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MA 1024 Laboratory 1: Displaying vector-valued functions

Purpose

The purpose of this lab is to give you practice using Maple to plot vector valued functions and their derivatives.

Maple procedures are available to study vectors and vector-valued functions. This lab will introduce commands that we have written here at WPI as part of the CalcP package. Before you can use any of these commands, you must load them using the with command. Since some of the commands use the linalg package, it is probably a good idea to load it as well.

  > with(linalg):

Warning: new definition for norm Warning: new definition for trace

  > with(CalcP):

The list below gives the names of the procedures we will be using, as well as brief descriptions. Maple help screens are available for all of these procedures, so refer to them for further examples. To refresh your memories, the commands introduced in the previous lab are also listed.

VDiff: Differentiates vector-valued functions.
VPlot: Plots vector-valued functions in two and three dimensions.
VMag: Computes the magnitude of a vector.
ParamPlot: Animates parametric curves in two dimensions.
ParamPlot3D: Animates parametric curves in three dimensions.
Speed: Computes the speed of a particle moving on a path defined by a vector-valued function ${\bf r}(t)$ .
unitvect: Computes the unit vector associated with a vector ${\bf v}$ .
tanvect: Computes the unit tangent vector, ${\bf T}(t)$ , for a vector-valued function ${\bf r}(t)$ .

Examples

For more examples, see the help screens for the individual procedures. Note that all of the procedures accept either Maple vectors or Maple lists as arguments. Note also that several of the procedures allow you to evaluate the result at a specific value of t by using a second argument of the form t=a (to evaluate at t=a). If the second argument is simply t, then the result of the procedure is an expression involving t. This does not apply to the VDiff command, however, where argumets after the first are used to indicate derivatives.

  > circ := t -> vector([12*sin(t),12*cos(t)]);

$\begin{maplelatex} \begin{displaymath} {\it circ} := {t} \rightarrow {\rm vector... ...rm sin}(\,{ t}\,), 12\,{\rm cos}(\,{t}\,)\,]\,)\end{displaymath}\end{maplelatex}$

  > VDiff(circ(t),t);

$\begin{maplelatex} \begin{displaymath}[\,12\,{\rm cos}(\,{t}\,)\, - 12\,{\rm sin}(\,{t}\,)\,]\end{displaymath}\end{maplelatex}$

  > r := t -> vector([t,t^2]);

$\begin{maplelatex} \begin{displaymath} {r} := {t} \rightarrow {\rm vector}(\,[\,{t}, {t}^{2}\,]\,)\end{displaymath}\end{maplelatex}$

  > tanvect(r(t),t);

$\begin{maplelatex} \begin{displaymath} \left[ \! \,{\displaystyle \frac {1}{\sq... ...+ 4\,{t}^{2}\, \! \right\vert }}}\, \! \right] \end{displaymath}\end{maplelatex}$

  > tanvect(r(t),t=1);

$\begin{maplelatex} \begin{displaymath} \left[ \! \,{\displaystyle \frac {1}{5}}... ...laystyle \frac {2}{5}}\,\sqrt {5}\, \! \right] \end{displaymath}\end{maplelatex}$

The next few commands deal with a simple example of a curve known as a helix. In one of the exercises, you will be asked to generate a curve similar to a helix. If we want to visualize the projection of a helix onto the x,y-plane, we can rotate the plot of the graph, so that the z axis, would be perpendicular to the computer monitor. In other words the the z component would be pointing in/out of the screen. Notice in this case the projection of the the helix onto the x,y-plane would be a circle.

  > h := t -> vector([cos(t),sin(t),t]);

$\begin{maplelatex} \begin{displaymath} {h} := {t} \rightarrow {\rm vector}(\,[\,{\rm cos}(\,{t}\,), {\rm sin}(\,{t}\,), {t}\,]\,)\end{displaymath}\end{maplelatex}$

  > VPlot(h(t),t=0..4*Pi);

  > ParamPlot3D(h(t),t=0..4*Pi);

  > Speed(h(t),t);

$\begin{maplelatex} \begin{displaymath} \sqrt {2}\end{displaymath}\end{maplelatex}$

Exercises

1.

Find a vector-valued function ${\bf r}(t)$ whose graph lies on the cone x²+y²=2z, such that the graph spirals around the cone and the $\hat{k}$ component is increasing. (You can think of this as motion of a particle spiraling around the cone)

(a): Plot the graph of your function.
(b): Find the derivative of your function and plot the graph of this derivative.
(c): Find and plot the magnitude of this derivative (which is Speed).
(d): If the speed is not a decreasing function, figure out a way to change your original function so that the speed is a decreasing function of t. Plot the speed for the new function to demonstrate that it is decreasing.

2.

Find a vector-valued function ${\bf r}(t)$ such that the projection of its graph onto the x,y-plane is a circle of radius 1 and the projection onto the y,z-plane is a circle of radius, 1 both of which are centered at the origin.

(a): Plot the graph of your function.
(b): Plot the projections onto the x,y and y,z-planes.
(c): Plot the projection onto the x,z-plane. Is it possible to have this projection be a circle too? Why? In fact, what can you say must be true about this projection, if the others are the circles we specified?

3.

Find a vector-valued function ${\bf r}(t)$ that is equal to its derivative. Hint: think about what scalar function has the property that its derivative is itself.

(a): Plot ${\bf r}(t)$ .
(b): Plot the second derivative of ${\bf r}(t)$ .
(c): Can you characterize all vector-valued functions that are equal to their derivatives? Of what form must they be?

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Next: About this document ... Up: Labs and Projects for Previous: Labs and Projects for

Christine M Palmer
3/17/1998

MA 1024 Laboratory 1: Displaying vector-valued functions

Purpose

Background

Examples

Exercises