Problem Set 5 Individual

Problem I5-1

Computer Guide, Problem 9.10, Page 108

ParaEllip  =  {a Cos[v] Cos[u], b Cos[v] Sin[u], 
          c Sin[v]};

To solve for the non-parametric equation, we can designate x, y, and z to be the three members of the vector valued function above.

X  =  ParaEllip[[1]];
Y  =  ParaEllip[[2]];
Z  =  ParaEllip[[3]];

Simple algebraic manipulations generate the equation of an ellipsoid.

FullSimplify[
      (X^2)/(a^2) +  (Y^2)/(b^2) +  (Z^2)/(c^2)]

(X^2)/(a^2) +  (Y^2)/(b^2) +  (Z^2)/(c^2) == 1 is the equation of this ellipsoid in Cartesian coordinates.

Now, to find a unit normal vector.  The cross product of two tangents will net a normal vector to a surface.  Two tangents can be the partial derivatives of the ellipsoid with respect to its two variables u and v.

<< Calculus`VectorAnalysis`

UEllip  =  D[ParaEllip, u]
VEllip  =  D[ParaEllip, v]

Cross[UEllip,  VEllip] // Simplify

Unfortunately, my computer's memory precludes me from including  ellipsoid graphics (one computer crash is enough...).   By varying the values a,b,c, it should be clear that the ellipsoid can change its shape and orientation quite a bit.  However, the center of the ellipsoid will always remain at (0,0,0) as long as the Cartesian variables (x,y,z) do not include integer shifts, ( e.g. (((x - 3))^2)or (((y + 7))^2), etc.).

Level set formulation.

Clear["Global`*"]

a = 1;
b = 2;
c = 3;

LevelSet = 
      Solve[(X^2)/(a^2) +  (Y^2)/(b^2) +  (Z^2)/(c^2)  == 1, Z]

Note how the following plot consists of concentric ellipses which exist as "shells" of a full ellipse.  The errors that result when the following command is processed occur because of the square root in the expression.  Here's one way to envision the level sets (in two dimensions)

ContourPlot[
        (3/2) (Sqrt[4 - 4 (X^2) - (Y^2)]), {X, -2, 2}, {Y, -2, 
          2}];

I found the following functions defined on the internet at the following link: (http://www.dimi.uniud.it/~gorni/Mma/Mma.html#LevelPlot3D).  Note you won't be able to properly evaluate this function again unless you click on "Click here to load package" which is embedded in the Mathematica notebook linked on the above website.
What's neat about this user-defined function is that it shows you the 2-D level set projections on the 3-D leve set surface.

LevelPlot3D[
        (3/2) (Sqrt[4 - 4 (X^2) - (Y^2)]),  {X, - .1,  .1}, {Y, - .1,  .1}, ShowProjectedLevelSets -> True];

Problem I5-2

Mass = density integrated over volume.

Clear["Global `*"]

Sigma  =  (x^2) + (y^2) + (z^2);

(∫_(-2)^2)(∫_(-3)^3)(∫_(-1)^1) 
            Sigma \[DifferentialD]x \[DifferentialD]y \[DifferentialD]z