Let $S$ be $$\{ (x,y,z) \in \mathbb{R}^3:x > 0, y \geq 1, z \geq 1, xyz\leq1 \}$$
and the variable change given as:
$$x = u, y = \frac{u+v}{u}, z = \frac{u+v+w}{u+v}$$
Problem is determining the set you get from that variable change.
I'm really not sure how to plot this set by hand(and it's a problem from a test, so I should be able to, apparently), it doesn't even seem finite, so I assume it wouldn't make any sense to try to do this using the boundary of $S$. Furthermore, I don't see how determining $u,v,w$ helps me at all. (Wolfram, for some reaosn, refuses to plot $xyz \leq 1$.) Unless I made some calculation mistakes, it should be: $$u=x, v = x(y-1), w = (z-1)xy$$
I'm really lost, cause I'm not sure what the exact approach here should be. Later on, I'm supposed to use this change of variables to compute $$\int_S 1$$ but I just don't see what I'm supposed to do with this.
\begin{gather} x > 0,\ y \geq 1,\ z \geq 1,\ xyz \leq 1 \\\iff u>0,\ \frac{u+v}{u} \geq 1,\ \frac{u+v+w}{u+v} \geq 1,\ u \cdot \frac{u+v}{u}\cdot \frac{u+v+w}{u+v} \leq 1 \\ \iff u > 0,\ u+v \geq u,\ u+v+w\geq u+v,\ u+v+w \leq 1\\ \iff u>0,\ v \geq 0,\ w \geq0,\ u + v + w \leq 1 \end{gather} Can you see how to sketch this set?