WEBVTT 00:00:00.000 --> 00:00:05.573 This is calculus in a single variable and I'm Robert Ghrist, professor of 00:00:05.573 --> 00:00:11.146 mathematics and electrical and systems engineering, at the University of 00:00:11.146 --> 00:00:18.321 Pennsylvania. We're about to begin lecture eleven, Bonus Material. What have we 00:00:18.321 --> 00:00:24.138 looked at? In lecture eleven we considered the various rules for computing 00:00:24.138 --> 00:00:29.719 derivatives. Linearity, product rule, chain rule. Other kinds of rules. Why 00:00:29.719 --> 00:00:36.243 spend so much time focusing on the rules? One reason is they help with computations. 00:00:36.243 --> 00:00:42.688 But another deeper reason is that there are structures in mathematics and related 00:00:42.688 --> 00:00:49.154 areas that obey similar types of rules. And when we find them and recognize them, 00:00:49.154 --> 00:00:55.797 we can come to a deeper understanding by knowing how calculus works. Let's look at 00:00:55.797 --> 00:01:02.766 two specific examples. One example involves spaces or domains over which you 00:01:02.766 --> 00:01:09.213 might do calculus. Here are some specific examples, one type of space might be a 00:01:09.213 --> 00:01:15.405 interval or a line segment. Another might be a circular disk. One thing that's 00:01:15.405 --> 00:01:21.428 interesting to talk about when you're looking at the geometry of spaces is the 00:01:21.428 --> 00:01:27.069 boundary. And we're going to think of boundary as an operator, not unlike a 00:01:27.069 --> 00:01:32.862 derivative. In fact, we're going to give it a symbol that is evocative of the 00:01:32.862 --> 00:01:37.893 differentiation symbol, called the script-e, d, dow, for the boundary 00:01:37.893 --> 00:01:45.032 operator. Now what is the boundary of an interval or a line segment? It consists 00:01:45.032 --> 00:01:51.491 precisely of two end points. What is the boundary of a circular disc? It consists 00:01:51.491 --> 00:01:58.928 precisely of those points that are on the boundary the circle. Now, observe how the 00:01:58.928 --> 00:02:05.435 boundary behaves with respect to other structures on spaces. There's a type of 00:02:05.435 --> 00:02:12.442 product called the Cartesian product that is somewhat familiar even if you've never 00:02:12.442 --> 00:02:18.865 seen it formally before. We denote it by a times sign. For example, a rectangle, 00:02:18.865 --> 00:02:25.205 solid, filled-in rectangle can be considered as the Cartesian product of two 00:02:25.205 --> 00:02:32.170 intervals. A solid cylinder can be considered as a product of an interval 00:02:32.170 --> 00:02:39.016 with a circular disc. I'm sure you recognize many other shapes as well that 00:02:39.016 --> 00:02:45.953 can be similarly decomposed. Now, what happens when we take the boundary of a 00:02:45.953 --> 00:02:53.255 product o f spaces? For a rectangle, what we get are four edges with those corners. 00:02:53.255 --> 00:03:00.057 Now notice that this perimeter of the rectangle can be decomposed, into a union 00:03:00.057 --> 00:03:06.962 of two pieces. The two edges on the side, and the two edges on the top. We can think 00:03:06.962 --> 00:03:12.704 of that as the boundary of the first interval times the second union, the 00:03:12.704 --> 00:03:19.320 boundary of the second interval times the first. Similarly with the solid cylinder, 00:03:19.320 --> 00:03:24.803 if we look at the boundary of that, we can decompose it into a couple of pieces. 00:03:24.803 --> 00:03:30.148 There's the left and right hand sides of the cylinder, the, the, the round disk 00:03:30.148 --> 00:03:35.631 part, that can be thought of as the disk times the boundary of the interval. And 00:03:35.631 --> 00:03:41.184 then there's that portion of the boundary of the cylinder that wraps around the 00:03:41.184 --> 00:03:46.460 sides. That can be thought of as the interval cross the boundary, of the disk. 00:03:47.400 --> 00:03:56.840 It is a general fact that the boundary of the product of two spaces, A times B, is 00:03:56.840 --> 00:04:05.773 equal to the boundary of A times B union A times the boundary of B. And if you look 00:04:05.773 --> 00:04:11.528 at that formula carefully, you'll see what looks just like a product rule. But with 00:04:11.528 --> 00:04:16.933 different operations involved. And indeed, there's a close and deep connection 00:04:16.933 --> 00:04:22.408 between the boundary operator, Dell, and the derivative operator D, that we know 00:04:22.408 --> 00:04:29.191 and love. Let's consider another example, this time coming from lists and computer 00:04:29.191 --> 00:04:34.802 science. Consider a list with five elements. I'm going to represent each 00:04:34.802 --> 00:04:40.814 element abstractly by a variable, x, so that the list of five elements might 00:04:40.814 --> 00:04:47.221 suggestively be called x to the fifth. Now consider the following operator, a 00:04:47.221 --> 00:04:53.502 deletion operator, d, that acts by deleting one of the terms in the list. 00:04:53.502 --> 00:05:00.225 What happens? Well, I could delete the first term, or I could have deleted the 00:05:00.225 --> 00:05:07.779 second term, or the third term, or the fourth term or the fifth. In this setting 00:05:07.779 --> 00:05:15.570 I get a list of four elements. Or another list of four elements or another or 00:05:15.570 --> 00:05:22.394 another or another. And if we call those lists of four elements algebraically as X 00:05:22.394 --> 00:05:29.050 to the fourth and if we represent the logical or as a formal addition, then what 00:05:29.050 --> 00:05:35.291 do we have? We really have that the deletion operat or D applied to X to the 00:05:35.291 --> 00:05:41.080 fifth equals five times X to the fourth. Now where have you seen something like 00:05:41.080 --> 00:05:46.317 that before? It turns out that there's an entire calculus for lists. Let's look at a 00:05:46.317 --> 00:05:53.277 simple example of how this works. Let us denote by x to the n a list of n elements. 00:05:53.277 --> 00:06:01.042 What would x to the zero be? A list of zero elements. Oh, that would be an empty 00:06:01.042 --> 00:06:07.909 list. And what should we call that mathematically? Let's call that one, 00:06:07.911 --> 00:06:12.389 because x to the zero really ought to be one. 00:06:12.391 --> 00:06:19.969 And now consider the set L of all finite lists. We'd like to know, what does L look 00:06:19.969 --> 00:06:26.870 like, symbolically. Here's a statement that I think you'll agree is true. Any 00:06:26.870 --> 00:06:33.774 list is either empty or it has a first entry. What does that translate to you, 00:06:33.774 --> 00:06:40.870 algebraically? Any list means L, is, connotes an equal sign, empty. Oh, we've 00:06:40.870 --> 00:06:48.559 seen what the empty list is, that's one. we've seen that OR, corresponds to a 00:06:48.559 --> 00:06:56.050 formal addition. And what does it mean, for a list to have a first entry? That 00:06:56.050 --> 00:07:03.282 means, you've got an X followed by some other finite list, or X times L. Now, 00:07:03.282 --> 00:07:09.856 let's just take that algebraic equation and start manipulating it. Forget about 00:07:09.856 --> 00:07:16.597 what these things mean grammatically, for the moment. If I move X, x L to the other 00:07:16.597 --> 00:07:23.004 side, factor out L and divide through by 1-X, then we'll get that L is equal to one 00:07:23.004 --> 00:07:28.940 over one minus x. Now what does that mean? I've come up with an algebraic expression 00:07:28.940 --> 00:07:35.178 for all finite lists and I recognize one over one minus X as the geometric series 00:07:35.178 --> 00:07:40.869 one plus X plus X squared plus X cubed etcetera. One way to interpret that, is 00:07:40.869 --> 00:07:46.848 that we have derived the fact. That the set of all finite lists consists of the 00:07:46.848 --> 00:07:52.780 empty list, or a list of one element, or a list of two elements, or a list of three 00:07:52.780 --> 00:07:59.079 elements, etc. That's a pretty remarkable derivation. It's using a type of calculus, 00:07:59.079 --> 00:08:04.572 that is very different from what we're learning in this class but has some 00:08:04.572 --> 00:08:10.272 strange similarities. And as you go on in your calculus education, you'll see that 00:08:10.272 --> 00:08:16.056 calculus applies not just to, problems in physics or economics, but in all kinds of 00:08:16.056 --> 00:08:21.700 fields and in all kinds of manners, which, a t first, might not have been so obvious.