1. change of variables

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.

2. Re: change of variables

Originally Posted by phycdude
how did the transformation end up so compact and simple? or how is that transformation done.
According to the well known theorem of change of variables:

$\displaystyle \int_0^{+\infty}\int_0^{+\infty}f(x,y)\;dxdy=\int_ 0^{\pi/2}d\phi \int_0^{+\infty}f(\rho \cos \phi,\rho\sin \phi)\rho\;d\rho$

3. Re: change of variables

wow,
Originally Posted by FernandoRevilla
According to the well known theorem of change of variables:

$\displaystyle \int_0^{+\infty}\int_0^{+\infty}f(x,y)\;dxdy=\int_ 0^{\pi/2}d\phi \int_0^{+\infty}f(\rho \cos \phi,\rho\sin \phi)\rho\;d\rho$
i wonder how i got this far without meeting this, what widely available books (or decently written webpages+ ebooks) talks about this?

4. Re: change of variables

Originally Posted by phycdude
wow, i wonder how i got this far without meeting this, what widely available books (or decently written webpages+ ebooks) talks about this?
Look here:

Jacobian matrix and determinant - Wikipedia, the free encyclopedia

especially Example 3. Then, decide yourself if the webpage is decently written.