Lecture 11: BONUS!

This is calculus in a single variable and
I'm Robert Ghrist, professor of

mathematics and electrical and systems
engineering, at the University of

Pennsylvania. We're about to begin lecture
eleven, Bonus Material. What have we

looked at? In lecture eleven we considered
the various rules for computing

derivatives. Linearity, product rule,
chain rule. Other kinds of rules. Why

spend so much time focusing on the rules?
One reason is they help with computations.

But another deeper reason is that there
are structures in mathematics and related

areas that obey similar types of rules.
And when we find them and recognize them,

we can come to a deeper understanding by
knowing how calculus works. Let's look at

two specific examples. One example
involves spaces or domains over which you

might do calculus. Here are some specific
examples, one type of space might be a

interval or a line segment. Another might
be a circular disk. One thing that's

interesting to talk about when you're
looking at the geometry of spaces is the

boundary. And we're going to think of
boundary as an operator, not unlike a

derivative. In fact, we're going to give
it a symbol that is evocative of the

differentiation symbol, called the
script-e, d, dow, for the boundary

operator. Now what is the boundary of an
interval or a line segment? It consists

precisely of two end points. What is the
boundary of a circular disc? It consists

precisely of those points that are on the
boundary the circle. Now, observe how the

boundary behaves with respect to other
structures on spaces. There's a type of

product called the Cartesian product that
is somewhat familiar even if you've never

seen it formally before. We denote it by a
times sign. For example, a rectangle,

solid, filled-in rectangle can be
considered as the Cartesian product of two

intervals. A solid cylinder can be
considered as a product of an interval

with a circular disc. I'm sure you
recognize many other shapes as well that

can be similarly decomposed. Now, what
happens when we take the boundary of a

product o f spaces? For a rectangle, what
we get are four edges with those corners.

Now notice that this perimeter of the
rectangle can be decomposed, into a union

of two pieces. The two edges on the side,
and the two edges on the top. We can think

of that as the boundary of the first
interval times the second union, the

boundary of the second interval times the
first. Similarly with the solid cylinder,

if we look at the boundary of that, we can
decompose it into a couple of pieces.

There's the left and right hand sides of
the cylinder, the, the, the round disk

part, that can be thought of as the disk
times the boundary of the interval. And

then there's that portion of the boundary
of the cylinder that wraps around the

sides. That can be thought of as the
interval cross the boundary, of the disk.

It is a general fact that the boundary of
the product of two spaces, A times B, is

equal to the boundary of A times B union A
times the boundary of B. And if you look

at that formula carefully, you'll see what
looks just like a product rule. But with

different operations involved. And indeed,
there's a close and deep connection

between the boundary operator, Dell, and
the derivative operator D, that we know

and love. Let's consider another example,
this time coming from lists and computer

science. Consider a list with five
elements. I'm going to represent each

element abstractly by a variable, x, so
that the list of five elements might

suggestively be called x to the fifth. Now
consider the following operator, a

deletion operator, d, that acts by
deleting one of the terms in the list.

What happens? Well, I could delete the
first term, or I could have deleted the

second term, or the third term, or the
fourth term or the fifth. In this setting

I get a list of four elements. Or another
list of four elements or another or

another or another. And if we call those
lists of four elements algebraically as X

to the fourth and if we represent the
logical or as a formal addition, then what

do we have? We really have that the
deletion operat or D applied to X to the

fifth equals five times X to the fourth.
Now where have you seen something like

that before? It turns out that there's an
entire calculus for lists. Let's look at a

simple example of how this works. Let us
denote by x to the n a list of n elements.

What would x to the zero be? A list of
zero elements. Oh, that would be an empty

list. And what should we call that
mathematically? Let's call that one,

because x to the zero really ought to be
one.

And now consider the set L of all finite
lists. We'd like to know, what does L look

like, symbolically. Here's a statement
that I think you'll agree is true. Any

list is either empty or it has a first
entry. What does that translate to you,

algebraically? Any list means L, is,
connotes an equal sign, empty. Oh, we've

seen what the empty list is, that's one.
we've seen that OR, corresponds to a

formal addition. And what does it mean,
for a list to have a first entry? That

means, you've got an X followed by some
other finite list, or X times L. Now,

let's just take that algebraic equation
and start manipulating it. Forget about

what these things mean grammatically, for
the moment. If I move X, x L to the other

side, factor out L and divide through by
1-X, then we'll get that L is equal to one

over one minus x. Now what does that mean?
I've come up with an algebraic expression

for all finite lists and I recognize one
over one minus X as the geometric series

one plus X plus X squared plus X cubed
etcetera. One way to interpret that, is

that we have derived the fact. That the
set of all finite lists consists of the

empty list, or a list of one element, or a
list of two elements, or a list of three

elements, etc. That's a pretty remarkable
derivation. It's using a type of calculus,

that is very different from what we're
learning in this class but has some

strange similarities. And as you go on in
your calculus education, you'll see that

calculus applies not just to, problems in
physics or economics, but in all kinds of

fields and in all kinds of manners, which,
a t first, might not have been so obvious.