Let $D$ be the region surrounded by the quadrilateral with vertices $O(0,0) , A(2,1), B(s,t)$ and $C(1,3)$ with $s$ and $t$ constants.
1- Using the change of variables $(x,y)=(au+bc+cuv, du +ev+fuv)$ Find the constants $a, b, c, d$ such as $E:0\le u \le 1 ; 0\le v\le 1$ corresponds to $D$.
2- Calculate $\iint_D xy \,dx\,dy$
I am stuck at the first question and I cannot seem to find $a, b, c$ and $d$.
I tried to write the region $D$ such as
$$ \left\{ \begin{array}{c} 0\le x\le 1 ; \frac x2\le y\le 3x \\ 1\le x\le 2 ; \frac x2\le y\le \frac{t-3}{s-1}(x-1)+3 \\ 2\le x\le s ; \frac{t-1}{s-2}(x-2)+1\le y\le \frac{t-3}{s-1}(x-1)+3 \\ \end{array} \right. $$
Then I tried to substitute with $u$ and $v$ but I was not able to find the constants.
Any hint will be very much appreciated
The vertices of $E$ are $(0,0), (0,1), (1,0), (1,1).$ The suggested change of variables is $$(x,y)=F(u,v)=(au+bc+cuv, du +ev+fuv)$$ Along the edges of $E$, one of the variables $u,v$ is constant, so if we look at the coordinate functions of $F$, that is, $$F_1(u,v)=au+bc+cuv\\F_2(u,v)=du +ev+fuv$$ they are simply linear functions along the edges, because the $uv$ term will disappear. So, it appears that $F$ will map the edges of $E$ to the edges of $D$.
We have $$\begin{align} F(0,0)&=(0,0)\\ F(0,1)&=(bc,ev)\\ F(1,0)&=(a+bc,d+ev)\\ F(1,1)&=(a+b+c,d+e+f) \end{align}$$
Now, let's guess that $$\begin{align} (bc, ev)&= (2,1)\\ (a+bc,d+ev)&=(1,3)\\ (a+b+c,d+e+f)&=(s,t) \end{align}$$
We must have $a=-1$, $d=2$, $bc=2$, $ev=1$. The last equation gives $$s=-1+b+\frac2b\\t=2+e+\frac1e$$ and we can solve for $b$ abd $e$ in terms of $s$ and $t$.
There's no guarantee that this will work, though I'd be surprised if it doesn't. Once you get $F$, check that it works. If it doesn't, and you're sure it doesn't, go back and make a different guess about which vertex maps to which vertex, and see if that works.
Good luck.