proof of Box–Muller transform (polar form) There're proofs of Box–Muller transform available online but my book (pattern recognition and machine learning) seems to have put it in a different form.


I didn't follow the derivation of equation 11.12, can anyone please help? Thanks!
EDIT
As mentioned in Nadiels's answer, there's a mistake in formula 11.10 and 11.11, as logarithm has to take in a positive number (PRML errata).

As a general rule, a pdf in one set of variables is given by the pdf in another set of variables multiplied by the determinant of the Jacobian matrix.
In you case, you must calculate:
$$\left|\left(\begin{array}(\partial z_1 / \partial y_1 & \partial z_2 / \partial y_1  \\ \partial z_1 / \partial y_2 & \partial z_2 / \partial y_2\end{array}\right)\right|$$
Now, this looks a bit complicated, since you are given $y_1,y_2$ in terms of $z_1,z_2$ and not the other way round. Luckily, we can use the Inverse function theorem to get that:
$$\left|\frac{\partial (z_1, z_2)}{\partial (y_1, y_2)}\right|= \left|\frac{\partial (y_1, y_2)}{\partial (z_1, z_2)}^{-1}\right|=\left|\frac{\partial (y_1, y_2)}{\partial (z_1, z_2)}\right|^{-1}$$
So you should be able to calculate the derivatives and then the inverse determinant. 
