,z)$ points inwards.