Math 3298  -  Hampton  -  Lab 7  -  Due 11/30.

Lab reports should conform to the guidelines explained on the handout.  Collaboration during the lab is strongly encouraged, although the write-up of the lab report must be done individually.

Sphere and Hypersphere volumes

In this lab you will determine the n-dimensional 'volume' of a hypersphere, or n-dimensional sphere.  To help you do this, I will compute volumes of  three- and four-dimensional spheres of radius a.

For the 3  D case,  
    we use spherical coordinates and the spherical volume element to find:

Integrate[
      p^2 Sin[phi], {p, 0, a}, {phi, 0, Pi}, {theta, 0, 2*Pi}]

In four dimensions, we need an extension of spherical coordinates and the Jacobian.  We introduce a new 'hyperpolar' angle phi2, which like phi has values in [0,Pi].

w[p_, phi_, theta_,  phi2_] := 
        p Cos[phi2];
z[p_, phi_, theta_, phi2_] := 
        p Sin[phi2] Cos[phi];
x[p_, phi_, theta_,  phi2_] := 
        p Sin[phi2] Sin[phi] Cos[theta] ;
y[p_, phi_, theta_,  phi2_] := 
        p Sin[phi2] Sin[phi] Sin[theta];

hypervariables = Map[#[p, phi, theta, phi2] &, {x, y, z, w}]

jacobian = 
      Det[Map[D[hypervariables, #] &, {p, phi, theta, phi2}]]

Lets make Mathematica simplify that:

FullSimplify[
      Abs[jacobian], {p ≥ 0, 
        0 ≤ phi ≤ Pi, 
        0 ≤ phi ≤ Pi, Element[phi, Reals], 
        Element[phi2, Reals], Element[p, Reals], 
        Element[theta, Reals]}]

Integrate[
      (p^3) Sin[phi] Sin[phi2]^2, {p, 0, a}, {phi, 0, Pi}, {theta, 0, 
        2*Pi}, {phi2, 0, Pi}]

So that's the volume of a four-dimensional sphere - a^4 Pi^2/2.

Exercise

1.  	a. Compute the 5-d volume of a 5-sphere.  You will need to introduce a new coordinate.  
     	b. What will the Jacobian for 6-dimensional spherical coordinates look like?  Use your guess to compute the 6-d volume of a 6-sphere.  
 
2.  	a. Given that the 7-dimensional volume of a sphere of radius a is a^7((16 (π^3))/105) and the 8-dimensional volume is a^8 ((π^4)/24), can you guess a formula for the n-dimensional volumes?  It will be different for even and odd n.  Remember to use the 1-d and 2-d volumes (2a and π a^2 respectively) to help you guess.
	b. Plot your guess for the n-dimensional volume for radius a = 1 for n going from 1 to 20.  What strikes you as odd about the graph?