Example

When we change an integral from rectangular to polar coordinates by using $x= r\cos \theta$ and $y=r\sin \theta$, the Jacobian is

$$\frac{ \partial (x,y) }{\partial(u, v)}= \left\vert \begin{a... ... \end{array}\right\vert = r \cos ^2 \theta + r\sin^2\theta = r$$

So the integral will be

$$\int \int_R f(x,y) dxdy = \int \int_S f( r\cos \theta, r\sin \theta ) r dr d\theta$$