Hi!
Consider the functions
f[Theta_, Phi_] :=
a*Cos[Theta]^3 + b*Cos[Theta] Sin[Theta]^2 -
c*(Cos[Phi] - 3 Cos[3 Phi]) Sin[Theta]^3;
and
Sin[Theta]*f[Theta, Phi]
I'm interessed in computing the volume inside the associated surface :
LeMin = Last@
Minimize[{Sin[Theta]*f[Theta, Phi] /. {a -> 1, b -> 1, c -> 1},
0 <= Theta <= Pi && 0 <= Phi <= 2 Pi}, {Theta, Phi}]
LeMax = Last@
Maximize[{Sin[Theta]*f[Theta, Phi] /. {a -> 1, b -> 1, c -> 1},
0 <= Theta <= Pi && 0 <= Phi <= 2 Pi}, {Theta, Phi}]
Show[{SphericalPlot3D[
Sin[Theta]*f[Theta, Phi] /. {a -> 1, b -> 1, c -> 1}, {Theta, 0,
Pi}, {Phi, 0, 2 Pi},
PlotStyle -> Directive[Cyan, Opacity[0.7], Specularity[White, 10]],
Mesh -> None, PlotPoints -> 300,
AxesLabel -> {Style["f", Bold, 15],
Style["\[CapitalTheta]", Bold, 15], Style["\[Phi]", Bold, 15]},
PlotRange -> All, AspectRatio -> Automatic, ImageSize -> Large],
Graphics3D[{PointSize[Large], Black,
Point[{Theta, Phi, Sin[Theta]*f[Theta, Phi]} /. {a -> 1, b -> 1,
c -> 1} /. LeMin]}],
Graphics3D[{PointSize[Large], Red,
Point[{Theta, Phi, Sin[Theta]*f[Theta, Phi]} /. {a -> 1, b -> 1,
c -> 1} /. LeMax]}]}]
I'm puzzeled by two facts:
(1) the computed extremes of the function are far from the border of the graph (even with a large number of points)
(2) the bounding box is centered on the origin, and does not correspond to the domain of interest, [0,Pi] X [0,2 Pi].
Is there something wrong in this plot? In addition, the extremes computed with FindMinimum (Maximum) are also too small (big), and far from the border, too.
Regards,
Claude