An intelligent way to see if this surface is orientable

I am trying to do this excercise: $$\begin{array} { c } { \text { Let } M \text { be the regular surface in } \mathbb { R } ^ { 3 } \text { parametrised by } } \\ { X : \mathbb { R } \times ( - 1,1 ) \rightarrow \mathbb { R } ^ { 3 } \text { with } } \\ { X ( u , v ) = 2 ( \cos u , \sin u , 0 ) + v \sin ( u / 2 ) ( 0,0,1 ) } \\ { + v \cos ( u / 2 ) ( \cos u , \sin u , 0 ) } \\ { \text { The curve } \gamma : \mathbb { R } \rightarrow M \text { defined by } } \\ { \gamma : t \mapsto X ( t , 0 ) } \end{array}$$

Determine whether $$\gamma$$ is a geodesic and if $$X(u,v)$$ is orientable or not.

Using the geodesic equations I have seen when $$\gamma$$ is geodesic, but I am having troubles prooving the orientability. I want to know if there is an easy way to see that. Thanks.

• Hint: This is a parametrization of the Möbius strip. Just out of curiosity, where did you get this exercise from? – MisterRiemann Oct 30 '18 at 21:24
• Oh! I haven't realized that. Thanks!!. The book is "An introduction to Gaussian Geometry" of S. Gudmundsson. matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf – J.Rodriguez Oct 30 '18 at 22:32
• Very nice, I recognized that exercise since the author of those notes is my very own teacher! It's always cool to see how many people use his notes! =) – MisterRiemann Oct 31 '18 at 9:08