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

\[\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}} = \left| {\begin{array}{*{20}{c}}{\displaystyle \frac{{\partial x}}{{\partial u}}}&\displaystyle {\frac{{\partial x}}{{\partial v}}}\\\displaystyle {\frac{{\partial y}}{{\partial u}}}&\displaystyle {\frac{{\partial y}}{{\partial v}}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}{2u{v^3}}&{3{u^2}{v^2}}\\{ - {u^{ - \,\,\frac{1}{2}}}}&0\end{array}} \right| = 0 - \left( { - 3{u^{\frac{3}{2}}}{v^2}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{3{u^{\frac{3}{2}}}{v^2}}}\]