Introduction

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Welcome…

…to Math 223, Calculus 3.

I’m Doug Baldwin.

I’ve been teaching at Geneseo for 30+ years, but most of them were in computer science. When Geneseo eliminated its computer science program, I moved to math, where I teach a mixture of traditional math courses and courses that explore mathematical foundations of computer science. My research interests are in computer graphics, which is an area that very nicely combines math and computing. Which leads to…

What This Course Is About

In very short form, it looks at what happens when you relax the idea that a function must have one input variable and one output variable.

For examples, consider computer animated snakes (because… why not?):

If you think about how you might describe such a thing, you run into all sorts of applications of “multivariable” and “vector” mathematics.

• First of all, the real world is 3-dimensional, so points on the snake are described by x, y, and z coordinates.
• The snake more or less looks like a curve in 3 dimensions. So it might be described by equations or a function that defines x, y, and z in terms of some other variable, say t. Now you have a function with multiple output variables, a so-called “vector-valued function”:

• But such an equation really only describes a line through the snake’s center, its spine if you will. To give it thickness you might think of each position along the spine having an associated circle around that spine. For this, the snake function needs a second input:

• Those circles aren’t necessarily circles around the x, y, or z axis they’re circles around tangents to the snake’s spine. This sounds like derivatives are creeping in.
• And it turns out the lighting on the snake, how bright or dark each point on its body looks, is calculated based on how light from a virtual light source reflects off a plane tangent to the body towards a virtual viewer. So here’s another notion of “tangent” — planes tangent to surfaces — that likely corresponds to some sort of derivative too.
• Finally, if you want a really accurately lit snake (which this one isn’t), you need to admit that light doesn’t come from just one point, it comes from all over. So you want to add up how incoming light from every direction around each point on the snake reflects towards the viewer — adding up values like this sounds like an integral, but over space rather than along an axis.

All of these ideas (i.e., higher dimensional coordinates, vector-valued and multi-variable functions, derivatives and integrals of such things, etc.) are the subject matter of this course.

Introductions

Please introduce yourselves to each other.

In particular, find someone sitting near you who you don’t already know, and introduce yourselves.

(This is more than just a nice social thing. This class will involve a lot of in-class problem-solving, and these introductions help you at least sort of know someone you can work with when that starts happening.)

Next

Course policies, as laid out in the syllabus.

Please read the syllabus by class time tomorrow. (Not as in “memorize every word of it,” but as in “identify things that are unclear, which we should talk about in class, or different enough from what you’re used to that they are worth remembering.”)

In class, we’ll answer questions you have about the syllabus, and I’ll probably ask you some questions that get at things that I think are important in it.

That pattern, of you reading something before class, and then using class time for me to answer your questions about it, and all of us to discuss some questions or problems that I think are central to the material, is how most class meetings will work.

The syllabus is online. You can find it through the link above, or through Canvas. Which suggests that a brief look at our Canvas space is worth taking. The main take-away from that tour is that the “Modules” section is where to go for just about everything for the course: daily class notes, handouts, etc.

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