I have already sought for help in a previous post concerning the problem :
let $X, Y \stackrel{}{\sim} \mathscr{N}(0,1)$ be two independent random variables.
find the joint PDF of $(U,Z) = (X,X^2+Y^2)$
so I proceeded as follows :
\begin{align} &\int_{\mathbb{R^2}}f_X(v)f_Y(w)dvdw = 1 = \int_{\mathbb{R^2}}f_{U,Z}(m,n)dmdn \\ & \int_{\mathbb{R^2}}f_X(v)f_Y(w)dvdw = \int_{\mathbb{R \times \mathbb{R_{+}}}}2f_X(v)f_X(w)dvdw =\int_{\mathbb{R \times \mathbb{R_{+}}}}\frac1{\pi}\exp[-{{(v^2+w^2)} \over {2}}]dvdw = J\\ \end{align}
using the the change of variable $n = v,\; m = v^2+w^2$ we have :
\begin{align} J & = \int_{\mathbb{R \times [n^2,+\infty[}}\frac1{2\pi\sqrt{m-n^2}}\exp[-{{m} \over {2}}]dmdn \\ & = \int_{\mathbb{R^2}}\frac1{2\pi\sqrt{m-n^2}}\exp[-{{m} \over {2}}] \cdot \chi _{[n^2,+\infty[}(m) dmdn \\ \end{align}
can I say that $f_{U,Z}(u,z) = \frac1{2\pi\sqrt{z-u^2}}\exp[-{{z} \over {2}}] \cdot \chi _{[u^2,+\infty[}(z) $ ?