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Mathematik-Online problems:

Problem 1093: Integration over a Simplex

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Find the integral of the function

$\displaystyle f(x,y,z) = (1-x-y)z $

over the simplex.

$\displaystyle S:\quad x,y,z \geq 0\,, \quad x+y+z\leq 1 \, . $

First you should show, that the transformation defined by

$\displaystyle \left( \begin{array}{r}x\\ y\\ z\end {array} \right) = g(u,v,w) ... ...v (1-u)\\ w (1-u)(1-v) \end {array} \right) \,, \qquad 0 \le u,v,w \le 1 \,, $

transfers the simplex $ S$ into the unit cube.

Using this transformation it is easy to calculate the integral.

(Author: Höfert)

see also:

[Examples]
  automatisch erstellt am 22.  7. 2008