Orientation of a surface

An orientation of a regular surface is a continuous choice of unit normal. Reversing orientation replaces $N$ by $-N$.

An orientable surface admits a consistent choice of local orientation. Choosing one makes it an oriented surface.

On the smooth boundary of an orientable surface, let $n$ be the chosen unit normal vector and $t$ the positively oriented unit tangent vector used by Stokes theorem. The outward conormal is $u=t\times n$. It is a unit vector, tangent to the surface and orthogonal to its boundary. This convention turns $(n\times v)\cdot d\mathbf{r}$ into $u\cdot vds$ by cyclic invariance of the scalar triple product.

An oriented surface is a surface equipped with a chosen orientation.

An orientation of a surface induces the boundary direction used in Stokes theorem. Walking in the positive boundary direction keeps an upward-oriented surface on the left.