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.

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.