 Normal reversal around the Möbius core

For the model $F$ of the Möbius band, the tangent vectors along its core circle of a Möbius band areTheir cross product, divided by its norm $R$, gives the continuous unit normalAlthough $F(2\pi ,0)=F(0,0)$, we have $n(2\pi )=-n(0)$. Any other continuous choice of unit normal along this parametrized interval is either $n$ everywhere or $-n$ everywhere: its relative sign is a continuous map into the discrete space $\{+1,-1\}$, and the interval is connected. Thus no choice can agree at the identified endpoints. This short calculation proves that the Möbius band has no global continuous unit normal and is not an orientable surface.

The plotted quantity is $n(\theta )\cdot n(0)=\cos (\theta /2)$. It compares ambient vectors at generally different points; after one circuit it is $-1$ at the starting point, and after two circuits it is $+1$ there. Only these endpoint comparisons are needed for the orientability argument.