Welcome to Calculus, I'm professor Ghrist. We're about to begin Lecture 11, bonus

material. In our main lesson, we covered the basic

rules for differentiation. Well, the question comes up, why spend so

much time on those rules? You might think that it is merely, so

that we can do computations more easily, more efficiently.

And although that's very helpful, there are other reasons why you want to know

these rules. Mathematics, and indeed, the sciences,

are full of interesting patterns, and sometimes you'll find something you

recognize hiding within a very different-looking system.

Knowing the rules helps you recognize patterns.

We're going to take a look at two examples of this from very different

fields. We'll begin by looking at spaces, or

geometric domains on which you might do Calculus.

Examples would include a simple interval. Or, maybe a circular disk.

Now, there's an operator that is not unlike differentiation, but it acts on

spaces as opposed to functions. This is the boundary operator.

And we denote it with a scriptie sort of d.

What does the boundary operator do? Well, it gives you boundary of a domain.

For example, the boundary of an interval is simply the two end points.

The boundary of a circular disc is simply the circle that is at the edge or

boundary of that space. Now, there are other operations that act

on spaces as well. For example, there's a way to multiply

two spaces together in something called the Cartesian product.

For example, a rectangle can be considered as the product of two

intervals. Or, a circular cylinder can be considered

as the product of an interval with a circular disc that would be a solid three

dimensional domain. Now, you might guess that there's some

interesting Mathematics contained inside of this product.

For example, the disc is two dimensional, the interval is one dimensional, and this

cylinder is three dimensional. 2 plus 1.

I wonder if that pattern continues. Well, let's consider what happens when we

compute the boundary of a product. What is the boundary of a rectangle?

Well, it consists of the edges along the boundary of the rectangle of course.

But we can think about that in terms of the product structure.

This boundary is really the boundary of the first interval times the second

interval. But we also have the other edges as well,

which can be viewed as the boundary of the second interval times the first.