Three Continuity Problems

<< Graphics`Graphics3D`

<< Graphics`ImplicitPlot`

Problem 16

F[x_, y_] := 
      If[x^2 + y^2 > 0, ((x y ((x^2) - (y^2)))/((x^2) + (y^2))), 
        0]

Fp1 = 
        Plot3D[F[x, y], {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {30, 30}, 
          Mesh -> False];

    
Clear[Pl1, c];

    
Pl1[c_] := 
        Plot3D[c, {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {30, 30}, Mesh -> False, 
          DisplayFunction -> Identity];

Table[
        Show[{Fp1, Pl1[z]}, 
          PlotRange -> {{-2, 2}, {-2, 2}, {-1.3, 1.3}}, 
          DisplayFunction -> $DisplayFunction], {z, -1, 
          1,  .1}];

Table[
        ImplicitPlot[F[x, y] == z, {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {30, 30}, Frame -> True], {z, -1, 
          1,  .1}];

ShadowPlot3D[F[x, y], {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> 25]

?ShadowPlot3D

ContourPlot[F[x, y], {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> {50, 50}, 
      Contours -> Range[-1, 1,  .1]]

Problem 17

F1[x_, y_] := 
      If[x^2 + y^2 > 0, ((x y)/((x^2) + (y^2))), 0]

Fp2 = 
        Plot3D[F1[x, y], {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {80, 80}, 
          Mesh -> False];

    
Clear[Pl1];

    
Pl1[c_] := 
        Plot3D[c, {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {30, 30}, Mesh -> False, 
          DisplayFunction -> Identity];

Table[
        Show[{Fp2, Pl1[z]}, 
          PlotRange -> {{-2, 2}, {-2, 2}, {- .7,  .7}}, 
          DisplayFunction -> $DisplayFunction], {z, -0.5, 
          0.5,  .1}];

Table[
        ImplicitPlot[F1[x, y] == z, {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {50, 50}, 
          Frame -> True], {z, - .5,  .5,  .05}];

ShadowPlot3D[F1[x, y], {x, -2, 2}, {y, -2, 2}, 
        Mesh -> False, PlotPoints -> 45];

Problem 18

F2[x_, y_] := 
      If[x^2 + y^2 > 0, (((x^2) y )/((x^4) + (y^2))), 
        0]

Fp3 = 
        Plot3D[F2[x, y], {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {60, 60}, 
          Mesh -> False];

Table[
        Show[{Fp3, Pl1[z]}, 
          PlotRange -> {{-2, 2}, {-2, 2}, {-1.3, 1.3}}, 
          DisplayFunction -> $DisplayFunction], {z, -0.5, 
          0.5,  .1}];

Table[
        ImplicitPlot[F2[x, y] == z, {x, -2, 2}, {y, -2, 2}, 
          PlotPoints -> {30, 30}, 
          Frame -> True], {z, - .5,  .5,  .05}];

ContourPlot[F2[x, y], {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> {50, 50}, 
      Contours -> Range[- .5,  .5,  .05]]