Jacobian

Definition:

The Jacobian of the transformation $x=g(u,v)$, $y=h(u,v)$ is:

\begin{align} \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| \end{align}

Change of Variable for a Double Integral

Suppose that we want to integrate $f(x,y)$ over the region $R.Underthetransformation$x=g(u,v)$,$y=h(u,v)$theregionbecomes$S$ and the integral becomes,

$$\begin{array}{r}\underset{R}{\iint }f\left(x,y\right)dA=\underset{S}{\iint }f\left(g\left(u,v\right),h\left(u,v\right)\right)\left|\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}\right|d\overline{A}\\ \\ dA=\left|\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}\right|d\overline{A}\end{array}$$