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!)

In[192]:=

  <<Graphics`ParametricPlot3D`

In[193]:=

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

In[194]:=

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

In[195]:=

  grad3[f][x,y,z]

Out[195]=

  {2 x, 2 y, 2 (1 - z)}

In[196]:=

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

Out[196]=

                2 x
  {------------------------------, 
           2      2            2
   Sqrt[4 x  + 4 y  + 4 (1 - z) ]
   
                 2 y
    ------------------------------, 
            2      2            2
    Sqrt[4 x  + 4 y  + 4 (1 - z) ]
   
              2 (1 - z)
    ------------------------------}
            2      2            2
    Sqrt[4 x  + 4 y  + 4 (1 - z) ]

In[197]:=

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

Out[197]=

         1        1
  {0, -------, -------}
      Sqrt[2]  Sqrt[2]

In[198]:=

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

In[199]:=

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

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

In[200]:=

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

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