find an immersion from mobius strip to $\mathbb{R^3}$ .
One way to represent the Möbius strip embedded in $\mathbb{R^3}$ is by the parametrization. $$ \begin{array}{l} x(u, v)=\left(1+\frac{v}{2} \cos \frac{u}{2}\right) \cos u \\ y(u, v)=\left(1+\frac{v}{2} \cos \frac{u}{2}\right) \sin u \\ z(u, v)=\frac{v}{2} \sin \frac{u}{2} \end{array} $$ for $0 \leq u<2 \pi$ and $-1 \leq v \leq 1$. so it's enough that we show above function is smooth . is this true ?