Build a surface normal vector (I use Mathematica)

[member 428835]

Homework Statement

Given $r(s) = \sin(c s)/c$ we define the surface $S = \langle r(s) \cos(\phi), r(s) \sin(\phi), (1 - \cos(cs))/c \rangle$. Compute a normal vector $n$ to $S$.

Relevant Equations

a surface normal is found for a parametric surface $S(s,\phi)$ via $n = d_sS \times d_\phi S$

Not HW, but seems to fit here.

I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$

I use the following in Mathematica:

r[s_, \[Alpha]_] := Sin[Cos[\[Alpha]] s]/Cos[\[Alpha]]
z[s_, \[Alpha]_] := (1 - Cos[Cos[\[Alpha]] s])/Cos[\[Alpha]]
x[s_, \[CurlyPhi]_, \[Alpha]_] := r[s, \[Alpha]] Cos[\[CurlyPhi]]
y[s_, \[CurlyPhi]_, \[Alpha]_] := r[s, \[Alpha]] Sin[\[CurlyPhi]]
z[s_, \[CurlyPhi]_, \[Alpha]_] := z[s, \[Alpha]]
S[s_, \[CurlyPhi]_, \[Alpha]_] := {x[s, \[CurlyPhi], \[Alpha]],
  y[s, \[CurlyPhi], \[Alpha]], z[s, \[CurlyPhi], \[Alpha]]}
Cross[D[S[s, \[CurlyPhi], \[Alpha]], s],
   D[S[s, \[CurlyPhi], \[Alpha]], \[CurlyPhi]]].S[
   s, \[CurlyPhi], \[Alpha]] // FullSimplify

If you copy-paste this you should get the same output, where $c = \cos(\alpha)$. What am I doing wrong? What is $n$? Any help is greatly appreciated!

 

[member 428835]

My mistake! Unsure why I was taking that final dot product: it obviously should not be zero except for rare spheres!