Welcome to Calculus, I'm professor Grist.
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. We need to add these together.
This is done through taking a union of those two sets of edges.
Hmm, it seems as though there's something I've seen before hiding within that
boundary computation. Le'ts see if the same thing works with
the cylinder. If I look at the boundary of this
cylinder, then what does it consist of? It consists of the two end caps which is
the circular disc, cross the boundary of the interval, but it also has the side
that wraps around. That is the interval times the boundary
of the circular disk. If we take the union of those
collections, we get the boundary of the cylinder.
It is a fact that if you have two spaces, A and B, you take their product and
compute the boundary of that. You can decompose that boundary as the
boundary of the first space, times the second union.
The first space times the boundary of the second.
Now, where have you seen something like that before?
That's really a product rule. But we're not differentiating anything,
we're simply computing boundaries. This gives you a hint that there's some
deep relationshiop between differentiation and boundaries.
That relationship will be fully exploited when you get to the end of multi-variable
Calculus. Let's consider another example.
A completely different setting. This time, working in computer science
and looking at lists. Let's say that we have a list of objects.
Let's say they're all the, the same type of object, so that we call them x.
So, a list of five items would be x x x x x.
Let's do some Mathematics. Let's combine those together and call
that list x to the 5th. That tells us there are five elements.
Now, how do you take the derivative of a list?
Well, that doesn't make any sense, but there are some things that we can do.
Consider the following deletion operator, D, that acts on a list and deletes one of
the items. What is D of this five item list?
Well, you could delete the first item in the list or you could delete the second
item, or the third, or the fourth, or the fifth.
Now, each of these is really a a list of four elements.
So, we would call it x to the 4th. One way to encode this logical or, is to
use a formal sort of addition. We'll just call that plus.
That means, or. So, deleting a five item list gives us
one of five different four item lists. We could, say, more compactly, that D of
x to the 5th is 5 times x to the 4th, and this is something that should look
familiar. I wonder if there's something deeper than
we can do with this intuition. So, let us denote by x to the N, a list
of N elements. What would X to the 0 be?
Oh, that would be an empty list. You have no items in your list.
But instead of calling that x to the 0, let's call it 1, like it ought to be, and
consider the collection of all finite lists.
We'll call that L. Let's do some Calculus.
Here's a statement. Any list is empty, or it has a first
entry. I hope you'll agree that that's
intuitively true. What does it mean in this language that
we're constructing? Well, any list certainly means L.
That's our collection of all finite lists.
What does it mean that it could be empty? Well, that means that it could be 1.
What does or mean? Oh, or is our formal addition.
So, L is equal to 1 plus, and here's the tricky one.
What does it mean to say that a list has a first entry?
Well, that means it has an x in it, followed by something, by some finite
list, maybe empty, maybe not empty. We can rewrite this statement as L equals
1 plus x times L. And now, let's do some Algebra.
See what comes out. If we move all of the L terms over to one
side and factor out that L, what happens when we solve for L?
And get an expression for all finite lists?
Well, as you can clearly see, L is 1 over 1 minus x.
What in the world does that mean? Well, we know what one over 1 minus x
really is. In terms of the geometric series.
What does that mean? Well, in this language we've constructed,
you can read this statement, as saying, that any finite list is empty, or, it is
one item, or it has two items, or it has three items, or 4, or 5 etcetera.
And so, we see yet another interpretation for the geometric series.
Now, some students are surprised by this example that you have derivatives and
Taylor series coming up in a more computer science context.
Well, that shouldn't be too much of a surprise.
Mathematics is full of patterns. Patterns that can describe all sorts of
things in the natural world. If you're interested in learning more
about some of these unusual examples. The first, concerning spaces, comes from
the subject of topology. The second, concerning lists, comes from
the subject of analytic combinatorics