Functions of 3 or more variables

p[x_, y_, 
        z_] := (E^(-α z)) ((Subscript[p, a]) - (Subscript[p, 1]) (E^(-((x^2) + (y^2)))))

Here are contour maps of p at two different heights z = 0 and z = 1, for α arbitrarily set to 1, (Subscript[p, a])set to 1, and (Subscript[p, 1]) set to 0.9.

α = 1;  (Subscript[p, a]) = 1; 
    (Subscript[p, 1]) = 0.9;
p1 = Plot3D[p[x, y, 0],{x, -3, 3},{y, -3, 3},PlotRange -> All,DisplayFunction -> Identity];
p2 = Plot3D[p[x, y, 1],{x, -3, 3},{y, -3, 3},PlotRange -> All,DisplayFunction -> Identity];
p3=ContourPlot[p[x, y, 0],{x, -3, 3},{y, -3, 3},DisplayFunction -> Identity];
p4=ContourPlot[p[x, y, 1],{x, -3, 3},{y, -3, 3},DisplayFunction -> Identity];
Show[
        GraphicsArray[{{p1, p2}, {p3, p4}}]];

Notice that despite the shape of the surfaces and contours looking the same, they are in fact different as the magnitudes are different (look at the magnitudes on the z axis of the surface plots).

Temperature in the sun

\(This\ is\ a\ very\ simplistic\ model\ of\ the\ temperature\ distribution\ in\ the\ sun . \ 
        It\ assumes\ that\ temperature\ is\ maximum\ right\ at\ the\ centre\ of\ the\ sun, \ 
    and\ then\ decays\ exponentially\ as\ the\ square\ of\ the\ radius\ of\ a\ sphere . \ 
        It\ is\ best\ to\ plot\ the\ level\ surfaces\ using\ parametric\ equations\ and\ spherical\ coordinates . \ 
        The\ plot\ that\ follows\ shows\ level\ surfaces\ for\ temperatures, \ T\  = \ 0.4\ T\_s, \ 0.6\ T\_s, \ 
    and\ 0.8\ T\_s . \ 
        The\ surfaces\ are\ not\ completely\ closed\ only\ for\ visualisation\ \(\(purposes\)\(.\)\)\)

T[x_, y_, 
        z_] := (Subscript[T, s]) (E^(-((x^2) + (y^2) + (z^2))))
(Subscript[T, s]) = 1;
r[c_] := Log[((Subscript[T, s])/c)]
c = 0.4  (Subscript[T, s]); p1 = ParametricPlot3D[{r[c] Sin[θ] Cos[ϕ], 
              r[c] Sin[θ] Sin[ϕ], r[c] Cos[θ]},{ϕ, ((-2 π)/10), 
              2  π - ((5  π)/10)},{θ, 0, π},DisplayFunction -> Identity];
c = 0.6  (Subscript[T, s]); p2 = ParametricPlot3D[{r[c] Sin[θ] Cos[ϕ], 
              r[c] Sin[θ] Sin[ϕ], r[c] Cos[θ]},{ϕ, ((-2 π)/10), 
              2  π - ((5  π)/10)},{θ, 0, π},DisplayFunction -> Identity];
c = 0.8  (Subscript[T, s]); p3 = ParametricPlot3D[{r[c] Sin[θ] Cos[ϕ], 
              r[c] Sin[θ] Sin[ϕ], r[c] Cos[θ]},{ϕ, ((-0 π)/10), 
              2  π - ((0  π)/10)},{θ, 0, π},DisplayFunction -> Identity];
Show[{p1, p2, p3}, 
        DisplayFunction -> $DisplayFunction];
Null

Linear function whose equation has a variable missing

If our equation has a variable missing, that variable can take any value. That is, regardless of what value we assign to the missing variable, the function will look the same. For example in the following function, z plays no part. Hence the function looks exactly the same for z = -∞, -10, 0, 1, 2, 3.14, 5.3, ....., etc, etc.

T[x_, y_, z_] := (x^2) + (y^2)

    
R = 1; 
    p1 = ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], z}, {ϕ, 0, 
          2  π}, {z, 0, 5}, DisplayFunction -> Identity];
R = 2; 
    p2 = ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], z}, {ϕ, 0, 
          2  π}, {z, 0, 5}, DisplayFunction -> Identity];
R = 3; 
    p3 = ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], z}, {ϕ, 0, 
          2  π}, {z, 0, 5}, DisplayFunction -> Identity];
Show[{p1, p2, p3}, 
        DisplayFunction -> $DisplayFunction];