Lab 2: Vector Valued Functions (Adapted from a lab by Stephen Greenfield.) Here we wish to investigate vector valued functions, both graphically and numerically. We will start by loading two Maple packages necessary for the rest of the lab. (Note that using a semicolon in the commands below makes Maple list out all the new commands you have loaded. If you use a colon instead of a semicolon, it will load the packages but suppress the lists given below.) with(plots); with(VectorCalculus);

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Each student will use individual data for this lab. It is usually convenient to save your assigned vector valued function as something initially, using the following type of command. (This method saves r as a function of t, so if you want to find r at t=a, you can just say r(a).) We will use the following function as an example: r:=t-> [cos(t),sin(t),t];

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r(t);

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r(2);

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For your individual function, you will investigate the following: 1. A plot of your curve from t=0 to t=2*Pi 2. The number of points of intersection. (It is usually important to view your plot from many angles to determine this.) 3. The speed at a function of time, as well as the specific speed when t=2. 4. The arclength of the curve from t=0 to t=2*Pi. 5. The arclength of the curve from t=0 to t=Pi. Here are some useful commands and hints: To plot a vector valued function in three-space, we use the "spacecurve" command. You can then turn this plot around and look for points of intersection. (If Maple does not let you do this automatically, right click and look at the "Manipulator" option.) spacecurve(r(t),t=0..12*Pi);

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However, since this is not a very informative view of this curve, we can specify certain additional options that affect the appearance of the curve and the axes. (Note that increasing the "numpoints" tells Maple to draw the curve by plotting out more points and therefore gives a smoother curve. Try experimeting with different values.) spacecurve(r(t),t=0..12*Pi, thickness=2, color=black, axes=boxed, labels=[x,y,z], numpoints=200);

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Now, we can also make some calculations with our space curve. We can differentiate a vector valued function using the command "diff". diff(r(t),t);

NyUsJC1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0YpISIiLUkkY29zR0YmRioiIiI=

We can find the length of a vector (or vector valued function) using the command "Norm". Norm(<1,1,1>);

KiQpIiIkIyIiIiIiI0Ym

b:=Norm(<r(t)>);

KiQsJiIiIkYkKiRJInRHNiIiIiNGJCNGJEYo

We can also integrate using the command "int". Note that this command gives the best exact expression it can produce, which sometimes leaves the integral completely untouched. To get a numerical value, we use the "evalf" command. int(b, t=0..2*Pi);

LCYqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiLCZGJkYmKiRGJCIiIyIiJSNGJkYpRiYtSSNsbkc2JEYlSShfc3lzbGliRzYiNiMsJkYkISIjKiRGJ0YrRiYjISIiRik=

evalf(int(b, t=0..2*Pi));

JCIrOiVIYzcjISIp