Let $S$ be an oriented surface with unit normal vector $\mathbf{n}$ and parameterization $$r(u,v)=(X(u,v),Y(u,v),Z(u,v)),\qquad (u,v)\in D\subset\mathbb R^2.$$
In the context of vector calculus, the definitions below are used to define the "positive orientation" for the border $\partial S$.
Definition 1. We say that $\partial S$ is positively oriented provided that its parameterization is given by $r\circ \gamma$, where $\gamma $ is a counterclockwise parameterization of $\partial D$, and the orientation of $S$ is given by $\mathbf{n}=\frac{r_u\times r_v}{\|r_u\times r_v\|}$.
Definition 2. We say that $\partial S$ is positively oriented provided that, as you walk in around $\partial S$ with your head pointing in the direction of $\mathbf{n}$, the surface is always on your left.
Note that definition 1 is the one that works in the proof of Stokes Theorem and definition 2 is geometrically useful in order to find the correct orientation in the exercises.
For some particular cases, where we can calculate $r$, $\gamma$ and $\mathbf{n}$ explicitly, we can see that these definitions agree.
Question: How could we justify, in general, that these definitions are equivalent?