Error in variable trying to perform SlicedVectorPlot

I have been trying to plot the normal vector over the Möbius strip, which I know I cannot do asking it to be a well-defined continuous map, nevertheless you can do it in a small region, which I am trying to illustrate, my code in the following:

x[u_, v_] := (1 - v Sin[u/2])*Cos[u]

y[u_, v_] := (1 - v Sin[u/2])*Sin[u]

z[u_, v_] := v Cos[u/2]

φ[u_, v_] := {x[u, v], y[u, v], z[u, v]}

φu[u_, v_] := D[φ[u, v], u]

φv[u_, v_] := D[φ[u, v], v]

n[u_, v_] := 
 FullSimplify[Cross[φu[u, v], φv[u, v]]/
  Norm[Cross[φu[u, v], φv[u, v]]], -1/2 < v < 1/2 && 0 < u]

SliceVectorPlot3D[
 n[ArcTan[y1/x1], 
  z1/Cos[ArcTan[y1/x1]/2]], {z1^2 - 1 - Sqrt[x1^2 + y1^2] == 
   z1^2/(Cos[ArcTan[y1/x1]/
      2])^2}, {x1, -3, -1}, {y1, -3, -1}, {z1, -3, -1}]

but I keep getting the error message General::ivar: z1 Sec[1/2 ArcTan[y1/x1]] is not a valid variable.

Any help would be appreciated.