# Finding a transformation that maps a unit square onto a rectangle

So, it is a two part question:

1a: Find a transformation that maps the unit square onto the rectangle whose vertices are (0,1), (1,0), (4,3), and (3,4).

1b: Use your transformation to compute double integral with bound R (x+y)dA where R is the rectangle in part a.

By transformation, does it simply mean algebraically how would you transfer the square to the rectangle? If so, I said that eh length of the short side of the rectangle is sqrt(2), and the long side is 3sqrt(2). Thus, we have to first multiply the top and bottom of the square by 3sqrt(2) and the right and left by sqrt(2). Afterwards, translate it 1 unit to the right along the x axis and use a 45? or 315? degree rotation matrix to rotate the new rectangle 45? or 315? degrees around the point (1,0).

I have no idea how to do part B. Can someone please explain that.

• I'm not sure how to convert it from xy to uv plane – Math19384 Apr 14 '18 at 19:52

a) Ok, well since you are trying to map a rectangle onto a rectangle, an affine change of variables should do the trick. You should therefore try to find a matrix and a vector $$A= \begin{bmatrix} a&b\\c&d \end{bmatrix}\qquad w=\begin{bmatrix} e\\f\end{bmatrix}$$ such that the transformation $T(u,v)= A \begin{bmatrix} u\\v\end{bmatrix} + w$ maps the unit rectangle in the $uv$-plane to $R$.
b) Writing $x$ and $y$ in terms of $u$ and $v$, we have $$x= au+bv+e\qquad y= cu+dv+f$$
(you should justify to yourself why this is the case) and therefore the change of variables formula tells us that $$\int_R (x+y) = \int_0^1 \int_0^1 (au+bv+e + cu+dv+f)\cdot | \det J_t| \ dudv \\ =|ad-bc|\int_0^1\int_0^1 ((a+c)u + (b+d)v + (e+f)) \ dudv$$ (here I have used $J_t$ to denote the Jacobian of $T$).