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Math Help - change of variables

  1. #1
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    change of variables

    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.
    Attached Thumbnails Attached Thumbnails change of variables-transform.png  
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: change of variables

    Quote Originally Posted by phycdude View Post
    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
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  3. #3
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    Re: change of variables

    wow,
    Quote Originally Posted by FernandoRevilla View Post
    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?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: change of variables

    Quote Originally Posted by phycdude View Post
    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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