programming - Why am I not getting a numeric response?

Code

In[830]:= curvature[x_, y_, z_, t_] :=

 With[{s = {x, y, z}},
  With[{v = D[s, t]},
   With[{ fT = v / Norm[v]},
    With[{fK = D[fT, t] / Norm[v]},
     Norm[fK]
     ]
    ]
   ]
  ]
curvature[ Cos[t], Sin[t], t, t] /. t -> 1 // N


Out[831]= \[Sqrt](0.5 Abs[-0.59501 - 
      0.0955129 (-0.909297 Derivative[1][Abs][0.540302] + 
         0.909297 Derivative[1][Abs][0.841471])]^2 + 
   0.5 Abs[-0.382051 + 
      0.148752 (-0.909297 Derivative[1][Abs][0.540302] + 
         0.909297 Derivative[1][Abs][0.841471])]^2 + 
   0.015625 Abs[-0.909297 Derivative[1][Abs][0.540302] + 
      0.909297 Derivative[1][Abs][0.841471]]^2)

Question

I have a // N there.

Why do I not get a numerical answer. Instead, why do I have Derivative[1][Abs] all over the place?

Thanks!

EDIT

The following "fixes" the code. Why?

myNorm[lst_] := Sqrt[lst . lst]
curvature[x_, y_, z_, t_] :=
 With[{s = {x, y, z}},
  With[{v = D[s, t]},

   With[{ fT = v / myNorm[v]},
    With[{fK = D[fT, t] / myNorm[v]},
     myNorm[fK]
     ]
    ]
   ]
  ]
curvature[ Cos[t], Sin[t], t, t] /. t -> 1 // N

This returns 0.5

Answer

According to the chain rule of differentiation, $(f\circ g)'(t)=f'(g(t))g'(t)$. Look at:

D[Abs[Sin[t]], t]
(* Cos[t] Derivative[1][Abs][Sin[t]] *)

You have several Abs[Sin[t]] cropping up everywhere due to Norm[v] in your code.

---

I would suggest re-writing the function as:

Clear[curvature2]
curvature2[vec : {x_, y_, z_}, t_] := 

    With[{norm = Simplify[Norm@#, t ∈ Reals] &},norm[D[#/norm@#, t]/norm@#] &@D[vec, t]]

curvature2[{Cos[t], Sin[t], t}, t]
(* 1/2 *)