How do I identify each surface in a multi-surface Plot3D?

Question

I am plotting three surfaces on a 3D plot. They differ in the value of one parameter. Is there a way to distinguish each surface for various values of the parameter.

My code is

Plot3D[{2*(Cosh[ h/ T]*(2 + Exp[2 /( T)]) + 
     Exp[-2  /( T)]*Cosh[3  h/T] ), 
  2*(Cosh[ h/ T]*(2 + Exp[2 10/( T)]) + 
     Exp[-2*10 /( T)]*Cosh[3  h/T] ), 
  2*(Cosh[ h/ T]*(2 + Exp[2 100/( T)]) + 
     Exp[-2*100 /( T)]*Cosh[3  h/T] )}, {T, 0.001, 1000}, {h, 0, 1}, 
 PlotLabel -> 
  Style[Framed["Partition Function Z"], 20, Red, 
   Background -> Lighter[LightYellow]], Background -> LightGray, 
 ColorFunction -> "BlueGreenYellow", 
 AxesLabel -> {T[kelvin], B[tesla]}, BoundaryStyle -> Thick, 
 ClippingStyle -> Opacity[0.5]]

EDIT

I am trying to use ShowLegend because PlotLegend cannot work with Plot3D in v8... The Code I am using is

Z[T_, h_, a_] := 
  2*(Cosh[h/T]*(2 + Exp[2 a/(T)]) + Exp[-2/(T)]*Cosh[3 h/T]);
params = {1, 10, 100};
ShowLegend[Plot3D[
  Evaluate[Table[
    Tooltip[Z[T, h, a], StringForm["a = ``", a]], {a, params}]], {T, 
   0.001, 1000}, {h, 0, 1}, 
  PlotLabel -> 
   Style[Framed["Partition Function Z"], 20, Red, 
    Background -> Lighter[LightYellow]], 
  MeshFunctions -> {(*#1&,*)#2 &, #3 &}, 
  Mesh -> {(*15,*)15, Range[7.5, 11, 0.1]}, 
  MeshStyle -> {(*Gray,*)Gray, Black}, Background -> LightGray, 
  PlotStyle -> {Red, Green, Blue}(*,PlotLegends->Table[StringForm[
  "a = ``",a],{a,params}]*), AxesLabel -> {T[kelvin], B[tesla]}, 
  BoundaryStyle -> Thick, ClippingStyle -> Opacity[0.5]], 
 Graphics3D[{Table[{Arrowheads[0.0004], 
     Arrow[{{900, 0.2 + 0.25 Log10[a], 9.5}, {50. + 5 a, 
        0.01 + 0.005 a, Z[50. + 5 a, 0.01 + 0.005 a, a]}}], 
     Text[Style[Framed[StringForm["J = ``", a]], Red, 
       Background -> Lighter[LightYellow]], {900, 0.2 + 0.25 Log10[a],
        9.5}]}, {a, params}]}],{{{Graphics[{Table[StringForm["a = ``", a], {a, params}]}]}}, 
 LegendPosition -> {1.1, -.4}}]

Answer 1

f[k_, h_, t_] := 2*(Cosh[h/t]*(2 + Exp[(2/t) 10^k]) + Exp[-(2 /t) 10^k]*Cosh[3 h/t]);

grad = {"LakeColors", "DarkRainbow", "NeonColors"};

Show@Table[
  Plot3D[f[k, h, t], {t, 0.001, 1000}, {h, 0, 1}, 
   ColorFunction -> grad[[k + 1]], ColorFunctionScaling -> True, 
   PlotRange -> {8, 8.8},
   PlotLabel -> Style[Framed["Partition Function Z"], 20, Red, Background -> Lighter[LightYellow]], 
   Background -> LightGray, AxesLabel -> {T[kelvin], B[tesla]}, BoundaryStyle -> Thick, 
   ClippingStyle -> Opacity[0.5]
   ], {k, {0, 1, 2}}]

Edit

Or you could do something like this:

f[k_, h_, t_] := 2*(Cosh[h/t]*(2 + Exp[(2/t) 10^k]) + Exp[-(2/t) 10^k]*Cosh[3 h/t]);

Needs["PlotLegends`"];
GraphicsRow[{
 Show@Table[Plot3D[f[k, h, t], {t, 0.001, 1000}, {h, 0, 1},
     ColorFunction -> 
      Function[{x,y,z}, RGBColor[Sequence @@ RotateLeft[{1,0,0}, k], Rescale[z, {7.8,8.8}]]],
     ColorFunctionScaling -> False,
     MeshFunctions -> {#3 &},
     PlotRange -> {8, 8.8},
     PlotLabel -> Style[Framed["Partition Function Z"], 20, Red],
     Background -> LightGray,
     AxesLabel -> {T[kelvin], B[tesla]},
     BoundaryStyle -> Thick,
     ClippingStyle -> Opacity[0.5]],
    {k, {0, 1, 2}}],
  Graphics@
   Legend[
      Table[{Graphics[{RGBColor[Sequence @@ RotateLeft[{1, 0, 0}, k]], Rectangle[]}], 10^k}, 
      {k, 0, 2}], LegendBorder -> None, LegendShadow -> None]}]

Answer 2

There are various things you might try, depending on your particular objectives.

1. If you aren't wedded to gradients, you can use #3& as a mesh function to indicate height. For your particular functions, you might wish to omit #1& as a mesh function, since the mesh lines are parallel.

2. If you need to distinguish graphs in an notebook and not in a printout, you can use tooltips.

3. You can insert 3D graphics to draw labels & arrows to the plots. Possibly unsatisfactory if the graph is to be rotated about.

Here is an example incorporating all three ideas (with belisarius's label).

Z[T_, h_, a_] := 
  2*(Cosh[h/T]*(2 + Exp[2 a/(T)]) + Exp[-2/(T)]*Cosh[3 h/T]);
params = {1, 10, 100};
Show[Plot3D[
  Evaluate[Table[
    Tooltip[Z[T, h, a], StringForm["a = ``", a]], {a, params}]], {T, 
   0.001, 1000}, {h, 0, 1}, 
  PlotLabel -> 
   Style[Framed["Partition Function Z"], 20, Red, 
    Background -> Lighter[LightYellow]], 
  MeshFunctions -> {(*#1&,*)#2 &, #3 &}, 
  Mesh -> {(*15,*)15, Range[7.5, 11, 0.1]}, 
  MeshStyle -> {(*Gray,*)Gray, Black}, Background -> LightGray, 
  PlotStyle -> {Red, Green, Blue}, 
  PlotLegends -> Table[StringForm["a = ``", a], {a, params}], 
  AxesLabel -> {T[kelvin], B[tesla]}, BoundaryStyle -> Thick, 
  ClippingStyle -> Opacity[0.5]],
 Graphics3D[{Table[{Arrowheads[0.0004], 
     Arrow[{{900, 0.2 + 0.25 Log10[a], 9.5}, {50. + 5 a, 
        0.01 + 0.005 a, Z[50. + 5 a, 0.01 + 0.005 a, a]}}], 
     Text[Style[Framed[StringForm["a = ``", a]], Red, 
       Background -> Lighter[LightYellow]], {900, 0.2 + 0.25 Log10[a],
        9.5}]}, {a, params}]}]
 ]