Paul's Online Notes
Paul's Online Notes
Home / Calculus III / Multiple Integrals / Change of Variables
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 4-8 : 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.

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