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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
> >>
> >
>
>
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