Question

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes as shown in the figure below.

R is bounded by y = 2x − 2, y = 2x + 2, y = 2 − x, y = 4 − x

Answer 1

The first step is to rewrite the equations given in the problem.
y = 2x - 2 → 2x - y = 2
y = 2x + 2 → 2x - y = -2
y = 2 - x → x + y = 2
y = 4 - x → x + y = 4
Then let u = 2x - y and v = x + y. Therefore we get the equations with the following bounds -2 ≤ u ≤ 2 and 2 ≤ v ≤ 4.