Computing volume inside a given surface

Questions
,
1

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

2

Hi Claude.
you have to express the two points in quaestion in spherical coordinates.

Show[{SphericalPlot3D[
   Sin[Theta]*f[Theta, Phi] /. {a -> 1, b -> 1, c -> 1}, {Theta, 0, 
    Pi}, {Phi, 0, 2 Pi}, 
   PlotStyle -> Directive[Cyan, Opacity[0.3], 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[({Sin[Theta] Cos[Phi], Sin[Theta] Sin[Phi], 
         Cos[Theta] f[Theta, Phi]} /. {a -> 1, b -> 1, c -> 1}) /. 
      Last@N[LeMin]]}], 
  Graphics3D[{PointSize[Large], Red, 
    Point[({Sin[Theta] Cos[Phi], Sin[Theta] Sin[Phi], 
         Cos[Theta] f[Theta, Phi]} /. {a -> 1, b -> 1, c -> 1}) /. 
      Last@N[ LeMax]]}]}]

Regards Roland