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This has a connection to the previous post here: Using a complicated function as color function Also, a recent post of mine appeared here: Create a colorbar plot legend but I should have solved the problem with the color function before asking the question about color bar. If opening a new thread goes against the forums policies I apologize.

Thanks to the answers I got, I have now what I wanted

h = ParametricPlot3D[lst, {u, 0, 2*Pi}, {v, 0, Pi}, Mesh -> False, 
    ColorFunction -> (Function[{x, y, z, u, v}, Hue[Rescale[dam, {0, 1}]]]), 
    ColorFunctionScaling -> False, ImageSize -> 800];
g = Plot3D[Rescale[dam, {0, 1}], {u, 0, 2*Pi}, {v, 0, Pi}, Mesh -> False, 
    ColorFunction -> (Function[{u, v, z}, Hue[Rescale[dam, {0, 1}]]]), 
    ColorFunctionScaling -> False, PlotRange -> All];
Row[{h, Show[g, ImageSize -> 400]}]

What I further want is to add a color bar to each plot. Something like

Legended[h, BarLegend[{colFun,{min,max}}]] (*version 9*)

which does not give the requested color bar.

Also,

colourBar = 
  DensityPlot[y, {x, 0, 1}, {y, 0, 1}, AspectRatio -> 5, 
   PlotRangePadding -> 0, 
   FrameTicks -> {{None, Range[0, 1, 0.1]}, {None, None}}, 
   ColorFunction -> colFun];
Row[{Show[gr1, ImageSize -> Large], 
  Show[colourBar, ImageSize -> Small]}] (*vesrion 8 alternative*)

does not work as well.

Thanks in advance for any advice.

Dimitris

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{min, max} ={-10, 10};
h2 = ParametricPlot3D[lst, {u, 0, 2*Pi}, {v, 0, Pi}, Mesh -> False, 
     ColorFunction -> (Function[{x, y, z, u, v}, Hue[Rescale[dam, {0, 1}]]]), 
     ColorFunctionScaling -> False, 
     ImageSize -> 800, 
     PlotLegends -> BarLegend[{Hue, {min, max}}, ColorFunctionScaling -> True]]

enter image description here

or

Legended[h, BarLegend[{Hue, {min, max}}, ColorFunctionScaling -> True]]

Using Hue instead of colFun as the ColorFunction setting in your colourBar gives something that you can use as the second argument of Legended in version 8:

colourBar2 = DensityPlot[v, {u, 0, 1}, {v, min, max},
                       ColorFunction -> Hue,
                       AspectRatio -> 10, Frame -> True, PlotRangePadding -> 0, 
                       ImagePadding -> 25,
                       FrameTicks -> {False, Automatic, False, Automatic}]

enter image description here

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