Re: How to integrate a function over a polygon

*To*: mathgroup at smc.vnet.net*Subject*: [mg123205] Re: How to integrate a function over a polygon*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 26 Nov 2011 05:08:47 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111250957.EAA11225@smc.vnet.net>

Well, perhaps you mean this. Let the function be: f[x_, y_, z_] := x^2 + y^2 + z^2 We want to integrate it over the simplex: x+y+z==1, 0<=x<=1,0<=y<=1,0<=y<=1 On the simplex the function can be expressed in terms of only x and y as follows: g[x_, y_] = Expand[Last[PolynomialReduce[f[x, y, z], {x + y + z - 1}, {z, x, z}]]] 2*x^2 + 2*x*y - 2*x + 2*y^2 - 2*y + 1 In terms of x and y the simplex can be described as: cond[x_, y_] := x + y <= 1 && 0 <= x <= 1 && 0 <= y <= 1 So now we simply compute: Integrate[Boole[cond[x, y]]*g[x, y], {x, 0, 1}, {y, 0, 1}] 1/4 Andrzej Kozlowski On 25 Nov 2011, at 10:57, Mikael wrote: > Well, as I wrote in my OP, it is a 2-diemnsional unit simplex so you can always re-parametrize the function to have 2 arguments. > > In any case, your answer is not useful unless you had also answered the original question apart from your remark. > >> First of all, f would need three arguments. >> >> Bobby >> >> On Wed, 23 Nov 2011 06:07:00 -0600, Mikael >> <mikaen.anderson.1969 at gmail.com> wrote: >> >>> The subject line asks the general question but to >> be more specific >>> suppose I have a 2-dimentional unit simplex defined >> as >>> >>> Polygon[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}]. >>> >>> I winder how I can integrate a function f(x,y) over >> this simplex. Thanks. >>> >> >> >> -- >> DrMajorBob at yahoo.com >> >

**References**:**Re: How to integrate a function over a polygon***From:*Mikael <mikaen.anderson.1969@gmail.com>