Special region of R}^n

  • #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
 

Answers and Replies

  • #2
13,807
10,956
If ##x_1 > x_i## for all ##i>1## then ##y_i=x_1-x_i## looks as a good coordinate system. This way you get simply a Cartesian quadrant.
 

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