PDF of $U,V$ where $U=|X-Y|$ and $V=\operatorname{min}(X,Y)$

Question

Let $X\perp Y$ two random variables with uniform distribution on $[0,1]$. I found that $f_U(u)=2-2u$ and $f_V(v)=2-2v$. Now I have to apply the formula for a joint density of a transformation when $X\perp Y$, that is $f_{UV}(u,v)=f_X(u)f_Y(v)|J|$.

So I should solve the system of equations $u=|x-y|$ and $v=\min(x,y)$ in order to calculate $|J|$ (I do apologize but I can't insert systems of equations and matrices in Latex: everything appears on the same row and I don't know why), but I'm stuck. Could you help me to solve it?

Thnaks in advance.