Lab 04 Vector-Valued Functions

[Your name(s) here!]

Examples to steal from

Here are examples of some of the kinds of things you’ll do in this lab.

Plotting a parametric vector function:

This is really a vector function (OverVector[r])(t) where the vector is a position vector.  It’s got 3 functions for x(t), y(t), and z(t) in its declaration, But I’m going to risk a little clarity by writing it without bothering to put the vector over top...

r[t_]:={t,t Sin[t],t Cos[t]}
ParametricPlot3D[r[t],{t,0,8π},PlotStyle->Blue]

Displaying a point on the curve, together with the derivative vector at that point.

Show[ 
ParametricPlot3D[r[t],{t,0,8π},PlotStyle->Blue],
Graphics3D[       {
(* Big red dot at r[t=3.5 π] *)
Red,PointSize[.04],Point[r[3.5 π]], 
(* Arrow[ ] requires coordinates of tail, then head *)
 Arrow[ {   r[3.5 π],r[3.5π]+r'[3.5 π]  } ] 
}   ]    
]

Animating the “particle”

Animate [
Show[ 
ParametricPlot3D[r[k],{k,0,8π},PlotStyle->Blue],
Graphics3D[       {
(* Big red dot at r[t=3.5 π] *)
Red,PointSize[.04],Point[r[t]], 
(* Arrows require coordinates of tail, then head *)
 Arrow[ {   r[t],r[t]+r'[t]  } ] 
}   ]    
],
{t,0,8 π}
]

Speed:  Speed is the magnitude (or “norm”) of the derivative vector:  |(OverVector[r])'(t) |.  Here’s how you can graph the (scalar) speed as a function of time:

Plot[     Norm[r'[t]]     ,{t,0,8π}]

Exercise 1

N[{E,e}]

4 pts Consider the two arcs a(t)=⟨(e^t)sin t,(e^t)cos t, (e^t)⟩,0≤t≤2π and b(t)=⟨t sin(ln t),t cos(ln t),t⟩,1≤t≤(e^(2π)).  [In Mathematica, type E (not  e) to get the base of the natural logarithm.] Plot both space curves and show that they trace out identical curves.  You can display two plots sides by enclosing them in list brackets, like { ParametricPlot3d[...], ParametricPlot3d[...]  }.

Exercise 2

6 pts In order to examine how a particle moves according to each function, create an animation of each like this:  First create a display of the arc and a large dot located at r(t). Then “Manipulate[..]” the plot to move the dot as t is varied with a slider.  Finally substitute Animate for Manipulate. This will change t at a constant rate.  (See if you can get the derivative vector to show as well).

Comment on differences that you observe between the two arc functions.

Exercise 3

4 pts Calculate a'(t), |a'(t)|, b'(t), and |b'(t)|.  Graph the speeds. Describe how these results support (or challenge) the observations you made above.

Exercise 4

10 pts Use Mathematica to calculate the arc length separately for each vector function. Use the formulae we developed in class, which involves integrating the “speed”...

({{({{L=(∫_a^b)|(OverVector[r])'(t)|dt}})}})

Calculate the arc-length using both a(t) and b(t)--they should agree if the space-curves are identical.

And then use Mathematica’s built-in ArcLength[...] function using both a(t) and b(t) and see if that function gives the same result.