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### Section 15.8 : Change of Variables

2. Compute the Jacobian of the following transformation.

$x = {u^2}{v^3}\hspace{0.25in}y = 4 - 2\sqrt u$ Show Solution

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

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