I'm having trouble with a change of variables problem in vector calculus. The question states $D$ is a parallelogram with vertices $(0,0), (1,0), (1,2)$ and $(2,2)$, and $R$ is the rectangle $[0,1] \times [0,2]$. It wants me to determine a linear transformation that maps $D$ in the $x$–$y$ plane to $R$ in the $u$–$v$ plane.
I've plotted the parallelogram $D$ and also plotted the rectangle $R$ but I'm having trouble seeing the necessary change of variables to change the points of $D$ to the points of $R$.
I did notice that most of the points of $D$ already resemble $R$ except for $(2,2)$ in $D$ and $(0,2)$ in $R$ but I'm not sure if I should just eye ball it and guess or if there are formulas for this type of thing, so any help would be greatly appreciated, thanks 🙂
Best Answer
$D$ is the region bounded by the lines $y=2x$, $y=2x-2$, $y=0$, $y=2$. Or:
$\frac{2x-y}{2}=0$, $\frac{2x-y}{2}=1$, $y=0$, $y=2$.
So let $u=\frac{2x-y}{2}$ and $v=y$.
The answer isn’t unique, notice that the boundary can also be written as: $\frac{y}{2}=0$, $\frac{y}{2}=1$, $2x-y=0$, $2x-y=2$.
So $u=\frac{y}{2}$ and $v=2x-y$ also works.