orientation of a surface

A consistent choice of which side of a surface counts as "outward," made everywhere at once — not every surface admits one.

A surface is orientable if a continuously varying unit normal $\mathbf{n}$ can be defined at every point such that $\mathbf{n}$ never flips discontinuously as it's carried around any closed loop on the surface. For a parametric patch this is automatic, since $\mathbf{n}=\dfrac{\mathbf{r}_u\times\mathbf{r}_v}{|\mathbf{r}_u\times\mathbf{r}_v|}$ varies continuously in $u,v$ by construction — but the same surface can sometimes only be covered by patches whose normals disagree where they overlap, and no relabeling fixes it.

Examples

A sphere is orientable — its outward normal never flips no matter how it's carried around a loop.
A Möbius strip is non-orientable — walk a normal vector once around the strip's center curve and it comes back pointing the opposite way it started.