Unit Normal for Surfaces

This follows in much the same way. Consider the cone x^2+y^2=(1-z)^2. Here's the plot. (I fancied it up for you!)


  <<Graphics`ParametricPlot3D`


  cone=ParametricPlot3D[{r Cos[u],
  r Sin[u],1-r},{u,0,2 Pi},{r,0,1}];


  Clear[f,x,y,z]
  f[x_,y_,z_]:=x^2+y^2-(1-z)^2


  grad3[f][x,y,z]


  u=grad3[f][x,y,z]/norm[grad3[f][x,y,z]]


  utest=u/.{x->0,y->1,z->0}


  vector=vectorPlot[{{0,1,0}},{{0,1,0}+utest}];


  Show[cone,vector,DisplayFunction->$DisplayFunction];

If that view does not convince you, look at it this way:


  Show[cone,vector,DisplayFunction->$DisplayFunction,
  ViewPoint->{1,0,0}];