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Special region of R}^n

  1. Dec 4, 2007 #1
    Hi,
    I am looking for a "good" way to parametrize the region of [tex]\mathbb{R}^n[/tex] where one coordinate, say [tex]x_1[/tex] is greater than all the others.

    I came up with a possiblity to do that in hyperspherical coordinates [tex]\{r,\varphi_1,\varphi_2,\dots ,\varphi_{n-1}\}[/tex]

    where

    [tex]0\leq r \leq\infty[/tex]
    [tex]0\leq \varphi_{n-1} \leq 2\pi[/tex]
    [tex]0\leq \varphi_\nu \leq\pi \quad\quad 1\leq\nu\leq n-2[/tex]

    Then for example, if I wanted to integrate over the region of [tex]\mathbb{R}^n[/tex] where
    [tex]x_1 \geq x_2 \dots \geq x_n[/tex]
    I could do it like this

    [tex]\int_0^\infty dr\int_{-\frac{3}{4}\pi}^{\frac{\pi}{4}}d\varphi_n \int_{0}^{\frac{\pi}{2}-ArcTan[Cos[\varphi_n]]}d\varphi_{n-1}\dots\int_{0}^{\frac{\pi}{2}-ArcTan[Cos[\varphi_2]]}d\varphi_1 r^{n-1}Sin[\varphi_1]^{n-2}\dots Sin[\varphi_{n-1}] + [/tex] [tex]+ \int_0^\infty dr\int_{-\frac{\pi}{4}}^{\frac{5}{4}\pi}d\varphi_n \int_{0}^{\frac{\pi}{2}-ArcTan[Sin[\varphi_n]]}d\varphi_{n-1}\dots\int_{0}^{\frac{\pi}{2}-ArcTan[Cos[\varphi_2]]}d\varphi_1 r^{n-1}Sin[\varphi_1]^{n-2}\dots Sin[\varphi_{n-1}] [/tex]


    If I then sum over all permutations of [tex]\{x_2,\dots ,x_n\}[/tex] I can integrate over the region where [tex]x_1[/tex] is greater than all the other coordinates. However the integration limits are somewhat unwieldy so my question is if anybody knows of a better way to parametrize the region I am interested in.

    Thanks

    Cheers,
    Pere
     
  2. jcsd
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