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Calculus: Some Non-Euclidean Geometry

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:55
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 85 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: The Basics (8 lessons, $11.88)
Calculus: Pre-Calculus Review (4 lessons, $9.90)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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The Basics
Precalculus Review
Some Non-Euclidean Geometry [1 of 1]
Well, as we're seeing early on in our discussions together, the shortest distance between two points, if you want to find that, is the length of the straight line. And, in fact, I can, I can sort of illustrate that for you right now, you know. If you take two points, one here and one here, this is a straight line distance between them. And, in fact, if you want to try an interesting little virtual experiment, would be sort of fun to try, I mean, just take the two points as these fingers here and look at the, the straight line distance. Can you visually see that straight line, if you - use your imagination and you can actually see it with your imagination - and look what happens: as I move, can you see the line changing? It gets shorter, it gets longer. Virtual reality, folks, right here. If you're watching really carefully, you can see it. So, neat. Shortest distance is always a straight line.
Or is it? Is the shortest distance always a straight line? It turns out that the answer actually is no. The shortest distance is not always a straight line. And you don't have to look too far for an example. In fact, you only have to look at our own world, which is the world of the sphere. So, in fact, here is a copy of, of the world right here. And, you see, if you want to actually go from, let's say, Williamstown, Massachusetts down to Sydney, Australia, say, well, the shortest distance actually would not be a straight line, but actually would be this curved, this curved, a path over here. That curved path is actually part of a circle. It's the circle that goes through Sydney, Australia, Williamstown, Massachusetts, and whose center is the very center of the earth. These are called great circles, or sometimes geodesics. And it turns out that you probably have already traveled along these things, if you've ever taken a plane ride because, in fact, pilots love to go on geodesics. These are the shortest paths and they love fuel. So, they want to conserve fuel as much as possible, and so they usually fly along the shortest distance, which is this part of a great circle. And geometry of this sort, geometry, where shortest distances aren't lines, but, in fact, circles or other things, are called non-Euclidean geometry, but who cares? The point is that shortest distance isn't always, isn't always a straight line.
Now, I know what you're saying. You're saying, "Well, sure, okay. Shortest distance isn't a straight line. That's because this thing is curved, it's round." Right? I know that's what you're saying. So, what I'd like now is to get a volunteer from the studio audience - let's see if I have... Oh, I have a volunteer right here. Want to come up here? This is great, this is great. And, and what is your name, sir?
My name is Jeffery Dan.
Jeffery Dan?
Yes, it is.
I see. And what do people call you?
Jim.
I see, Jim, Jim. And have we met before?
Never.
Well, actually, this is our technician here, folks, right here in the studios, and he's here to help us out, but we haven't rehearsed this at all and this is all ...
True.
True. That's all true. Okay, good. Now, did you, did you see that, by the way, that the shortest distance here is part of this great circle?
Yes, I did.
Yes. And so, of course, but the circle is round, so it's not that interesting. Right?
True.
Yeah. So, let's get rid of that. Let's get rid of that right now. Okay.
[laughter]
So get rid of that. And, instead - live video, folks, right here. Real video. Instead, what I did was I actually made something to show you and my video virtual student over there. And it's a room, actually. And let me show you this room - move these things here. It's a room. I want you to take a look at it and I want you to take a look at, also, where you are. And it's a transparent room, so you can see inside of it. And you can see that on one side, we have a spider on one side here, and on the other side, I hope you can get this, we actually have a fly. So, we have a fly on one side and on the opposite wall right here, we have a spider. Now, it turns out that the, the spider is, is hungry, you know, and wants to eat the fly. Okay? So, the, the spider wants to get from, from where it's located to, to the fly. Now, the spider, of course, can't fly himself, but can crawl on all the walls, can also crawl, of course, on the floor, and can even crawl on the ceiling. So, it can crawl everywhere, but it just can't swing across, it can't fly. It can't fly. And so, the question is - and the fly is not going to move, by the way. The fly is not going to move. I glued the fly in myself. The fly is not moving. Okay?
So, the spider wants to get from here to the fly. And the question is what is the shortest path, what is the shortest distance that is required for the spider to get to the fly? Now, I'm going to give Jim here a chance to actually think about that for a second, and while he's thinking, the important thing here... In fact, he may guess. And in mathematics, of course, that is the key thing, not to be afraid to take guesses, take risks and possibly make mistakes. See, because if you make a mistake, you then discover that, in fact, the solution was more interesting than you first may have thought. The mistakes, of course, are very valuable. They lead to insight. Anyway, I was trying to stall a little bit there to give you a chance.
Okay.
Do you have a, do you have a path that you'd like to share with the group?
Yes.
Okay, why don't you show us?
The path ...
He did see it, he did see here, by the way [inaudible].
The spider is going to crawl this way.
Okay.
And then crawl this way.
All right.
And then crawl that way.
Okay, now, I want to make sure that everyone can see that path, those of you who, who are watching out there. So, so, Jim's idea is for the spider to come right up along the edge here, and then go across the diagonal. Now, of course, I like the diagonal because this is sort of a right triangle, and so the diagonal cutting this is a really clever idea, I think, and then it goes down and captures the, the fly. Okay, now let's measure that and see what the distance actually is. Okay?
Okay.
Okay, do you have a measuring thing?
No.
Okay, I happen to have one, don't worry.
Okay.
Because I have one right here. Here we go. So, let's measure this and we'll measure head of spider to head of bug and, just to keep us on the up and up, I'll start here so you can read it off there. I'm going to start here with the fly here. Right there, come across, okay, now I'm going to go down. And why don't you read out exactly what you see there at the head of the spider.
22 inches.
22 inches, ladies and gentlemen, right here. 22 inches. Okay. Well, that's a great guess. That's a great guess. Shortest distance - Jim was thinking a straight line. It, it turns out actually that, that there is a shorter path, but, but great guess, because, of course, what he was trying to do was minimize [inaudible]. Actually, the actual path is much more exotic than you may think. It turns out that the, the spider first has to walk a little bit on this wall, and then walk on this wall, and then walk down this wall, and then finally get to here. It's sort of a hard path. Let me see if I can actually illustrate it for you. The path looks something like this. It starts up here, goes around, spirals around, sort of corkscrews in, you know, sort of corkscrews in like that. Can you see that path? Sort of an exotic path. Looks like this, so it corkscrews in. Now, yours was 22 inches. Let's measure this to see if this really is shorter or not. Yours was 22 inches. Let's measure this one. So, again, I start here at the head of the fly, go up this sort of exotic spiral kind of thing, and what does that distance say there? Can you read that off?
27 inches.
By the way, this is the former technician here. Why don't you read off what it really says there.
20 and one-half inches.
20 and one-half inches, folks, 20 and one-half inches. We saved an entire inch and one-half, at the very least, by going along the shortest, the shortest path. Now, now, the interesting thing about Jim's though, the think I like about Jim's is that his sort of minimized sideage. Right? It just went along this thing, right along this one side here, and then came down. That was it. My path actually sort of maximizes sideage. Right? It goes around almost every single side. In fact, it goes on all the sides, but one, goes on all the sides, but one. And it turns out, though, that my box is a high tech box. And it turns out that if you actually open my box up, which I can do, and if I do that, watch what happens when I open this box up. What do you see? You see ... Look at that. Shortest distance, straight line. So, Jim's guess actually was a good guess. He was trying to get a straight line and the shortest distance is a straight line. So, give him a hand! Wasn't that great! Give him a hand! Are you giving him a hand? There you go. Thank you, thank you very much. And as a consolation prize, a lovely consolation prize, I have to give to you the crab. So, now you have a crab. So, anyway, congratulations and thanks for being here.
Thank you.
Thanks a lot.
Thank you.
Anyway, shortest distance not always a straight line. Really interesting, and let's go back to calculus and see where we are. Okay, `bye for now.

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