im trying to understand a change of variables during an integration:

see the attached scan.

Now if you have the following:

$\displaystyle 4 \int^{\infty}_{0} \int^{\infty}_{0} x^{2u-1} y^{2v-1} e^{-(x^2+y^2)} dx dy $

and transform $\displaystyle x=\rho cos\phi $ will dx not be $\displaystyle dx= cos\phi d\rho - \rho sin\phi d\phi $ and for y, $\displaystyle y=\rhosin\phi $ similarly $\displaystyle dy= sin\phi d\rho + \rho cos\phi d\phi $

how did the transformation end up so compact and simple? or how is that transformation done.