Question: By an appropriate choice of new variables evaluate the integral $\int\int_R(x^2+y^2)\,dx\,dy$ over the interior of the square bounded by $y=\pm x$ and $y=\pm (x-2)$.
I sketched the square region (symmetric about $x$-axis, with vertices $(0,0)$, $(1,-1)$, $(1,1)$, $(2,0)$).
I need to use the Jacobian matrix for this variable substitution (even if it is possible another way, this is the method I would like to use). I realise that an appropriate substitution would be to rotate the square by 45 degrees, either clockwise or anticlockwise. In order to do this I used the rotation matrix and get the following:
$$u=\frac{1}{\sqrt{2}}x+\frac{1}{\sqrt{2}}y;$$ $$v=\frac{1}{\sqrt{2}}y-\frac{1}{\sqrt{2}}x$$
Rearranging to get:
$$x=\frac{u-v}{\sqrt{2}}$$ $$y=\frac{u+v}{\sqrt{2}}$$
However, according to the answer sheet I have been given, at this stage I am already wrong and all $\sqrt{2}$ should actually just be $2$. Why is this?