Notebook 1

\(r\  = \ {Cos[t], \ Sin[t], \ Sqrt[4\  - \ Sin[t]^2]}; 

    v\  = \ D[r, \ t]\)

Simplify[v . v  -  1]

v  /.  {{t  ->  0},  {t  ->  Pi/2}}

\(a\  = \ D[v, \ t]; 
a\  /. \ {{t\  -> \ 0}, \ {t\  -> \ Pi/2}}\)

a . a  /.  {{t  ->  0},  {t  ->  Pi/2}}

\(vCrossa\  = \ Cross[v, \ a]; 

    vCrossa\  /. \ {{t\  -> \ 0}, \ {t\  -> \ Pi/2}}\)

Sqrt[vCrossa . vCrossa]/Sqrt[v . v]^3  /. 
       {{t  ->  0},  {t  ->  Pi/2}}

\(aPrime\  = \ D[a, t]; 

    \((vCrossa . aPrime)\)/\((vCrossa . vCrossa)\)\  /. 
      \ {{t\  -> \ 0}, \ {t\  -> \ Pi/2}}\)

ParametricPlot3D[Evaluate[r],  {t,  0,  2  Pi}];