Re: Hypercube

[Bob Sciamanda <trebor@VELOCITY.NET>]

----- Original Message -----
From: "David Bowman" <David_Bowman@GEORGETOWNCOLLEGE.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Sunday, July 16, 2000 1:49 PM
Subject: Re: Hypercube
> . . .
> The simplest derivation I have seen involves taking the product of
> D distinct simple Gaussian integrals (whose value is known) and then
> converting resulting multidimensional integral via a coordinate
> transformation to a generalization of spherical polar coordinates.
>  . . .
> David Bowman
> David_Bowman@georgetowncollege.edu

The above derivation using Gaussian integrals can be found in Kittel's
"Elementary Statistical Physics".

Another, heuristic approach might be summarized as a generalization of the
calculation of V3(R), a sphere,  from V2(R), a circle:

V3(R) = INT(0 to PI) {PI (R*sin[Theta])^2} * R *sin(Theta) *dTheta

This generalizes to

Vn(R) = INT(0 to PI) {Vn-1(R*sin[Theta])}* R*sin[Theta] *dTheta

Then, applying Vn(R) = R^n * Vn(1) to Vn-1(R*sin[Theta]) in that integral,
one gets

Vn(R) = R^n * Vn-1(1) * INT(0 to PI) {sin(Theta)}^n * dTheta

The problem is then reduced to solving the above definite integral.  This
can be developed into a calculable series, etc.

Bob

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor