Solution

Plot[dz2[θ, 0], {θ, 0, π}]

-Graphics -

Solve[dz2[θ, φ] == 0, {θ, φ}]

θ_1 = ArcCos[1/Sqrt[3.]]

θ_2 = ArcCos[-1/Sqrt[3.]]

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

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

donut + 2 lobe

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

lobes = Show[{figure5, figure5b}] ;

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