Re: How to integrate a function over a polygon

*To*: mathgroup at smc.vnet.net*Subject*: [mg123227] Re: How to integrate a function over a polygon*From*: Mikael <mikaen.anderson.1969 at gmail.com>*Date*: Sun, 27 Nov 2011 04:14:29 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

Many thanks indeed for your elegant solution, Andrzej. May I ask a related question regarding the last argument in Expand[Last[PolynomialReduce[f[x, y, z], {x + y + z - 1}, {z, x, z}]]]. I wonder what is the role of {z, x, z} there. I get the same answer if I change it to {z, x} or {z} and I could not figure it out from the help page for PolynomialReduce either. /Mikael > 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 > 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 > >> > > > >

**Follow-Ups**:**Re: How to integrate a function over a polygon***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>