Solution

Plot[dz2[θ, 0], {θ, 0, π}] ;

[Graphics:../HTMLFiles/MATH5_TUT_1_703.gif]

Solve[dz2[θ, φ] == 0, {θ, φ}]

Solve :: ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. More…

Solve :: svars : Equations may not give solutions for all \"solve\" variables. More…

{{θ→ -ArcCos[-1/3^(1/2)]}, {θ→ArcCos[-1/3^(1/2)]}, {θ→ -ArcCos[1/3^(1/2)]}, {θ→ArcCos[1/3^(1/2)]}}

θ_1 = ArcCos[1/Sqrt[3.]]

θ_2 = ArcCos[-1/Sqrt[3.]]

0.955317

2.18628

ParametricPlot[{dz2[θ, 0] Sin[θ], dz2[θ, 0] Cos[θ]}, {θ, 0, 2 π}, PlotRange→ {{-.65, .65}, {-.65, .65}}, Frame→True] ;

[Graphics:../HTMLFiles/MATH5_TUT_1_713.gif]

lobe = ∫_0^θ_1∫_0^(2 π) (dz2[θ, φ])^2 * Sin[θ] φθ

donut = ∫_θ_1^θ_2∫_0^(2 π) (dz2[θ, φ])^2 * Sin[θ] φθ

0.30755

0.3849

donut + 2 lobe

1.

norm = ∫_0^π∫_0^(2 π) (dz2[θ, φ])^2 * Sin[θ] φθ

1

[Graphics:../HTMLFiles/MATH5_TUT_1_723.gif]

[Graphics:../HTMLFiles/MATH5_TUT_1_725.gif]

[Graphics:../HTMLFiles/MATH5_TUT_1_727.gif]

lobes = Show[{figure5, figure5b}] ;

[Graphics:../HTMLFiles/MATH5_TUT_1_729.gif]

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