Regions

In the tables, we give integration formulas for four different regions. The regions are defined for dimension n 2.

• C_n -- The n-dimensional cube

  Definition:
  -1 x_i 1         i = 1,..., n

  Volume:

  V = 2ⁿ
  C_n has 2ⁿ vertices (± 1, ± 1, ..., ± 1 )

  Weight function:

  w( x₁,..., x_n ) = 1

• S_n -- The n-dimensional sphere

  Definition:

  n i =1

  xi2

  1

  Volume:

  V =	2 n/2 n (n/2)

  Weight function:

  w( x₁,..., x_n ) = 1

• T_n -- The n-dimensional simplex

  Definition:

  n i =1

  xi

  1         xi 0       i = 1,...,n

  Volume:

  V =

  1 n!

  T_n has n + 1 vertices

  ( 0, 0, ... ,0 )

  ( 1, 0, ... ,0 )
  .
  .
  .
  ( 0, 0, ... ,1 )
  

  Weight function:
  w( x₁,..., x_n ) = 1

• E_n^-r2 -- The n-dimensional space with weight function ( -x₁² - ... -x_n² )

  Definition:
  - < x_i <         i = 1 , ... , n

  Weight function:

  w( x₁,..., x_n ) = exp( -x₁² - x₂² - ... - x_n² )

  ...

  En-r2

  w( x1,..., xn ) dx1, ... , dxn = n/2

• E_n^-r -- The n-dimensional space with weight function ( - ( x₁² - ... -x_n² ) )

  Definition:
  - < x_i <         i = 1 , ... , n

  Weight function:

  w( x₁,..., x_n ) = exp( -( -x₁² - x₂² - ... - x_n² ) )

  ...

  En-r

  w( x1,..., xn ) dx1, ... , dxn =

  2 ( n - 1 )! n/2 (n/2)

---

This page was last modified on Wednesday 07 November 2001, 09:19:36 CET.