change of variables

**phycdude** · November 16th 2011, 02:00 AM

im trying to understand a change of variables during an integration:
see the attached scan.
Now if you have the following:
$4 \int^{\infty}_{0} \int^{\infty}_{0} x^{2u-1} y^{2v-1} e^{-(x^2+y^2)} dx dy$
and transform $x=\rho cos\phi$ will dx not be $dx= cos\phi d\rho - \rho sin\phi d\phi$ and for y, $y=\rhosin\phi$ similarly $dy= sin\phi d\rho + \rho cos\phi d\phi$
how did the transformation end up so compact and simple? or how is that transformation done.

**FernandoRevilla** · November 16th 2011, 02:20 AM

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:

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

**phycdude** · November 16th 2011, 02:39 AM

wow,

Originally Posted by FernandoRevilla

According to the well known theorem of change of variables:

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

i wonder how i got this far without meeting this, what widely available books (or decently written webpages+ ebooks) talks about this?

**FernandoRevilla** · November 16th 2011, 03:33 AM

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.

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