Thomas Calculus 13th [Solutions]

958 Chapter 13 Vector-Valued Functions and Motion in Space
27-30. Example CAS commands:
Maple:
with( LinearAlgebra );
r : t*cos(t) | t*sin(t) | t ;
t0 : sqrt(3);
rr : eval( r, t t0 );
v : map( diff, r, t );
vv : eval( v, t t0 );
a : map( diff, v, t );
aa : eval( a, t t0 );
s : simplify(Norm( v, 2 )) assuming t::real;
ss : eval( s, t t0 );
T : v/s;
TT : vv/ss ;
q1: map( diff, simplify(T), t ):
NN : simplify(eval( q1/Norm(q1,2), t t0 ));
BB : CrossProduct( TT, NN );
kappa : Norm(CrossProduct(vv,aa),2)/ss^3;
tau : simplify( Determinant(< vv, aa, eval(map(diff,a,t),t t0) >)/Norm(CrossProduct(vv,aa),2)^3 );
a_
t : eval( diff( s, t ), t t0 );
a_
n : evalf[4]( kappa*ss^2 );
Mathematica: (assigned functions and value for t0 will vary)
Clear[t, v, a, t]
mag[vector _]: Sqrt[vector.vector]
Print["The position vector is", r[t _] {t Cos[t], t Sin[t], t}]
Print["The velocity vector is", v[t _] r'[t]]
Print["The acceleration vector is",a[t _] v'[t]]
Print["The speed is",speed[t _]
mag[v[t]]//Simplify]
Print["The unit tangent vector is", utan[t _]
v[t]/speed[t]//Simplify]
_
3
Print["The curvature is",curv[t ] mag[Cross[v[t],a[t]]] / speed[t] //Simplify]
_
2
Print["The torsion is", torsion[t ] Det[{v[t], a[t], a'[t]}] / mag[Cross[v[t],a[t]]] //Simplify]
Print["The unit normal vector is",unorm[t _]
utan'[t] / mag[utan'[t]//Simplify]
Print["The unit binormal vector is", ubinorm[t _]
Cross[utan[t],unorm[t]]//Simplify]
Print["The tangential component of the acceleration is",at[t _] a[t].utan[t]//Simplify]
Print["The normal component of the acceleration is",an[t _] a[t].unorm[t]//Simplify]
You can evaluate any of these functions at a specified value of t.
t0 Sqrt[3]
{utan[t0], unorm[t0], ubinorm[t0]}
N[{utan[t0], unorm[t0], ubinorm[t0]}]
{curv[t0], torsion[t0]}
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2014 Pearson Education, Inc.