There really isn’t much to do here other than compute the Jacobian.

\[\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}} = \left| {\begin{array}{*{20}{c}}{\displaystyle \frac{{\partial x}}{{\partial u}}}&\displaystyle {\frac{{\partial x}}{{\partial v}}}\\\displaystyle {\frac{{\partial y}}{{\partial u}}}&\displaystyle {\frac{{\partial y}}{{\partial v}}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}{ - \frac{v}{{{u^2}}}}&{\frac{1}{u}}\\{2u}&{ - 8v}\end{array}} \right| = \frac{{8{v^2}}}{{{u^2}}} - \left( 2 \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{8{v^2}}}{{{u^2}}} - 2}}\]