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

Â material. In our main lesson, we covered the basic

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

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

Â that we can do computations more easily, more efficiently.

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

Â these rules. Mathematics, and indeed, the sciences,

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

Â recognize hiding within a very different-looking system.

Â Knowing the rules helps you recognize patterns.

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

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

Â geometric domains on which you might do Calculus.

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

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

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

Â And we denote it with a scriptie sort of d.

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

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

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

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

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

Â two spaces together in something called the Cartesian product.

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

Â intervals. Or, a circular cylinder can be considered

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

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

Â interesting Mathematics contained inside of this product.

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

Â cylinder is three dimensional. 2 plus 1.

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

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

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

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

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

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

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

Â 3:41

We need to add these together. This is done through taking a union of

Â those two sets of edges. Hmm, it seems as though there's something

Â I've seen before hiding within that boundary computation.

Â Le'ts see if the same thing works with the cylinder.

Â If I look at the boundary of this cylinder, then what does it consist of?

Â It consists of the two end caps which is the circular disc, cross the boundary of

Â the interval, but it also has the side that wraps around.

Â That is the interval times the boundary of the circular disk.

Â If we take the union of those collections, we get the boundary of the

Â cylinder. It is a fact that if you have two spaces,

Â A and B, you take their product and compute the boundary of that.

Â You can decompose that boundary as the boundary of the first space, times the

Â second union. The first space times the boundary of the

Â second. Now, where have you seen something like

Â that before? That's really a product rule.

Â But we're not differentiating anything, we're simply computing boundaries.

Â This gives you a hint that there's some deep relationshiop between

Â differentiation and boundaries. That relationship will be fully exploited

Â when you get to the end of multi-variable Calculus.

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Let's consider another example. A completely different setting.

Â This time, working in computer science and looking at lists.

Â Let's say that we have a list of objects. Let's say they're all the, the same type

Â of object, so that we call them x. So, a list of five items would be x x x x

Â x. Let's do some Mathematics.

Â Let's combine those together and call that list x to the 5th.

Â That tells us there are five elements. Now, how do you take the derivative of a

Â list? Well, that doesn't make any sense, but

Â there are some things that we can do. Consider the following deletion operator,

Â D, that acts on a list and deletes one of the items.

Â What is D of this five item list? Well, you could delete the first item in

Â the list or you could delete the second item, or the third, or the fourth, or the

Â fifth. Now, each of these is really a a list of

Â four elements. So, we would call it x to the 4th.

Â One way to encode this logical or, is to use a formal sort of addition.

Â We'll just call that plus. That means, or.

Â So, deleting a five item list gives us one of five different four item lists.

Â We could, say, more compactly, that D of x to the 5th is 5 times x to the 4th, and

Â this is something that should look familiar.

Â I wonder if there's something deeper than we can do with this intuition.

Â So, let us denote by x to the N, a list of N elements.

Â What would X to the 0 be? Oh, that would be an empty list.

Â You have no items in your list. But instead of calling that x to the 0,

Â let's call it 1, like it ought to be, and consider the collection of all finite

Â lists. We'll call that L.

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Let's do some Calculus. Here's a statement.

Â Any list is empty, or it has a first entry.

Â I hope you'll agree that that's intuitively true.

Â What does it mean in this language that we're constructing?

Â Well, any list certainly means L. That's our collection of all finite

Â lists. What does it mean that it could be empty?

Â Well, that means that it could be 1. What does or mean?

Â Oh, or is our formal addition. So, L is equal to 1 plus, and here's the

Â tricky one. What does it mean to say that a list has

Â a first entry? Well, that means it has an x in it,

Â followed by something, by some finite list, maybe empty, maybe not empty.

Â We can rewrite this statement as L equals 1 plus x times L.

Â And now, let's do some Algebra. See what comes out.

Â If we move all of the L terms over to one side and factor out that L, what happens

Â when we solve for L? And get an expression for all finite

Â lists? Well, as you can clearly see, L is 1 over

Â 1 minus x. What in the world does that mean?

Â Well, we know what one over 1 minus x really is.

Â In terms of the geometric series. What does that mean?

Â Well, in this language we've constructed, you can read this statement, as saying,

Â that any finite list is empty, or, it is one item, or it has two items, or it has

Â three items, or 4, or 5 etcetera. And so, we see yet another interpretation

Â for the geometric series. Now, some students are surprised by this

Â example that you have derivatives and Taylor series coming up in a more

Â computer science context. Well, that shouldn't be too much of a

Â surprise. Mathematics is full of patterns.

Â Patterns that can describe all sorts of things in the natural world.

Â If you're interested in learning more about some of these unusual examples.

Â The first, concerning spaces, comes from the subject of topology.

Â