I'm studying probability by myself, and I encounter this problem:
given $f_{r,\theta} = \frac{r}{2\pi\sigma^2}exp(\frac{-r^2}{2\sigma^2})$ find $f_{xy}$
I know that, $x^2 + y^2 = r^2 $, so I can just replace r:
$f_{xy} = \frac{\sqrt{x^2 + y^2}}{2\pi\sigma^2}exp(-\frac{x^2 + y^2}{2\sigma^2})$ |J|
What I'm having trouble is to find the Jacobian, I think it should be
$J = \begin{bmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y} \end{bmatrix}$ where $\theta = tan^{-1}(\frac{y}{x})$
Is this correct?