Vector Functions and their Graphs
> with(linalg):The list below gives the names of the procedures we will be using as well as brief descriptions of what each does. Maple help screens are available for all of these procedures, so refer to them for further examples.
> circ := t -> vector([12*sin(t),12*cos(t)]);
> r := t -> vector([t,t^2]);
Computing the unit normal vector is always more complicated than computing the unit tangent vector . In addition, Maple likes to do computations as generally as possible, which can cause complications. In particular, Maple assumes that all variables can be complex numbers. This isn't a good assumption in calculus, so the tanvect, normalvect, and Curvature procedures had to be written to give results that are real numbers. A side-effect of this is that results of the commands, especially the normalvect command, do not always appear in the simplest form. Consider the following example.
The notation appears because Maple had to differentiate the absolute value function to obtain the normal vector. Recall that the deriviative of is 1 if x > 0 and -1 if x < 0. Maple's notation stands for the derivative of , evaluated at . Because Maple assumes that t can be complex, it allows for to be negative. In our calculations, can never be negative so the value of is simply 1.
This may seem like a pain, but it won't affect your ability to compute normal vectors at fixed values of t, as shown in the following examples, or plot normal vectors. Putting the absolute value in several of the CalcP package functions for curve comptations was a necessary evil, because leaving it out produced answers that were just plain wrong. By putting it in, the procedures give the correct answer, but in a form that is more complicated than we would like.
The absolute value also appears in the output of the Curvature procedure, as shown above. This means that if you differentiate the output of the Curvature function, for example to find extreme values, you may see the Maple abs procedure in the results. You shouldn't be alarmed by this, just remember that the value will always be 1. Fortunately, the Maple solve command usually handles cases involving the abs procedure just fine.
In the following example, plots of a vector-valued function and its curvature are generated. You should compare the two plots and try to understand their relationships.
The next few commands deal with a simple example of a curve known as a helix.
> h := t -> vector([cos(t),sin(t),t]);
> with(CalcP): with(linalg):
You may use options to include more information on your plots, or to make it easier to read. Use accents (not single quotes) to designate a string for a title.
> VPlot(circ(t), t=0..2*Pi, color=RED, axes=NORMAL, labels=[x,y], title=`circ(t)`);
This command will find the tangent vector at a given point, but it will not give you a tangent line.
> slope:= tanvect(circ(t),t=Pi/3);
Maple's tangentline command does not work for vector valued functions, so you need to make the line yourself. Here is a relatively painless way to do this. Basically tanline is of the form slope*t + c, where c is the point of tangency (here circ(t) evaluated at t=Pi/3). Note that tanline is simply an expression, not a function. You should not type tanline(t). You can arbitrarily choose your interval for the plot of tanline to make it longer or shorter. Try -Pi..Pi for example.
> tanline:= evalm((scalarmul(slope,t)) +
In order to display both plots on the same graph, you need to store the instructions for plotting into a variable, then use the plots[display] command.
> VPlotcirc:=VPlot(circ(t), t=0..2*Pi, color=RED,
labels=[x,y], title=`circ(t) with tangentline at t=Pi/3`):
For 3D graphs, you should use the right mouse button to rotate or pivot the image for a clearer view. Use the middle mouse key to redisplay. To compare two 3D graphs, you should rotate them all into the same position. If, after rotating, you notice the scaling is different, you may want to try Projection(Constrained) on the VPlot menu bar, in order to get an accurate comparison. Also, choose a type of axes for reference e.g. Normal.
2. Explore graphically the behavior of the helix
Set b=1. Plot the helix R(t) together with the tangent line to the curve at t=3/2 for a = 1, 2, and 4 over the interval 0<=t<=4. Describe in your own words what happens to the graph of the helix and the position of the tangent line as a increases through these positive values.
Let a=1. Plot the helix R(t) together with the tangent line to the curve
at t=3/2 for b = 1/2, 1, and 2 over the interval
in your own words what happens to the graph of the helix and the position
of the tangent line as b increases through these positive values.
Given the animated picture, what surface do you think these graphs may belong to? Justify your answer analytically.