Find the preimage of set $\Omega$ given by a transformation from cartesian $(XYZ)$ to canonical cylindrical coordinates $(R \Theta Z)$ and also find the values of $r$, $\theta$ and $z$ such that this transformation is one to one. $$\Omega = \left \{(x, y, z): x^2 + y^2 + z^2 \leq 3, z \geq \sqrt{x^2+y^2} \right \}$$
After some calculations, I managed to draw a cylinder cut in half with height at most $\sqrt{3}$ in the $R \Theta Z$-plane, but I was not able to find/restrict the values $r$, $\theta$ and $z$ to make this transformation one to one.
Firstly let me note, that $z^2=x^2+y^2$ is cone, so our figure is inside upper part of cone bounded by sphere with center in $(0,0,0)$ and radius $\sqrt{3}$. Intersection of given surfaces gives circle $x^2+y^2 = \frac{3}{2}$ on $OXY$ plane.
Taking cylindrical coordinates $x=\rho \cos \theta, y= \rho \sin \theta,z=z$ gives boundaries $$\left\lbrace \begin{array}{l}0 \leqslant \theta \lt 2\pi \\ 0 \leqslant \rho \leqslant \sqrt{\frac{3}{2}} \\ \rho \leqslant z \leqslant \sqrt{3-\rho^2}\end{array}\right\rbrace$$