Orientable surface

Definition

Let $S \subseteq \mathbb{R}^3$ be a surface that has a tangent plane at each point. The surface $S$ is orientable if there is a continuous map $n \colon S \to \mathbb{R}^3$ with the following property. For each $p \in S$, $n(p)$ is a unit normal vector of $S$ at $p$. Such a map $n$ is an orientation of $S$, and $S$ together with $n$ is an oriented surface.

Remarks

If $S$ is connected and orientable, then it has exactly two orientations, $n$ and $-n$. A regular injective parametrisation $r$ gives the orientation $n = (r_u \times r_v) / \| r_u \times r_v \|$. Swapping $u$ and $v$ gives the opposite orientation. Spheres, planes and tori are orientable. The boundary of a bounded region in $\mathbb{R}^3$ has the outward normal as its standard orientation. The Möbius strip is not orientable. Integrals of vector fields across a surface, such as flux, need an orientation. Reversing the orientation changes their sign. Integrals of scalar functions do not need one.

Explanations

L1 human

An orientable surface has two sides, an inside and an outside, or a top and a bottom. On a Möbius strip, an ant can walk from one side to the other without crossing an edge. So there is no consistent choice of side.