i'm having problems to resolve this type of question, So given $a>0$ i have $$\psi:\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}^3, (u,v) \rightarrow(x(u,v),y(u,v),z(u,v))$$
where $$x(u,v) = (a+cos(\frac{u}{2})sin(v)-sin(\frac{u}{2})sin(2v))cos(u)$$ $$y(u,v) = (a+cos(\frac{u}{2})sin(v)-sin(\frac{u}{2})sin(2v))sin(u)$$ $$z(u,v) = sin(\frac{u}{2})sin(v)+cos(\frac{u}{2})sin(2v)$$ The set $M = \psi(\mathbb{R} \times\mathbb{R})$ is a realization of the Klein bottle in $\mathbb{R}^3$. Determine the points where $\psi$ is an immersion.