# change of variables

• Nov 16th 2011, 02:00 AM
phycdude
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.
• Nov 16th 2011, 02:20 AM
FernandoRevilla
Re: change of variables
Quote:

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:

$\displaystyle \int_0^{+\infty}\int_0^{+\infty}f(x,y)\;dxdy=\int_ 0^{\pi/2}d\phi \int_0^{+\infty}f(\rho \cos \phi,\rho\sin \phi)\rho\;d\rho$
• Nov 16th 2011, 02:39 AM
phycdude
Re: change of variables
wow,
Quote:

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?
• Nov 16th 2011, 03:33 AM
FernandoRevilla
Re: change of variables
Quote:

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. :)