On the bottom of the triangle and
using this we get in the u-v plance
so as x goes from 0 to 1
u and v go to 1. so we get the equation
Now on the hypotenuse so we get
so as x goes from 1 to 0
So we get the horizontal line segment as u goes from -1 to 1
Finally on the last part we get x=0 y =0..1
so as y goes from 1 to 0
we get the line segment
So we get the triangle in my above post.
always a sketch will help to clarify things.
for the region you have a triangle whose coordinates are thus under the transformation it's for the region.
from there things are a bit clear to take the new region where the double integral is taken.
of course, don't forget the Jacobian.