Ratio-product coordinates ($u=x/y,v=xy,dxdy=(2u{)}^{-1}dudv$)

On the positive quadrant, the inverse coordinate change is $x=\sqrt{uv}$, $y=\sqrt{v/u}$. Its Jacobian determinant is $1/(2u)>0$. Regions bounded by positive constant-ratio lines and constant-product hyperbolas become rectangles in these coordinates. An integrand $x/y=u$ then cancels the Jacobian factor, making weighted area integrals particularly simple. The positivity restriction selects a unique square-root branch and fixes orientation.