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Section 15.8 : Change of Variables

1. Compute the Jacobian of the following transformation.

\[x = 4u - 3{v^2}\hspace{0.25in}y = {u^2} - 6v\] Show Solution

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

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