What is the transformation set?

Question

Suppose I have the following triangle in $\mathbb{R}^2$, $ A = \{ (x,y) \, | \, 0 \leq y \leq x \leq 1 \}$. I now perform the coordinate transformation $ z = \frac{y}{x}, u= x $. I now want to express the set $A$ as coordinates $(u,z)$, thus $ A = \{ (u,z) \, | \, \dots\}$. What will be the expression for this?

Answer 1

Consider the map $[0,1]\times[0,1]\to A$ given by $$ (u,z)\to (z, uz)\ .$$

• It is well defined,
• maps $(0,0)$ to $(0,0)$, but also maps each $(u,0)$ to $(0,0)$, this is for the purposes of an integration irrelevant,
• it is injective on the part of all $(u,z)$ with $z\ne 0$, since from $(z,uz)=(z',u'z')$ we get immediately $z=z'$, then $u=u'$,
• it is surjective.

It is thus simpler to write the coordinate transformation on the part of $A$ of all $(x,y)$ with $0<y\le x\le 1$.