Example 3

In three dimensions it is not as easy to visualize streamlines, but it can be done. Here the vector field is {3x^2, -y, z^3}.


  Clear[x,y,z]


  p5=PlotVectorField3D[{3 x^2,-y,z^3},{x,-5,5},{y,-5,5},{z,-5,5},Axes->True];


  Show[p5,ViewPoint->{0,1,0}];


  Clear[x,y,z]
  DSolve[{x'[t]/(3 x[t]^2)==y'[t]/-y[t],x[0]==2,y[0]==1},x,t];
  x=Evaluate[x[t]/.Flatten[%][[1]]]/.{y[t]->y,y[0]->1}


  Clear[y,z]
  DSolve[{y'[t]/-y[t]==z'[t]/z[t]^3,y[0]==1,z[0]==6},z,t];
  z=Evaluate[z[t]/.Flatten[%][[1]]]/.{y[t]->y,y[0]->1}


  p6=ParametricPlot3D[{x,y,z,RGBColor[1,0,0]},{y,-10,10}];


  Show[p5,p6];


  Show[p5,p6,ViewPoint->{0,1,0}];