How can I calculate the partial derivative of this parametric function?

Question

Given the Function:

$f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ , $r \in \mathbb{R}.$

$f(x,y) = \begin{cases} x=r\cos(\theta)\\ y= r\sin(\theta) \end{cases}$

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Calculate this Partial Derivative:

\begin{equation} \frac{\partial (x,y)}{\partial (r,\theta)} \end{equation}

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I do really need some help on this lads, any help would be really appreciated.

Answer 1

This symbol is shorthand for the Jacobian matrix $$\left[\begin{matrix}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta}\\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\\ \end{matrix}\right] $$