Formal definition of an outbound vector

Question

I've seen many references to the term outbound vector, in particular the outbound normal vector; for example, in $\mathbb{R}^2$, given a level set of a function G(x), if x>0, the outbound normal vector to the surface defined by G(x)=0 is, intuitively, the vector perpendicular to the surface and pointing in a north-easterly direction. Is there a general, formal way to define this intuitive concept?