Let $\mathcal{M} := \{ (x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = 1 \}$ and $p_0 = (1, 0, 0)$. Show that there is an open neighborhood $W \subset \mathbb{R}^3$ of $p_0$ and an embedding $f: \mathbb{R}^2 \supset U \rightarrow f (U) = \mathcal{M} \cap W \subset \mathbb{R}^3$.
My attempt: $U := [0, 1) \times (\frac{\pi}{2}, \frac{\pi}{2})$, $W = B_\varepsilon (p_0)$, and
$f : \begin{pmatrix} \varepsilon \\ \vartheta \\ \end{pmatrix} \mapsto \begin{pmatrix} \cos \vartheta \\ \sin \vartheta \\ \pm \sqrt{\varepsilon^2 + 2 (\cos \vartheta - 1)} \end{pmatrix} $
By construction, $f (U) = \mathcal{M} \cap W$ (in particular, $p_0 = f (0)$) and injective. Problem: $D f (0)$ doesn't exist, so I can't show that $f$ is an immersion and thus embedding. I would also like to use the inverse function theorem at $f^{-1} (p_0)$.
The Cartesian representation didn't bring me anywhere either. I would appreciate any hints.
References: For plots, http://www.grad.hr/geomteh3d/prodori/prodor_sf_eng.html. For finding $f$, https://en.wikipedia.org/wiki/Sphere%E2%80%93cylinder_intersection#Intersection_is_a_single_closed_curve, https://en.wikipedia.org/wiki/Sphere#Intersection_of_a_sphere_with_a_more_general_surface.
Let $g:\mathbb R^3\to \mathbb R:\ (x,y,z)\mapsto x^2+y^2-1$. Then, the cylinder $\mathcal M$ is given by $g^{-1}(\{0\})$. By the regular level set theorem, $\mathcal M$ is embedded in $\mathbb R^3$.
You do not have to write out explicitly the slice chart, because seeing this in this special case is not hard: if we take $F:=(g,y,z)$ we note that since the Jacobian of $F$ is not zero at $(1,0,0),$ the inverse function theorem gives a chart $(W_{(1,0,0)},(g,y,z))$ in the atlas of $\mathbb R^3$ and then $\mathcal (\mathcal M\cap W_{(1,0,0)},(y,z))$ is a slice chart for $\mathcal M$ at $(1,0,0)$ (because $g$ vanishes on $\mathcal M$).
If $\pi$ is projection onto the $y-z$ plane in $\mathbb R^3$ and if we set $U:=\pi(W_{(1,0,0)})$, then $f(y,z)=(\sqrt{1-y^2},y,z)$ maps $U$ onto $\mathcal M\cap W_{(1,0,0)}$.