Deos during Lecture on Curves, Math s21a, Summer 2019, 7/3/2019

Plotting a curve on a sphere

r[t_,s_]:={Cos[t] Sin[s],Sin[t] Sin[s],Cos[s]}; 
S1=ParametricPlot3D[ r[t,s],{t,0,2Pi},{s,0,Pi},Mesh->False];
S2=Graphics3D[{Red,Tube[Table[r[7t,13t],{t,0,2Pi,0.002}],0.03]}];
Show[{S1,S2},Boxed->False,Axes->False]

Curvature Identity

r[t_]:={x[t],y[t],z[t]}; s[t_]:=Sqrt[r'[t].r'[t]];T[t_]:=r'[t]/s[t];  v[t_]:=Cross[r'[t],r''[t]]; A=Sqrt[T'[t].T'[t]]/s[t];  B=Sqrt[v[t].v[t]]/s[t]^3;Simplify[A^2==B^2]