Functions of Two Variables

Professor Mohammad Ghomi
Honors Calculus III, Georgia Tech, Spring 2006

Basic Plots; Animation of (x^2) + n (y^2)

Plot3D[x^2 + y^2,  {x, -1,  1},  {y, -1, 1}, 
        BoxRatios -> {1, 1, 1}];

ContourPlot[
        x^2 + y^2,  {x, -1,  1},  {y, -1, 1}];

Plot3D[x^2 - y^2,  {x, -1,  1},  {y, -1, 1}, 
        BoxRatios -> {1, 1, 1}];

ContourPlot[
        x^2 - y^2,  {x, -1,  1},  {y, -1, 1}];

Table[
        Plot3D[x^2 + n y^2,  {x, -1,  1},  {y, -1, 1}, 
          BoxRatios -> {1, 1, 1}, PlotLabel -> Chop[n,  .001],  
          Axes -> False], {n, -1, 1, 0.1}];

Limit of (((x^4) + (y^4))/((((x^2) + (y^4)))^3))along various directions

Plot3D[
        x^4  y^4/(x^2 + y^4)^3,  {x, -1, 1},  {y, -1, 1},  
        PlotPoints -> {75, 75}];

f[x_, y_] := x^4  y^4/(x^2 + y^4)^3;
ParametricPlot3D[{t, t,  f[t, t]},  {t, -1, 1},  
        BoxRatios -> {1, 1, 0.5}];

f[x_, y_] := x^4  y^4/(x^2 + y^4)^3;
ParametricPlot3D[{t^2, t,  f[t^2, t]},  {t, -1, 1},  
        BoxRatios -> {1, 1, 0.5}, PlotPoints -> 20];

f[x_, y_] := x^4  y^4/(x^2 + y^4)^3;
Table[
        ParametricPlot3D[{t Cos[θ], t Sin[θ],  
            f[t Cos[θ], t Sin[θ]]},  {t, -1, 1},  
          PlotRange -> {{-1, 1}, {-1, 1}, {0,  .15}}, 
          BoxRatios -> {1, 1, 0.5}], {θ, -Pi, Pi, 
          Pi/20}];

f[x_, y_] := x^4  y^4/(x^2 + y^4)^3;
Table[
        Plot[f[t Cos[θ], t Sin[θ]],  {t, -1, 1},  
          PlotRange -> {{-1, 1}, {0,  .15}}], {θ, -Pi, Pi, 
          Pi/20}];

Tangent Planes

T[f_, a_, b_][x_, y_] := 
      f[a, b] + ((∂_x) f[x, y] /. {x -> a, 
                y -> b}) (x - 
              a) + ((∂_y) f[x, y] /. {x -> a, 
                y -> b}) (y - b)

f[x_, y_] := x^2 + y^2;

Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, 
        BoxRatios -> {1, 1, 1}, 
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 
              2}}];

Plot3D[T[f,  .5,  .5][x, y] // Evaluate, {x, -1, 
          1}, {y, -1, 1}, 
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 2}}, 
        BoxRatios -> {1, 1, 1}, ClipFill -> None];

Show[%, %%];