Bxdty

The following function generates a plot of the 3d function indicated in the example.

Plot3D[(x^2 + y^2) Exp[1 - x^2 - y^2], {x, -3, 3}, {y, -3, 3}, 

 Mesh -> None, ImageSize -> Large, PlotPoints -> 35, 

 PlotStyle -> {Texture[

    StreamPlot[

     Evaluate[-D[(x^2 + y^2) Exp[1 - x^2 - y^2], {{x, y}}]], {x, -3, 

      3}, {y, -3, 3}, Frame -> None, ImageSize -> Large, 

     StreamStyle -> Black]]}]

However, when I choose a different ColorFunction parameter the texture (that only consists of arrows) disappears. Any idea how to correct this? I tried to make the background transparent, combine two 3D plots etc without success. Also, I have no idea why this is happening.

Here is the 3D plot without the gradient field.

Plot3D[(x^2 + y^2) Exp[1 - x^2 - y^2], {x, -3, 3}, {y, -3, 3}, 

 Mesh -> None, ImageSize -> Large, PlotPoints -> 35, 

 PlotStyle -> {Texture[

    StreamPlot[

     Evaluate[-D[(x^2 + y^2) Exp[1 - x^2 - y^2], {{x, y}}]], {x, -3, 

      3}, {y, -3, 3}, Frame -> None, ImageSize -> Large, 

     StreamStyle -> Black]]}, ColorFunction -> "Rainbow"]

You can use StreamDensityPlot (which accepts the ColorFunction option) to produce the texture:

sdp = StreamDensityPlot[Evaluate[{-D[(x^2 + y^2) Exp[1 - x^2 - y^2], {{x, y}}], 

    (x^2 + y^2) Exp[1 - x^2 - y^2]}], {x, -3, 3}, {y, -3, 3}, 

   StreamStyle -> Black, 

   ColorFunction -> "Rainbow", 

   ColorFunctionScaling -> False, Frame -> False, Axes -> False, 

   PlotRangePadding -> None];

Plot3D[(x^2 + y^2) Exp[1 - x^2 - y^2], {x, -3, 3}, {y, -3, 3}, 

 Mesh -> None, ImageSize -> Large, PlotPoints -> 35, 

 PlotStyle -> Texture[Lighter@sdp], Lighting -> "Neutral"]

The color is not quite right but the idea seems to work. Edit: much closer now.

dp = DensityPlot[(x^2 + y^2) Exp[1 - x^2 - y^2], {x, -3, 3}, {y, -3, 3}, 

   ColorFunction -> "Rainbow", PlotPoints -> 100];



sp = StreamPlot[

   Evaluate[-D[(x^2 + y^2) Exp[1 - x^2 - y^2], {{x, y}}]], {x, -3, 3}, {y, -3, 3}, 

   Frame -> None, ImageSize -> Large, StreamStyle -> Black];



tex = Show[dp, sp, Frame -> None, PlotRangePadding -> 0, ImageSize -> 500];



Plot3D[(x^2 + y^2) Exp[1 - x^2 - y^2], {x, -3, 3}, {y, -3, 3}, Mesh -> None, 

 ImageSize -> Large, PlotPoints -> 35

 , PlotStyle -> {Texture[Lighter[tex, 0.15]]}

 , Lighting -> "Neutral"

]

PlotStyle -> Texture[...] relies on VertexTextureCoordinates to map the texture to polygon vertices.

ColorFunction -> colorfunction relies on VertexColors to associate colors with the polygon vertices.

Only one of them actually gets to style the polygon. In my case, it seems to be the texture:

Graphics3D[{Texture[RandomImage[1, 100]], 

  Polygon[{{0, 0, 0}, {1, 0, 0}, {1, 1, 0}}, 

   VertexColors -> {Red, Green, Blue}, 

   VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}}]}, 

 Lighting -> "Neutral", BoxRatios -> {1, 1, 1}]

It sounds like the color function is winning in your case. It wouldn't surprise me if that was dependent on things like OS, software version, phase of the moon, etc...