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

\[\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}} = \left| {\begin{array}{*{20}{c}}{\displaystyle \frac{{\partial x}}{{\partial u}}}&\displaystyle {\frac{{\partial x}}{{\partial v}}}\\\displaystyle {\frac{{\partial y}}{{\partial u}}}&\displaystyle {\frac{{\partial y}}{{\partial v}}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}4&{ - 6v}\\{2u}&{ - 6}\end{array}} \right| = - 24 - \left( { - 12uv} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 24 + 12uv}}\]