Change of Variables — Question 6

Question 6 – Solution

Strategy. Track multiplicity: the two halves of the source fold onto the same target rectangle.

Step 1: Image and folding

The image is $0\le x\le 1$, $0\le y\le 1$. Except along $u=0$, the points $(u,v)$ and $(-u,v)$ have the same image, so $T$ is two-to-one.

Step 2: Diagnose the overcount Since $$J_T=\frac{\partial(x,y)}{\partial(u,v)} =\begin{vmatrix}2u&0\\0&1\end{vmatrix}=2u,$$ integrating its absolute value over the full source gives $$\int_{-1}^1\int_0^1|2u|\,dv\,du=2,$$ twice the actual image area $1$. The absolute value corrects orientation, not multiplicity.

Step 3: Repair Restrict to the one-to-one half $0\le u\le 1$: $$\operatorname{area}(T(S))=\int_0^1\int_0^1 2u\,dv\,du =\boxed{1}.$$ Using $-1\le u\le 0$ alone gives the same result after taking $|J_T|$.