Problem:
Let $B$ be the region in the first quadrant of $\mathbb R^2$ restricted by the curves: $xy=1, xy=3, x^2-y^2=1, x^2-y^2=4$.
Calculate $\int_B(x^2+y^2)dxdy$.
Hint: Substitute $u=xy$ and $v=x^2-y^2$
Solution:
The jacobian-determinant of the map $\begin{pmatrix}u\\v\end{pmatrix}(x,y)$ is
$det\begin{pmatrix}y&x\\2x&-2y\end{pmatrix}=-2(x^2+y^2)$
So we get
$\int_B(x^2+y^2)dxdy=\frac{1}{2}\int_1^3du\int_1^4dv=3$
Question: I think I do get the basic principle and whats going on. What I don't get is the very first sentence of the solution. They say they take the jacobian-determinant of the map $\begin{pmatrix}u\\v\end{pmatrix}(x,y)$.
Now for me, that's: $\begin{pmatrix}u\\v\end{pmatrix}(x,y)=\begin{pmatrix}ux & uv \\ vx & vy\end{pmatrix}$ but that confuses me. Is that the correct intepretation of $\begin{pmatrix}u\\v\end{pmatrix}(x,y)$? nd if so, how do we get that?