SUNY Geneseo Department of Mathematics

Introduction to Vector Valued Functions

Tuesday, September 20

Math 223 01
Fall 2022
Prof. Doug Baldwin

Anything You Want to Talk About?

AWM

Geneseo’s student chapter of the Association for Women in Mathematics (AWM) is getting started for the year.

Vector-Valued Functions

Based on “Definition of a Vector-Valued Function” and “Graphing Vector-Valued Functions” in section 2.1 of the textbook.

Questions?

Key Ideas

The general form of a vector-valued function is: r(t) = ⟨ x(t), y(t), z(t) ⟩

Finding and Plotting Vector-Valued Functions

Plot a vector-valued function in Mathematica with the ParametricPlot3D function. It generally looks like

where x(t), y(t), and z(t) are expressions for the x, y, and z components of the vector-valued function in terms of some variable (I generically use t here, but it can be anything), and low and high are the low and high bounds on that variable’s values.

A standard helix. The simplest equation for such a thing is r(t) = ⟨cos t, sin t, t⟩. You could plot it in Mathematica with a command such as

We experimented with the basic helix, including things like scaling the arguments to sine and cosine to control how many turns of the helix there are, scaling t in the expression for z to control how tightly spaced the turns are, etc. Using some standard range for t, e.g., 0 to 1, and then scaling the expressions that use it, is often convenient when coming up with vector-valued functions. You can download the notebook in which we tried these things.

We also noticed that multiplying the sine and cosine terms by something controls the radius of the helix. If you multiply by different values, you get an elliptical cross section.

Can you make the helix pinch down to a point at the bottom and expand as it rises?

Using the idea that multiplying sine and cosine by something changes radius, make that “something” depend on t. In the simplest case, just multiply by t itself.

The computer-animated snake, or at least its center line: see if you can make it vary sinusoidally back and forth across one of the axes, with the variations getting bigger as you get closer to one end (the head), and have that end raised in the third dimension, as if the snake is lifting its head to look around.

Hints: Have t range from 0 to 1, and scale it as needed to get the number and amplitude of wiggles you want. To raise the head, take advantage of the fact that over the interval [0,1], tⁿ for large n (e.g., around 10 or 20) will be nearly 0 until t is close to 1, whereupon tⁿ will suddenly rise to nearly 1.

We ended up having the x coordinate vary with t, to lay the snake out along the x axis. The y coordinate varies sinusoidally (actually “cosinusoidally” the way we did it), scaled by √t to make the size of the sinusoid grow slowly (but notice that for 0 < t < 1, t is actually less than √t, so might be better for slow growth). The z coordinate uses the tⁿ idea, to get the body to mostly look like it lies flat, with a sudden rise at the end.

You can download the Mathematica notebook, with our solutions to and explorations of all of the above, including the snake, from Canvas.

Since the component functions in a vector-valued function are ordinary scalar functions, they can do all the things that make limits interesting.

For example, how does r(t) = ⟨1/(t-2)², (t²-4)/(t-2), ln(t-2)⟩ behave as t approaches 2?

So we’ll talk about limits (and continuity) of vector-valued functions tomorrow.

Please read “Limits and Continuity of a Vector-Valued Function” in section 2.1 of the textbook.