[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.58,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.58,0:00:02.33,Default,,0000,0000,0000,,PROFESSOR: Any\Nquestions about theory
Dialogue: 0,0:00:02.33,0:00:05.30,Default,,0000,0000,0000,,that gave you headaches\Nregarding homework
Dialogue: 0,0:00:05.30,0:00:07.74,Default,,0000,0000,0000,,you'd like to talk about?
Dialogue: 0,0:00:07.74,0:00:10.95,Default,,0000,0000,0000,,Anything related\Nto what we covered
Dialogue: 0,0:00:10.95,0:00:13.78,Default,,0000,0000,0000,,from chapter nine and today?
Dialogue: 0,0:00:13.78,0:00:15.61,Default,,0000,0000,0000,,STUDENT: Can we\Ndo some problems?
Dialogue: 0,0:00:15.61,0:00:17.41,Default,,0000,0000,0000,,PROFESSOR: I can\Nfix from problems
Dialogue: 0,0:00:17.41,0:00:20.10,Default,,0000,0000,0000,,like the ones in the\Nhomework, but also I
Dialogue: 0,0:00:20.10,0:00:24.55,Default,,0000,0000,0000,,can have you tell me what\Nbothers you in the homework.
Dialogue: 0,0:00:24.55,0:00:25.88,Default,,0000,0000,0000,,STUDENT: Oh, I have [INAUDIBLE].
Dialogue: 0,0:00:25.88,0:00:29.01,Default,,0000,0000,0000,,PROFESSOR: What bothered\Nme about my own homework
Dialogue: 0,0:00:29.01,0:00:35.33,Default,,0000,0000,0000,,was that I realized that I\Ndid not remind you something
Dialogue: 0,0:00:35.33,0:00:37.47,Default,,0000,0000,0000,,I assume you should\Nknow, which is
Dialogue: 0,0:00:37.47,0:00:42.80,Default,,0000,0000,0000,,the equation of a sphere of\Ngiven center and given radius.
Dialogue: 0,0:00:42.80,0:00:47.50,Default,,0000,0000,0000,,And since I trust you so much,\NI said, OK they know about it.
Dialogue: 0,0:00:47.50,0:00:50.93,Default,,0000,0000,0000,,And then somebody asked\Nme by email what that was,
Dialogue: 0,0:00:50.93,0:00:52.44,Default,,0000,0000,0000,,and I said, oh, yeah.
Dialogue: 0,0:00:52.44,0:00:54.24,Default,,0000,0000,0000,,I did not review that in class.
Dialogue: 0,0:00:54.24,0:01:14.75,Default,,0000,0000,0000,,So review the equation\Nin r3 form that's x, y, z
Dialogue: 0,0:01:14.75,0:01:35.52,Default,,0000,0000,0000,,of the sphere of radius r and\Ncenter p of coordinates x0, y0,
Dialogue: 0,0:01:35.52,0:01:37.58,Default,,0000,0000,0000,,z0.
Dialogue: 0,0:01:37.58,0:01:43.46,Default,,0000,0000,0000,,One of you asked me by email,\Ndoes-- of course you do,
Dialogue: 0,0:01:43.46,0:01:48.01,Default,,0000,0000,0000,,and then if you know it,\Ncan you help me-- can you
Dialogue: 0,0:01:48.01,0:01:50.48,Default,,0000,0000,0000,,help remind what that was?
Dialogue: 0,0:01:50.48,0:01:54.84,Default,,0000,0000,0000,,
Dialogue: 0,0:01:54.84,0:01:56.83,Default,,0000,0000,0000,,STUDENT: x minus x0--
Dialogue: 0,0:01:56.83,0:02:06.99,Default,,0000,0000,0000,,PROFESSOR: x minus x0 squared\Nplus y minus y0 squared
Dialogue: 0,0:02:06.99,0:02:15.06,Default,,0000,0000,0000,,plus z minus z0 squared\Nequals R squared.
Dialogue: 0,0:02:15.06,0:02:16.28,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:02:16.28,0:02:18.23,Default,,0000,0000,0000,,When you ask, for\Nexample, what is
Dialogue: 0,0:02:18.23,0:02:22.92,Default,,0000,0000,0000,,the equation of a units sphere,\Nwhat do I mean by unit sphere?
Dialogue: 0,0:02:22.92,0:02:23.63,Default,,0000,0000,0000,,STUDENT: Radius--
Dialogue: 0,0:02:23.63,0:02:28.33,Default,,0000,0000,0000,,PROFESSOR: Radius 1, and\Ncenter 0, standard unit sphere,
Dialogue: 0,0:02:28.33,0:02:30.08,Default,,0000,0000,0000,,will be.
Dialogue: 0,0:02:30.08,0:02:35.10,Default,,0000,0000,0000,,There is a notation for that\Nin mathematics called s2.
Dialogue: 0,0:02:35.10,0:02:36.98,Default,,0000,0000,0000,,I'll tell you why its called s2.
Dialogue: 0,0:02:36.98,0:02:41.31,Default,,0000,0000,0000,,x squared plus y squared\Nplus z squared equals 1.
Dialogue: 0,0:02:41.31,0:02:44.34,Default,,0000,0000,0000,,
Dialogue: 0,0:02:44.34,0:02:48.03,Default,,0000,0000,0000,,s2 stands for the dimension.
Dialogue: 0,0:02:48.03,0:02:53.00,Default,,0000,0000,0000,,That means the number\Nof the-- the number
Dialogue: 0,0:02:53.00,0:02:54.54,Default,,0000,0000,0000,,of degrees of freedom.
Dialogue: 0,0:02:54.54,0:03:00.17,Default,,0000,0000,0000,,
Dialogue: 0,0:03:00.17,0:03:04.32,Default,,0000,0000,0000,,you have on a certain manifold.
Dialogue: 0,0:03:04.32,0:03:07.63,Default,,0000,0000,0000,,
Dialogue: 0,0:03:07.63,0:03:08.90,Default,,0000,0000,0000,,What is a manifold?
Dialogue: 0,0:03:08.90,0:03:10.41,Default,,0000,0000,0000,,It's a geometric structure.
Dialogue: 0,0:03:10.41,0:03:12.80,Default,,0000,0000,0000,,I'm not going to\Ngo into details.
Dialogue: 0,0:03:12.80,0:03:15.64,Default,,0000,0000,0000,,It's a geometric structure\Nwith some special properties.
Dialogue: 0,0:03:15.64,0:03:18.70,Default,,0000,0000,0000,,
Dialogue: 0,0:03:18.70,0:03:21.15,Default,,0000,0000,0000,,I'm not talking about\Nother fields of algebra,
Dialogue: 0,0:03:21.15,0:03:22.41,Default,,0000,0000,0000,,anthropology.
Dialogue: 0,0:03:22.41,0:03:26.50,Default,,0000,0000,0000,,I'm just talking about geometry\Nand calculus math 3, which
Dialogue: 0,0:03:26.50,0:03:28.93,Default,,0000,0000,0000,,is multivariable calculus.
Dialogue: 0,0:03:28.93,0:03:35.20,Default,,0000,0000,0000,,Now, how do I think\Nof degrees of freedom?
Dialogue: 0,0:03:35.20,0:03:37.45,Default,,0000,0000,0000,,Look at the table.
Dialogue: 0,0:03:37.45,0:03:42.91,Default,,0000,0000,0000,,What freedom do I have to move\Nalong one of these sticks?
Dialogue: 0,0:03:42.91,0:03:45.44,Default,,0000,0000,0000,,I have one degree of\Nfreedom in the sense
Dialogue: 0,0:03:45.44,0:03:50.00,Default,,0000,0000,0000,,that it's given by a\Nparameter like time.
Dialogue: 0,0:03:50.00,0:03:50.67,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:50.67,0:03:53.02,Default,,0000,0000,0000,,It's a 1-parameter\Nmanifold in the sense
Dialogue: 0,0:03:53.02,0:03:55.34,Default,,0000,0000,0000,,that maybe I have\Na line, maybe I
Dialogue: 0,0:03:55.34,0:04:00.94,Default,,0000,0000,0000,,have the trajectory of the\Nparking space in terms of time.
Dialogue: 0,0:04:00.94,0:04:05.11,Default,,0000,0000,0000,,The freedom that the bag has\Nis to move according to time,
Dialogue: 0,0:04:05.11,0:04:09.68,Default,,0000,0000,0000,,and that's considered only\None degree of freedom.
Dialogue: 0,0:04:09.68,0:04:13.62,Default,,0000,0000,0000,,Now if you were on a\Nplane or another surface,
Dialogue: 0,0:04:13.62,0:04:18.01,Default,,0000,0000,0000,,why would you have more\Nthan one degrees of freedom?
Dialogue: 0,0:04:18.01,0:04:23.69,Default,,0000,0000,0000,,Well, I can move towards\Nyou, or I can move this way.
Dialogue: 0,0:04:23.69,0:04:26.72,Default,,0000,0000,0000,,I can draw a grid the way\Nthe x and y coordinate.
Dialogue: 0,0:04:26.72,0:04:29.98,Default,,0000,0000,0000,,And those are my\Ndegrees of freedom.
Dialogue: 0,0:04:29.98,0:04:32.96,Default,,0000,0000,0000,,Practically, the basis\NIJ gives me that kind
Dialogue: 0,0:04:32.96,0:04:35.84,Default,,0000,0000,0000,,of two degrees of freedom.
Dialogue: 0,0:04:35.84,0:04:36.34,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:04:36.34,0:04:41.67,Default,,0000,0000,0000,,If I'm in three coordinates, I\Nhave without other constraints,
Dialogue: 0,0:04:41.67,0:04:43.38,Default,,0000,0000,0000,,because I could be\Nin three coordinates
Dialogue: 0,0:04:43.38,0:04:46.74,Default,,0000,0000,0000,,and constrained to be on\Na cylinder, in which case
Dialogue: 0,0:04:46.74,0:04:48.86,Default,,0000,0000,0000,,I still have two\Ndegrees of freedom.
Dialogue: 0,0:04:48.86,0:04:53.44,Default,,0000,0000,0000,,But if I am a bug\Nwho is free to fly,
Dialogue: 0,0:04:53.44,0:04:59.82,Default,,0000,0000,0000,,I have the freedom to go with\Nthree degrees of freedom,
Dialogue: 0,0:04:59.82,0:05:00.83,Default,,0000,0000,0000,,right?
Dialogue: 0,0:05:00.83,0:05:03.70,Default,,0000,0000,0000,,I have three degrees of\Nfreedom, but if the bug
Dialogue: 0,0:05:03.70,0:05:09.08,Default,,0000,0000,0000,,is moving-- not flying,\Nmoving on a surface,
Dialogue: 0,0:05:09.08,0:05:11.56,Default,,0000,0000,0000,,then he has two\Ndegrees of freedom.
Dialogue: 0,0:05:11.56,0:05:16.64,Default,,0000,0000,0000,,So to again review, lines\Nand curves in general
Dialogue: 0,0:05:16.64,0:05:20.19,Default,,0000,0000,0000,,are one dimensional\Nthings, because you
Dialogue: 0,0:05:20.19,0:05:21.46,Default,,0000,0000,0000,,have one degree of freedom.
Dialogue: 0,0:05:21.46,0:05:24.17,Default,,0000,0000,0000,,Two dimensional\Nthings are surfaces,
Dialogue: 0,0:05:24.17,0:05:28.01,Default,,0000,0000,0000,,three dimensional things are\Nspaces, like the Euclidean
Dialogue: 0,0:05:28.01,0:05:31.80,Default,,0000,0000,0000,,space, and we are not\Ngoing to go beyond,
Dialogue: 0,0:05:31.80,0:05:33.66,Default,,0000,0000,0000,,at least for the time\Nbeing, we are not
Dialogue: 0,0:05:33.66,0:05:36.23,Default,,0000,0000,0000,,going to go beyond that.
Dialogue: 0,0:05:36.23,0:05:41.35,Default,,0000,0000,0000,,However, where anybody is\Ninterested in relativity,
Dialogue: 0,0:05:41.35,0:05:47.30,Default,,0000,0000,0000,,say or let's say four\Ndimensional spaces, or things
Dialogue: 0,0:05:47.30,0:05:52.20,Default,,0000,0000,0000,,of x, y, z spatial coordinates\Nand t as a fourth coordinate,
Dialogue: 0,0:05:52.20,0:05:56.65,Default,,0000,0000,0000,,then we can go into higher\Ndimensions, as well.
Dialogue: 0,0:05:56.65,0:05:57.15,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:05:57.15,0:06:01.08,Default,,0000,0000,0000,,
Dialogue: 0,0:06:01.08,0:06:02.82,Default,,0000,0000,0000,,I want to ask you a question.
Dialogue: 0,0:06:02.82,0:06:07.55,Default,,0000,0000,0000,,If somebody gives you on\NWeBWorK or outside of WeBWorK,
Dialogue: 0,0:06:07.55,0:06:11.49,Default,,0000,0000,0000,,on the first quiz or\Non the final exam,
Dialogue: 0,0:06:11.49,0:06:18.52,Default,,0000,0000,0000,,let's say you have\Nthis equation,
Dialogue: 0,0:06:18.52,0:06:28.53,Default,,0000,0000,0000,,x squared plus y squared plus\Nz squared plus 2x plus 2y
Dialogue: 0,0:06:28.53,0:06:32.47,Default,,0000,0000,0000,,equals 9.
Dialogue: 0,0:06:32.47,0:06:37.19,Default,,0000,0000,0000,,What is this identified as?
Dialogue: 0,0:06:37.19,0:06:38.35,Default,,0000,0000,0000,,It's a quadric.
Dialogue: 0,0:06:38.35,0:06:40.93,Default,,0000,0000,0000,,Why would this be a quadric?
Dialogue: 0,0:06:40.93,0:06:43.29,Default,,0000,0000,0000,,Well, there is no x, y, y, z.
Dialogue: 0,0:06:43.29,0:06:45.78,Default,,0000,0000,0000,,Those terms are missing.
Dialogue: 0,0:06:45.78,0:06:54.56,Default,,0000,0000,0000,,But I have something of the\Ntype of quadric x squared
Dialogue: 0,0:06:54.56,0:06:59.51,Default,,0000,0000,0000,,plus By squared plus\Nc squared plus dxy
Dialogue: 0,0:06:59.51,0:07:07.81,Default,,0000,0000,0000,,plus exz plus fyz plus, those\Nare, oh my God, so many.
Dialogue: 0,0:07:07.81,0:07:09.01,Default,,0000,0000,0000,,Degree two.
Dialogue: 0,0:07:09.01,0:07:16.30,Default,,0000,0000,0000,,Degree one I would\Nhave ax plus by plus cz
Dialogue: 0,0:07:16.30,0:07:22.21,Default,,0000,0000,0000,,plus a little d constant, and\Nwhew, that was a long one.
Dialogue: 0,0:07:22.21,0:07:23.05,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:07:23.05,0:07:27.47,Default,,0000,0000,0000,,Now, is this of the\Ntype of a project?
Dialogue: 0,0:07:27.47,0:07:28.20,Default,,0000,0000,0000,,Yes, it is.
Dialogue: 0,0:07:28.20,0:07:31.84,Default,,0000,0000,0000,,Of course there are some terms\Nthat are missing, good for us.
Dialogue: 0,0:07:31.84,0:07:35.58,Default,,0000,0000,0000,,How are you going to try to\Nidentify the type of quadric
Dialogue: 0,0:07:35.58,0:07:37.57,Default,,0000,0000,0000,,by looking at this?
Dialogue: 0,0:07:37.57,0:07:41.41,Default,,0000,0000,0000,,As you said very well,\NI think it's-- you say,
Dialogue: 0,0:07:41.41,0:07:45.74,Default,,0000,0000,0000,,I think of a sphere, maybe I can\Ncomplete the squares, you said.
Dialogue: 0,0:07:45.74,0:07:49.24,Default,,0000,0000,0000,,How do we complete the squares?
Dialogue: 0,0:07:49.24,0:07:52.90,Default,,0000,0000,0000,,x squared plus 2x plus\Nsome missing number,
Dialogue: 0,0:07:52.90,0:07:57.01,Default,,0000,0000,0000,,a magic number-- yes sir?
Dialogue: 0,0:07:57.01,0:08:02.05,Default,,0000,0000,0000,,STUDENT: So, basically I'll\Nhave to take x plus 2 times 4
Dialogue: 0,0:08:02.05,0:08:04.45,Default,,0000,0000,0000,,will go outside.
Dialogue: 0,0:08:04.45,0:08:07.84,Default,,0000,0000,0000,,It's like x min-- x plus 2--
Dialogue: 0,0:08:07.84,0:08:09.23,Default,,0000,0000,0000,,PROFESSOR: Why x plus 2?
Dialogue: 0,0:08:09.23,0:08:10.32,Default,,0000,0000,0000,,STUDENT: Because it's 2x--
Dialogue: 0,0:08:10.32,0:08:12.14,Default,,0000,0000,0000,,STUDENT: It's 2x.
Dialogue: 0,0:08:12.14,0:08:14.75,Default,,0000,0000,0000,,PROFESSOR: But if\NI take x plus 2,
Dialogue: 0,0:08:14.75,0:08:18.32,Default,,0000,0000,0000,,then that's going to give\Nme x squared plus 4x plus 4,
Dialogue: 0,0:08:18.32,0:08:19.55,Default,,0000,0000,0000,,so it's not a good idea.
Dialogue: 0,0:08:19.55,0:08:20.88,Default,,0000,0000,0000,,STUDENT: On the x plus 1
Dialogue: 0,0:08:20.88,0:08:21.89,Default,,0000,0000,0000,,PROFESSOR: x plus 1.
Dialogue: 0,0:08:21.89,0:08:25.70,Default,,0000,0000,0000,,So I'm going to complete\Nx plus 1 squared.
Dialogue: 0,0:08:25.70,0:08:29.42,Default,,0000,0000,0000,,What did I invent\Nthat wasn't there?
Dialogue: 0,0:08:29.42,0:08:30.20,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,0:08:30.20,0:08:31.82,Default,,0000,0000,0000,,PROFESSOR: I invented\Nthe 1, and I have
Dialogue: 0,0:08:31.82,0:08:34.18,Default,,0000,0000,0000,,to compensate for my invention.
Dialogue: 0,0:08:34.18,0:08:38.08,Default,,0000,0000,0000,,I added the 1, created\Nthe 1 out of nothing,
Dialogue: 0,0:08:38.08,0:08:41.15,Default,,0000,0000,0000,,so I have to compensate\Nby subtracting it.
Dialogue: 0,0:08:41.15,0:08:42.40,Default,,0000,0000,0000,,How much is from here to here?
Dialogue: 0,0:08:42.40,0:08:45.56,Default,,0000,0000,0000,,Is it exactly the\Nthing that I underlined
Dialogue: 0,0:08:45.56,0:08:49.38,Default,,0000,0000,0000,,with a wiggly line, a\Nlight wiggly line thing,
Dialogue: 0,0:08:49.38,0:08:52.98,Default,,0000,0000,0000,,plus what is the\Nblue wiggly line,
Dialogue: 0,0:08:52.98,0:09:01.73,Default,,0000,0000,0000,,the blue wiggly line\Nthat doesn't show--
Dialogue: 0,0:09:01.73,0:09:06.37,Default,,0000,0000,0000,,I have y plus 1 squared, and\Nagain, I have to compensate
Dialogue: 0,0:09:06.37,0:09:08.47,Default,,0000,0000,0000,,for what I invented.
Dialogue: 0,0:09:08.47,0:09:13.35,Default,,0000,0000,0000,,I created a 1 out of nothing,\Nso this is y squared plus 2y.
Dialogue: 0,0:09:13.35,0:09:17.59,Default,,0000,0000,0000,,
Dialogue: 0,0:09:17.59,0:09:21.86,Default,,0000,0000,0000,,The z squared is all by himself,\Nand he's crying, I'm so lonely,
Dialogue: 0,0:09:21.86,0:09:26.17,Default,,0000,0000,0000,,I don't know, there is\Nnobody like me over there.
Dialogue: 0,0:09:26.17,0:09:29.62,Default,,0000,0000,0000,,So in the end, I can rewrite\Nthe whole thing as x plus 1
Dialogue: 0,0:09:29.62,0:09:33.64,Default,,0000,0000,0000,,squared plus y plus 1\Nsquared plus z squared, if I
Dialogue: 0,0:09:33.64,0:09:39.14,Default,,0000,0000,0000,,want to work them out in\Nthis format, equals what?
Dialogue: 0,0:09:39.14,0:09:39.64,Default,,0000,0000,0000,,STUDENT: 10.
Dialogue: 0,0:09:39.64,0:09:40.99,Default,,0000,0000,0000,,
Dialogue: 0,0:09:40.99,0:09:41.89,Default,,0000,0000,0000,,PROFESSOR: 11.
Dialogue: 0,0:09:41.89,0:09:47.38,Default,,0000,0000,0000,,11 is the square\Nroot of 11 squared.
Dialogue: 0,0:09:47.38,0:09:49.31,Default,,0000,0000,0000,,Like my son said the other day.
Dialogue: 0,0:09:49.31,0:09:52.56,Default,,0000,0000,0000,,So that the radius\Nwould be square foot
Dialogue: 0,0:09:52.56,0:09:57.31,Default,,0000,0000,0000,,11 of a sphere of what circle?
Dialogue: 0,0:09:57.31,0:10:00.13,Default,,0000,0000,0000,,What is the-- or the\Nsphere of what center?
Dialogue: 0,0:10:00.13,0:10:00.88,Default,,0000,0000,0000,,STUDENT: Minus 1--
Dialogue: 0,0:10:00.88,0:10:03.96,Default,,0000,0000,0000,,PROFESSOR: Minus\N1, minus 1, and 0.
Dialogue: 0,0:10:03.96,0:10:06.08,Default,,0000,0000,0000,,So I don't want to insult you.
Dialogue: 0,0:10:06.08,0:10:07.87,Default,,0000,0000,0000,,Of course you know how\Nto complete squares.
Dialogue: 0,0:10:07.87,0:10:10.63,Default,,0000,0000,0000,,
Dialogue: 0,0:10:10.63,0:10:15.69,Default,,0000,0000,0000,,However, I have discovered in an\Nupper level class at some point
Dialogue: 0,0:10:15.69,0:10:19.46,Default,,0000,0000,0000,,that my students didn't know\Nhow to complete squares, which
Dialogue: 0,0:10:19.46,0:10:22.48,Default,,0000,0000,0000,,was very, very heartbreaking.
Dialogue: 0,0:10:22.48,0:10:24.75,Default,,0000,0000,0000,,All right, now.
Dialogue: 0,0:10:24.75,0:10:28.44,Default,,0000,0000,0000,,
Dialogue: 0,0:10:28.44,0:10:39.43,Default,,0000,0000,0000,,Any questions regarding--\Nwhile I have a few of yours,
Dialogue: 0,0:10:39.43,0:10:41.35,Default,,0000,0000,0000,,I'm going to wait\Na little bit longer
Dialogue: 0,0:10:41.35,0:10:43.65,Default,,0000,0000,0000,,until I give\Neverybody the chance
Dialogue: 0,0:10:43.65,0:10:47.94,Default,,0000,0000,0000,,to complete the extra credit.
Dialogue: 0,0:10:47.94,0:10:52.77,Default,,0000,0000,0000,,I have the question\Nby email saying,
Dialogue: 0,0:10:52.77,0:10:56.90,Default,,0000,0000,0000,,you mentioned that\Ngenius guy in your class.
Dialogue: 0,0:10:56.90,0:11:00.43,Default,,0000,0000,0000,,This is a 1-sheeted hyperboloid.
Dialogue: 0,0:11:00.43,0:11:04.25,Default,,0000,0000,0000,,x squared plus y squared minus\Nz squared minus 1 equals 0.
Dialogue: 0,0:11:04.25,0:11:08.19,Default,,0000,0000,0000,,The question was, by\Nemail, how in the world,
Dialogue: 0,0:11:08.19,0:11:19.59,Default,,0000,0000,0000,,did he figure out what the two\Nfamilies of generatrices are?
Dialogue: 0,0:11:19.59,0:11:22.04,Default,,0000,0000,0000,,So you have one family\Nand another family,
Dialogue: 0,0:11:22.04,0:11:27.51,Default,,0000,0000,0000,,and both together generate\Nthe 1-sheeted hyperboloid.
Dialogue: 0,0:11:27.51,0:11:30.20,Default,,0000,0000,0000,,
Dialogue: 0,0:11:30.20,0:11:33.07,Default,,0000,0000,0000,,Let me give you a little\Nbit more of a hint,
Dialogue: 0,0:11:33.07,0:11:35.82,Default,,0000,0000,0000,,but I'm still going to stop.
Dialogue: 0,0:11:35.82,0:11:41.87,Default,,0000,0000,0000,,So last time I said, he\Nnoticed you can root together
Dialogue: 0,0:11:41.87,0:11:46.97,Default,,0000,0000,0000,,the y squared minus 1 and the\Nx squared minus z squared,
Dialogue: 0,0:11:46.97,0:11:48.35,Default,,0000,0000,0000,,and you can separate them.
Dialogue: 0,0:11:48.35,0:11:52.85,Default,,0000,0000,0000,,So you're going to have x\Nsquared minus z squared equals
Dialogue: 0,0:11:52.85,0:11:53.80,Default,,0000,0000,0000,,1 minus y squared.
Dialogue: 0,0:11:53.80,0:11:56.42,Default,,0000,0000,0000,,
Dialogue: 0,0:11:56.42,0:12:00.35,Default,,0000,0000,0000,,You can't hide the\Ndifference of two squares
Dialogue: 0,0:12:00.35,0:12:03.99,Default,,0000,0000,0000,,as product of sum\Nand difference.
Dialogue: 0,0:12:03.99,0:12:13.39,Default,,0000,0000,0000,,x plus z times x minus z equals\N1 plus y times 1 minus y.
Dialogue: 0,0:12:13.39,0:12:19.52,Default,,0000,0000,0000,,So how can you\Neventually arrange stuff
Dialogue: 0,0:12:19.52,0:12:25.35,Default,,0000,0000,0000,,to be giving due\Nto the lines that
Dialogue: 0,0:12:25.35,0:12:28.81,Default,,0000,0000,0000,,are sitting on the surface?
Dialogue: 0,0:12:28.81,0:12:30.91,Default,,0000,0000,0000,,The lines that are\Nsitting on the surface
Dialogue: 0,0:12:30.91,0:12:33.71,Default,,0000,0000,0000,,are infinitely many,\Nand I would like
Dialogue: 0,0:12:33.71,0:12:39.32,Default,,0000,0000,0000,,at least a 1-parameter\Nfamily of such lines.
Dialogue: 0,0:12:39.32,0:12:40.83,Default,,0000,0000,0000,,You can have choices.
Dialogue: 0,0:12:40.83,0:12:43.23,Default,,0000,0000,0000,,One of the choices\Nwould be-- this
Dialogue: 0,0:12:43.23,0:12:46.18,Default,,0000,0000,0000,,is a product, of\Ntwo numbers, right?
Dialogue: 0,0:12:46.18,0:12:51.32,Default,,0000,0000,0000,,So you can write it as an\Nequality of two fractions.
Dialogue: 0,0:12:51.32,0:12:56.09,Default,,0000,0000,0000,,So you would have something\Nlike x plus z on top, x minus
Dialogue: 0,0:12:56.09,0:12:57.42,Default,,0000,0000,0000,,z below.
Dialogue: 0,0:12:57.42,0:13:01.57,Default,,0000,0000,0000,,Observe that you are\Ncreating singularities here.
Dialogue: 0,0:13:01.57,0:13:11.69,Default,,0000,0000,0000,,So you have to take x minus\Nz case equals 0 separately,
Dialogue: 0,0:13:11.69,0:13:16.54,Default,,0000,0000,0000,,and then you have, let's\Nsay you have 1 minus y here,
Dialogue: 0,0:13:16.54,0:13:21.12,Default,,0000,0000,0000,,and 1 plus y here.
Dialogue: 0,0:13:21.12,0:13:26.87,Default,,0000,0000,0000,,What else do you have to impose\Nwhen you impose x minus z
Dialogue: 0,0:13:26.87,0:13:27.94,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,0:13:27.94,0:13:29.75,Default,,0000,0000,0000,,You cannot have 7 over 0.
Dialogue: 0,0:13:29.75,0:13:31.50,Default,,0000,0000,0000,,That is undefined.
Dialogue: 0,0:13:31.50,0:13:34.63,Default,,0000,0000,0000,,but if you have 0 over\N0, that's still possible.
Dialogue: 0,0:13:34.63,0:13:38.43,Default,,0000,0000,0000,,So whenever you take x\Nminus z equals 0 separately,
Dialogue: 0,0:13:38.43,0:13:42.60,Default,,0000,0000,0000,,that will imply that the\Nnumerator corresponding to it
Dialogue: 0,0:13:42.60,0:13:44.84,Default,,0000,0000,0000,,will also have to be 0.
Dialogue: 0,0:13:44.84,0:13:47.43,Default,,0000,0000,0000,,And together these\Nguys are friends.
Dialogue: 0,0:13:47.43,0:13:49.42,Default,,0000,0000,0000,,What are they?
Dialogue: 0,0:13:49.42,0:13:49.92,Default,,0000,0000,0000,,2--
Dialogue: 0,0:13:49.92,0:13:51.21,Default,,0000,0000,0000,,STUDENT: A system of equations.
Dialogue: 0,0:13:51.21,0:13:53.13,Default,,0000,0000,0000,,PROFESSOR: It's a\Nsystem of equations.
Dialogue: 0,0:13:53.13,0:13:57.84,Default,,0000,0000,0000,,They both represent planes, and\Nthe intersection of two planes
Dialogue: 0,0:13:57.84,0:14:00.53,Default,,0000,0000,0000,,is a line.
Dialogue: 0,0:14:00.53,0:14:07.79,Default,,0000,0000,0000,,It's a particular line, which\Nis part of the family-- which
Dialogue: 0,0:14:07.79,0:14:10.34,Default,,0000,0000,0000,,is part of a family.
Dialogue: 0,0:14:10.34,0:14:13.32,Default,,0000,0000,0000,,
Dialogue: 0,0:14:13.32,0:14:14.10,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:14:14.10,0:14:21.10,Default,,0000,0000,0000,,Now, on the other hand, in case\Nyou have 1 plus y equals 0--
Dialogue: 0,0:14:21.10,0:14:25.40,Default,,0000,0000,0000,,so if it happens that you\Nhave this extreme case
Dialogue: 0,0:14:25.40,0:14:28.59,Default,,0000,0000,0000,,that the denominator\Nwill be 0, you absolutely
Dialogue: 0,0:14:28.59,0:14:34.83,Default,,0000,0000,0000,,have to impose x plus z to be 0,\Nand then you have another life.
Dialogue: 0,0:14:34.83,0:14:36.60,Default,,0000,0000,0000,,It's not easy for\Nme to draw those,
Dialogue: 0,0:14:36.60,0:14:39.39,Default,,0000,0000,0000,,but I could if you\Nasked me privately
Dialogue: 0,0:14:39.39,0:14:43.05,Default,,0000,0000,0000,,to draw those and show you\Nwhat the lines look like.
Dialogue: 0,0:14:43.05,0:14:43.66,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:14:43.66,0:14:46.18,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:14:46.18,0:14:51.65,Default,,0000,0000,0000,,So you have two special lines\Nthat are part of that picture.
Dialogue: 0,0:14:51.65,0:14:55.53,Default,,0000,0000,0000,,They are embedded\Nin the surface.
Dialogue: 0,0:14:55.53,0:15:00.42,Default,,0000,0000,0000,,How do you find a\Nfamily of planes?
Dialogue: 0,0:15:00.42,0:15:03.16,Default,,0000,0000,0000,,Oh my god, I only\Nhad one choice,
Dialogue: 0,0:15:03.16,0:15:06.08,Default,,0000,0000,0000,,but I could have\Nyet another choice
Dialogue: 0,0:15:06.08,0:15:08.32,Default,,0000,0000,0000,,of how to pick the parameters.
Dialogue: 0,0:15:08.32,0:15:11.54,Default,,0000,0000,0000,,Let's take lambda to be\Na real number parameter.
Dialogue: 0,0:15:11.54,0:15:15.74,Default,,0000,0000,0000,,
Dialogue: 0,0:15:15.74,0:15:20.22,Default,,0000,0000,0000,,And lambda could be\Nanything-- if lambda is 0,
Dialogue: 0,0:15:20.22,0:15:22.01,Default,,0000,0000,0000,,what have I got to have, guys?
Dialogue: 0,0:15:22.01,0:15:22.80,Default,,0000,0000,0000,,STUDENT: The top.
Dialogue: 0,0:15:22.80,0:15:24.53,Default,,0000,0000,0000,,PROFESSOR: The top\Nguys will be 0,
Dialogue: 0,0:15:24.53,0:15:29.14,Default,,0000,0000,0000,,and I still have 1 minus y\Nequals 0, a plane, intersected
Dialogue: 0,0:15:29.14,0:15:33.92,Default,,0000,0000,0000,,with x plus z equals 0,\Nanother plane, so still a line.
Dialogue: 0,0:15:33.92,0:15:37.30,Default,,0000,0000,0000,,So lambda equals 0 will give\Nme yet another line, which
Dialogue: 0,0:15:37.30,0:15:39.08,Default,,0000,0000,0000,,is not written big.
Dialogue: 0,0:15:39.08,0:15:41.10,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:15:41.10,0:15:42.49,Default,,0000,0000,0000,,Could lambda ever\Ngo to infinity?
Dialogue: 0,0:15:42.49,0:15:45.61,Default,,0000,0000,0000,,
Dialogue: 0,0:15:45.61,0:15:49.44,Default,,0000,0000,0000,,Lambda wants to go to\Ninfinity, and when does lambda
Dialogue: 0,0:15:49.44,0:15:50.41,Default,,0000,0000,0000,,go to infinity?
Dialogue: 0,0:15:50.41,0:15:52.12,Default,,0000,0000,0000,,STUDENT: When the\Nbottoms would equal 0--
Dialogue: 0,0:15:52.12,0:15:55.47,Default,,0000,0000,0000,,PROFESSOR: When both\Nthe bottoms would be 0.
Dialogue: 0,0:15:55.47,0:16:01.74,Default,,0000,0000,0000,,
Dialogue: 0,0:16:01.74,0:16:06.44,Default,,0000,0000,0000,,So this is-- I can call it L\Ninfinity, the line of infinity.
Dialogue: 0,0:16:06.44,0:16:07.08,Default,,0000,0000,0000,,You see?
Dialogue: 0,0:16:07.08,0:16:09.57,Default,,0000,0000,0000,,But still those\Nwould be two planes.
Dialogue: 0,0:16:09.57,0:16:11.80,Default,,0000,0000,0000,,There's an intersection,\Nit's a line.
Dialogue: 0,0:16:11.80,0:16:12.48,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:16:12.48,0:16:19.95,Default,,0000,0000,0000,,Can we write this family--\Njust one family of lines?
Dialogue: 0,0:16:19.95,0:16:24.08,Default,,0000,0000,0000,,A line is always an intersection\Nof two planes, right?
Dialogue: 0,0:16:24.08,0:16:28.15,Default,,0000,0000,0000,,So which are the planes\Nthat I'm talking about?
Dialogue: 0,0:16:28.15,0:16:32.60,Default,,0000,0000,0000,,x plus z equals\Nlambda times 1 plus y.
Dialogue: 0,0:16:32.60,0:16:36.54,Default,,0000,0000,0000,,This is not in the book,\Nbecause, oh my God, this is
Dialogue: 0,0:16:36.54,0:16:38.65,Default,,0000,0000,0000,,too hard for the book, right?
Dialogue: 0,0:16:38.65,0:16:43.34,Default,,0000,0000,0000,,But it's a nice example to\Nlook at in an honors class.
Dialogue: 0,0:16:43.34,0:16:47.47,Default,,0000,0000,0000,,1 minus y equals\Nlambda times x minus z.
Dialogue: 0,0:16:47.47,0:16:48.68,Default,,0000,0000,0000,,It's not in the book.
Dialogue: 0,0:16:48.68,0:16:55.30,Default,,0000,0000,0000,,It's not in any book that I know\Nof at the level of calculus.
Dialogue: 0,0:16:55.30,0:16:57.08,Default,,0000,0000,0000,,All right, OK.
Dialogue: 0,0:16:57.08,0:16:58.59,Default,,0000,0000,0000,,What are these animals?
Dialogue: 0,0:16:58.59,0:17:00.13,Default,,0000,0000,0000,,The first animal is a plane.
Dialogue: 0,0:17:00.13,0:17:02.53,Default,,0000,0000,0000,,The second animal is a plane.
Dialogue: 0,0:17:02.53,0:17:05.04,Default,,0000,0000,0000,,How many planes\Nare in the picture?
Dialogue: 0,0:17:05.04,0:17:11.63,Default,,0000,0000,0000,,For each lambda, you have a--\Nfor each lambda value in R,
Dialogue: 0,0:17:11.63,0:17:15.67,Default,,0000,0000,0000,,you have a couple of planes\Nthat intersect along your line.
Dialogue: 0,0:17:15.67,0:17:18.45,Default,,0000,0000,0000,,This is the line L lambda.
Dialogue: 0,0:17:18.45,0:17:21.56,Default,,0000,0000,0000,,And shut up, Magdalena,\Nyou told people too much.
Dialogue: 0,0:17:21.56,0:17:25.22,Default,,0000,0000,0000,,If you still want them to do\Nthis for 2 extra credit points,
Dialogue: 0,0:17:25.22,0:17:28.21,Default,,0000,0000,0000,,give them the chance\Nto finish the exercise.
Dialogue: 0,0:17:28.21,0:17:32.01,Default,,0000,0000,0000,,So I zip my lips, but\Nonly after I ask you,
Dialogue: 0,0:17:32.01,0:17:33.66,Default,,0000,0000,0000,,how do you think\Nyou are going to get
Dialogue: 0,0:17:33.66,0:17:37.98,Default,,0000,0000,0000,,the other family of rulers?
Dialogue: 0,0:17:37.98,0:17:42.21,Default,,0000,0000,0000,,The ruling guys are\Ntwo families, you see?
Dialogue: 0,0:17:42.21,0:17:47.24,Default,,0000,0000,0000,,So this family is\Ngoing in one direction.
Dialogue: 0,0:17:47.24,0:17:48.69,Default,,0000,0000,0000,,How am I going to\Nget two families?
Dialogue: 0,0:17:48.69,0:17:51.56,Default,,0000,0000,0000,,
Dialogue: 0,0:17:51.56,0:17:55.03,Default,,0000,0000,0000,,I have another choice\Nthat-- how did I take this?
Dialogue: 0,0:17:55.03,0:17:57.88,Default,,0000,0000,0000,,More or less, I made my choice.
Dialogue: 0,0:17:57.88,0:18:02.27,Default,,0000,0000,0000,,Just like having two\Npeople that would
Dialogue: 0,0:18:02.27,0:18:04.92,Default,,0000,0000,0000,,be prospective job candidates.
Dialogue: 0,0:18:04.92,0:18:07.63,Default,,0000,0000,0000,,You pick one of them.
Dialogue: 0,0:18:07.63,0:18:10.24,Default,,0000,0000,0000,,STUDENT: Now, we can put 1\Nminus y in the denominator.
Dialogue: 0,0:18:10.24,0:18:12.52,Default,,0000,0000,0000,,The denominator in\Nplace of 1 plus y.
Dialogue: 0,0:18:12.52,0:18:16.68,Default,,0000,0000,0000,,PROFESSOR: So I could have\Ndone-- I could have taken this,
Dialogue: 0,0:18:16.68,0:18:19.58,Default,,0000,0000,0000,,and put 1 plus y here,\Nand 1 minus y here.
Dialogue: 0,0:18:19.58,0:18:21.46,Default,,0000,0000,0000,,I'm going to let\Nyou do the rest,
Dialogue: 0,0:18:21.46,0:18:24.89,Default,,0000,0000,0000,,and get the second\Nfamily of generators
Dialogue: 0,0:18:24.89,0:18:28.46,Default,,0000,0000,0000,,for the whole surface.
Dialogue: 0,0:18:28.46,0:18:29.17,Default,,0000,0000,0000,,That's enough.
Dialogue: 0,0:18:29.17,0:18:31.18,Default,,0000,0000,0000,,You're not missing your credit.
Dialogue: 0,0:18:31.18,0:18:35.76,Default,,0000,0000,0000,,Just, you wanted help,\Nand I helped you.
Dialogue: 0,0:18:35.76,0:18:39.93,Default,,0000,0000,0000,,And I'm not mad whatsoever\Nwhen you ask me things.
Dialogue: 0,0:18:39.93,0:18:44.05,Default,,0000,0000,0000,,The email I got sounded like--\Nsays, this is not in the book,
Dialogue: 0,0:18:44.05,0:18:47.17,Default,,0000,0000,0000,,or in any book, or\Non the internet.
Dialogue: 0,0:18:47.17,0:18:48.79,Default,,0000,0000,0000,,How shall I approach this?
Dialogue: 0,0:18:48.79,0:18:51.17,Default,,0000,0000,0000,,How shall I start thinking\Nabout this problem?
Dialogue: 0,0:18:51.17,0:18:54.33,Default,,0000,0000,0000,,This is a completely\Nlegitimate question.
Dialogue: 0,0:18:54.33,0:18:58.28,Default,,0000,0000,0000,,How do I start on this problem?
Dialogue: 0,0:18:58.28,0:18:59.20,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:18:59.20,0:19:02.24,Default,,0000,0000,0000,,On the homework-- maybe it's\Ntoo easy-- you have two or three
Dialogue: 0,0:19:02.24,0:19:04.83,Default,,0000,0000,0000,,examples involving spheres.
Dialogue: 0,0:19:04.83,0:19:06.87,Default,,0000,0000,0000,,Those will be too easy for you.
Dialogue: 0,0:19:06.87,0:19:10.86,Default,,0000,0000,0000,,I only gave you a very thin\Namong of homework this time.
Dialogue: 0,0:19:10.86,0:19:15.18,Default,,0000,0000,0000,,You Have plenty of time until\NMonday at 1:30 or something PM.
Dialogue: 0,0:19:15.18,0:19:18.79,Default,,0000,0000,0000,,
Dialogue: 0,0:19:18.79,0:19:21.28,Default,,0000,0000,0000,,I would like to draw\Na little bit more,
Dialogue: 0,0:19:21.28,0:19:25.02,Default,,0000,0000,0000,,because in this homework\Nand the next homework,
Dialogue: 0,0:19:25.02,0:19:33.16,Default,,0000,0000,0000,,I'm building something special\Ncalled the Frenet Trihedron.
Dialogue: 0,0:19:33.16,0:19:36.58,Default,,0000,0000,0000,,And I told you a little bit\Nabout this Frenet Trihedron,
Dialogue: 0,0:19:36.58,0:19:39.70,Default,,0000,0000,0000,,but I didn't tell you much.
Dialogue: 0,0:19:39.70,0:19:42.60,Default,,0000,0000,0000,,
Dialogue: 0,0:19:42.60,0:19:45.91,Default,,0000,0000,0000,,Many textbooks in\Nmultivariable calculus
Dialogue: 0,0:19:45.91,0:19:48.81,Default,,0000,0000,0000,,don't say much about it,\Nwhich I think is a shame.
Dialogue: 0,0:19:48.81,0:19:52.67,Default,,0000,0000,0000,,
Dialogue: 0,0:19:52.67,0:19:57.00,Default,,0000,0000,0000,,You have a position\Nvector that gives you
Dialogue: 0,0:19:57.00,0:19:59.25,Default,,0000,0000,0000,,the equation of a regular curve.
Dialogue: 0,0:19:59.25,0:20:05.53,Default,,0000,0000,0000,,
Dialogue: 0,0:20:05.53,0:20:07.93,Default,,0000,0000,0000,,x of t, y of t, z of t.
Dialogue: 0,0:20:07.93,0:20:09.83,Default,,0000,0000,0000,,Again, what was a regular curve?
Dialogue: 0,0:20:09.83,0:20:13.87,Default,,0000,0000,0000,,I'm just doing review of\Nwhat we did last time.
Dialogue: 0,0:20:13.87,0:20:17.24,Default,,0000,0000,0000,,A very nice curve\Nthat is differentiable
Dialogue: 0,0:20:17.24,0:20:21.84,Default,,0000,0000,0000,,and whose derivative is\Ncontinuous everywhere
Dialogue: 0,0:20:21.84,0:20:23.03,Default,,0000,0000,0000,,on the interval.
Dialogue: 0,0:20:23.03,0:20:30.92,Default,,0000,0000,0000,,But moreover, the r prime\Nof t never becomes 0.
Dialogue: 0,0:20:30.92,0:20:35.52,Default,,0000,0000,0000,,So continuously differentiable,\Nand r prime of t
Dialogue: 0,0:20:35.52,0:20:40.98,Default,,0000,0000,0000,,never becomes 0 for any--\Ndo you know this name,
Dialogue: 0,0:20:40.98,0:20:43.20,Default,,0000,0000,0000,,any for every or for any?
Dialogue: 0,0:20:43.20,0:20:43.70,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:20:43.70,0:20:46.97,Default,,0000,0000,0000,,This is the symbolistics\Nof mathematics.
Dialogue: 0,0:20:46.97,0:20:50.09,Default,,0000,0000,0000,,You know because you\Nare as nerdy as me.
Dialogue: 0,0:20:50.09,0:20:51.97,Default,,0000,0000,0000,,But everybody else doesn't.
Dialogue: 0,0:20:51.97,0:20:53.65,Default,,0000,0000,0000,,You guys will learn.
Dialogue: 0,0:20:53.65,0:20:55.71,Default,,0000,0000,0000,,This is what\Nmathematicians like.
Dialogue: 0,0:20:55.71,0:21:00.26,Default,,0000,0000,0000,,You see, mathematicians hate\Nwriting lots of words down.
Dialogue: 0,0:21:00.26,0:21:05.01,Default,,0000,0000,0000,,If we liked writing essays\Nand lots of blah, blah, blah,
Dialogue: 0,0:21:05.01,0:21:06.85,Default,,0000,0000,0000,,we would do something else.
Dialogue: 0,0:21:06.85,0:21:08.76,Default,,0000,0000,0000,,We wouldn't do mathematics.
Dialogue: 0,0:21:08.76,0:21:11.49,Default,,0000,0000,0000,,We would do debates,\Nwe would do politics,
Dialogue: 0,0:21:11.49,0:21:14.40,Default,,0000,0000,0000,,we would do other things.
Dialogue: 0,0:21:14.40,0:21:16.88,Default,,0000,0000,0000,,Mathematicians like\Nideas, but when
Dialogue: 0,0:21:16.88,0:21:19.42,Default,,0000,0000,0000,,it comes to writing\Nthem down, they
Dialogue: 0,0:21:19.42,0:21:23.45,Default,,0000,0000,0000,,want to right them down in\Nthe most compact way possible.
Dialogue: 0,0:21:23.45,0:21:26.48,Default,,0000,0000,0000,,That's why they created\Nsort of their own language,
Dialogue: 0,0:21:26.48,0:21:30.73,Default,,0000,0000,0000,,and they have all sorts\Nof logical quantifiers.
Dialogue: 0,0:21:30.73,0:21:34.06,Default,,0000,0000,0000,,And it's like your\Nsecret language
Dialogue: 0,0:21:34.06,0:21:37.31,Default,,0000,0000,0000,,when it comes to your\Nless nerdy friends.
Dialogue: 0,0:21:37.31,0:21:45.04,Default,,0000,0000,0000,,So you go for every--\Nfor any or for every--
Dialogue: 0,0:21:45.04,0:21:45.96,Default,,0000,0000,0000,,do you know this sign?
Dialogue: 0,0:21:45.96,0:21:48.86,Default,,0000,0000,0000,,
Dialogue: 0,0:21:48.86,0:21:49.54,Default,,0000,0000,0000,,There exists.
Dialogue: 0,0:21:49.54,0:21:54.69,Default,,0000,0000,0000,,
Dialogue: 0,0:21:54.69,0:21:57.08,Default,,0000,0000,0000,,And do you know this thing?
Dialogue: 0,0:21:57.08,0:22:01.02,Default,,0000,0000,0000,,Because one of the-- huh?
Dialogue: 0,0:22:01.02,0:22:02.66,Default,,0000,0000,0000,,STUDENT: Is that factorial?
Dialogue: 0,0:22:02.66,0:22:05.45,Default,,0000,0000,0000,,PROFESSOR: Factorial,\Nbut in logic,
Dialogue: 0,0:22:05.45,0:22:09.02,Default,,0000,0000,0000,,that means there exists\Na unique-- a unique.
Dialogue: 0,0:22:09.02,0:22:11.32,Default,,0000,0000,0000,,So there exists a unique.
Dialogue: 0,0:22:11.32,0:22:14.72,Default,,0000,0000,0000,,There exists a unique number.
Dialogue: 0,0:22:14.72,0:22:18.78,Default,,0000,0000,0000,,There is a unique number.
Dialogue: 0,0:22:18.78,0:22:20.87,Default,,0000,0000,0000,,So we have our own language.
Dialogue: 0,0:22:20.87,0:22:23.56,Default,,0000,0000,0000,,Of course, empty set,\Neverybody knows that.
Dialogue: 0,0:22:23.56,0:22:28.16,Default,,0000,0000,0000,,And it's used in\Nmathematical logic a lot.
Dialogue: 0,0:22:28.16,0:22:33.34,Default,,0000,0000,0000,,You know most of the symbols\Nfrom unit intersection,
Dialogue: 0,0:22:33.34,0:22:35.48,Default,,0000,0000,0000,,or, and.
Dialogue: 0,0:22:35.48,0:22:38.57,Default,,0000,0000,0000,,I'm going to use some\Nof those as well.
Dialogue: 0,0:22:38.57,0:22:40.48,Default,,0000,0000,0000,,Coming back to the\NFrenet Trihedron,
Dialogue: 0,0:22:40.48,0:22:44.06,Default,,0000,0000,0000,,we have that velocity\Nvector at every point.
Dialogue: 0,0:22:44.06,0:22:44.94,Default,,0000,0000,0000,,We are happy with it.
Dialogue: 0,0:22:44.94,0:22:49.06,Default,,0000,0000,0000,,We have our prime of t\Nthat is referred from 0.
Dialogue: 0,0:22:49.06,0:22:51.08,Default,,0000,0000,0000,,I said I want to\Nmake it uniform,
Dialogue: 0,0:22:51.08,0:22:53.81,Default,,0000,0000,0000,,and then I divided\Nby the magnitude,
Dialogue: 0,0:22:53.81,0:22:57.86,Default,,0000,0000,0000,,and I have this wonderful t\Nvector we just talked about.
Dialogue: 0,0:22:57.86,0:23:03.58,Default,,0000,0000,0000,,Mr. t is r prime over the\Nmagnitude of r prime, which
Dialogue: 0,0:23:03.58,0:23:06.72,Default,,0000,0000,0000,,is called it's peak right?
Dialogue: 0,0:23:06.72,0:23:09.86,Default,,0000,0000,0000,,We divide by its peak.
Dialogue: 0,0:23:09.86,0:23:12.61,Default,,0000,0000,0000,,What's the name of t, again?
Dialogue: 0,0:23:12.61,0:23:13.57,Default,,0000,0000,0000,,STUDENT: Tangent unit--
Dialogue: 0,0:23:13.57,0:23:16.31,Default,,0000,0000,0000,,PROFESSOR: Tangent\Nunit vector, very good.
Dialogue: 0,0:23:16.31,0:23:19.69,Default,,0000,0000,0000,,How did you remember\Nthat so quickly?
Dialogue: 0,0:23:19.69,0:23:22.30,Default,,0000,0000,0000,,Tangent unit vector.
Dialogue: 0,0:23:22.30,0:23:28.07,Default,,0000,0000,0000,,There is also another\Nguy who is famous.
Dialogue: 0,0:23:28.07,0:23:32.76,Default,,0000,0000,0000,,I wanted to make him\Ngreen, but let's see
Dialogue: 0,0:23:32.76,0:23:35.17,Default,,0000,0000,0000,,if I can make him blue.
Dialogue: 0,0:23:35.17,0:23:42.38,Default,,0000,0000,0000,,t is defined-- should I\Nwrite the f on top of here?
Dialogue: 0,0:23:42.38,0:23:43.50,Default,,0000,0000,0000,,Do you know what that is?
Dialogue: 0,0:23:43.50,0:23:45.63,Default,,0000,0000,0000,,STUDENT: I thought n\Nwas the normal vector.
Dialogue: 0,0:23:45.63,0:23:48.14,Default,,0000,0000,0000,,PROFESSOR: t prime\Ndivided by the length of--
Dialogue: 0,0:23:48.14,0:23:48.72,Default,,0000,0000,0000,,STUDENT: Wait.
Dialogue: 0,0:23:48.72,0:23:53.34,Default,,0000,0000,0000,,I thought the vector\Nn was the normal.
Dialogue: 0,0:23:53.34,0:23:56.45,Default,,0000,0000,0000,,PROFESSOR: n-- there\Nare many normals.
Dialogue: 0,0:23:56.45,0:24:01.44,Default,,0000,0000,0000,,It's a very good thing, because\Nwe don't say that in the book.
Dialogue: 0,0:24:01.44,0:24:04.97,Default,,0000,0000,0000,,OK, this is the t along my r.
Dialogue: 0,0:24:04.97,0:24:09.11,Default,,0000,0000,0000,,Now when I go through a point,\Nthis is the normal plane,
Dialogue: 0,0:24:09.11,0:24:09.98,Default,,0000,0000,0000,,right?
Dialogue: 0,0:24:09.98,0:24:14.54,Default,,0000,0000,0000,,There are many normals to\Nthe surface-- to the curve.
Dialogue: 0,0:24:14.54,0:24:15.76,Default,,0000,0000,0000,,Which one am I taking?
Dialogue: 0,0:24:15.76,0:24:19.67,Default,,0000,0000,0000,,All of them are perpendicular\Nto the direction, right?
Dialogue: 0,0:24:19.67,0:24:20.27,Default,,0000,0000,0000,,STUDENT: tf.
Dialogue: 0,0:24:20.27,0:24:22.06,Default,,0000,0000,0000,,PROFESSOR: So I take\Nthis one, or this one,
Dialogue: 0,0:24:22.06,0:24:25.46,Default,,0000,0000,0000,,or this one, or this one, or\Nthis one, or this one, there.
Dialogue: 0,0:24:25.46,0:24:26.83,Default,,0000,0000,0000,,I have to make up my mind.
Dialogue: 0,0:24:26.83,0:24:31.39,Default,,0000,0000,0000,,And that's how people came up\Nwith the so-called principal
Dialogue: 0,0:24:31.39,0:24:33.45,Default,,0000,0000,0000,,unit normal.
Dialogue: 0,0:24:33.45,0:24:36.08,Default,,0000,0000,0000,,And this is the one\NI'm talking about.
Dialogue: 0,0:24:36.08,0:24:39.02,Default,,0000,0000,0000,,And you are right, it is normal.
Dialogue: 0,0:24:39.02,0:24:42.33,Default,,0000,0000,0000,,Principal unit normal.
Dialogue: 0,0:24:42.33,0:24:45.29,Default,,0000,0000,0000,,Remember this very\Nwell for your exam,
Dialogue: 0,0:24:45.29,0:24:48.27,Default,,0000,0000,0000,,because it's a very\Nimportant notion.
Dialogue: 0,0:24:48.27,0:24:50.10,Default,,0000,0000,0000,,How do I get to that?
Dialogue: 0,0:24:50.10,0:24:54.04,Default,,0000,0000,0000,,I take t, I differentiate\Nit, and I divide
Dialogue: 0,0:24:54.04,0:24:59.02,Default,,0000,0000,0000,,by the lengths of t prime.
Dialogue: 0,0:24:59.02,0:25:07.22,Default,,0000,0000,0000,,Now, can you prove to me\Nthat indeed this fellow
Dialogue: 0,0:25:07.22,0:25:09.71,Default,,0000,0000,0000,,is perpendicular to t?
Dialogue: 0,0:25:09.71,0:25:12.38,Default,,0000,0000,0000,,Can you do that?
Dialogue: 0,0:25:12.38,0:25:14.45,Default,,0000,0000,0000,,STUDENT: That n is\Nperpendicular to t?
Dialogue: 0,0:25:14.45,0:25:16.47,Default,,0000,0000,0000,,PROFESSOR: Mm-hmm.
Dialogue: 0,0:25:16.47,0:25:17.69,Default,,0000,0000,0000,,So a little exercise.
Dialogue: 0,0:25:17.69,0:25:22.63,Default,,0000,0000,0000,,
Dialogue: 0,0:25:22.63,0:25:30.59,Default,,0000,0000,0000,,Prove that-- Prove that I don't\Nhave a good marker anymore.
Dialogue: 0,0:25:30.59,0:25:37.51,Default,,0000,0000,0000,,Prove that n, the unit\Nprincipal vector field,
Dialogue: 0,0:25:37.51,0:25:44.84,Default,,0000,0000,0000,,is perpendicular-- you\Nsee, I'm a mathematician.
Dialogue: 0,0:25:44.84,0:25:48.69,Default,,0000,0000,0000,,I swear, I hate to write down\Nthe whole word perpendicular.
Dialogue: 0,0:25:48.69,0:25:51.61,Default,,0000,0000,0000,,I would love to\Nsay, perpendicular.
Dialogue: 0,0:25:51.61,0:25:57.84,Default,,0000,0000,0000,,That's how I write perpendicular\Nreally fast-- to t fore
Dialogue: 0,0:25:57.84,0:26:01.47,Default,,0000,0000,0000,,every value of t.
Dialogue: 0,0:26:01.47,0:26:03.01,Default,,0000,0000,0000,,For every value of t.
Dialogue: 0,0:26:03.01,0:26:03.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:26:03.91,0:26:06.34,Default,,0000,0000,0000,,How in the world can I do that?
Dialogue: 0,0:26:06.34,0:26:08.56,Default,,0000,0000,0000,,I have to think about it.
Dialogue: 0,0:26:08.56,0:26:11.82,Default,,0000,0000,0000,,This is hard.
Dialogue: 0,0:26:11.82,0:26:12.98,Default,,0000,0000,0000,,Wish me luck.
Dialogue: 0,0:26:12.98,0:26:15.80,Default,,0000,0000,0000,,So do I know\Nanything about Mr. t?
Dialogue: 0,0:26:15.80,0:26:18.15,Default,,0000,0000,0000,,What do I know about Mr. t?
Dialogue: 0,0:26:18.15,0:26:20.39,Default,,0000,0000,0000,,I'll take it and I'll\Ndifferentiate it later.
Dialogue: 0,0:26:20.39,0:26:25.05,Default,,0000,0000,0000,,It Mr. t is magic in the\Nsense that he's a unit vector.
Dialogue: 0,0:26:25.05,0:26:27.86,Default,,0000,0000,0000,,I'm going to write that down.
Dialogue: 0,0:26:27.86,0:26:31.91,Default,,0000,0000,0000,,t in absolute value equals 1.
Dialogue: 0,0:26:31.91,0:26:33.12,Default,,0000,0000,0000,,It's beautiful.
Dialogue: 0,0:26:33.12,0:26:36.92,Default,,0000,0000,0000,,If I squared that-- and\Nyou're going to say,
Dialogue: 0,0:26:36.92,0:26:38.60,Default,,0000,0000,0000,,why would you want\Nto square that?
Dialogue: 0,0:26:38.60,0:26:40.50,Default,,0000,0000,0000,,You're going to see in a minute.
Dialogue: 0,0:26:40.50,0:26:43.14,Default,,0000,0000,0000,,If I squared that,\Nthen I'm going
Dialogue: 0,0:26:43.14,0:26:50.58,Default,,0000,0000,0000,,to have the dot product\Nbetween t and itself equals 1.
Dialogue: 0,0:26:50.58,0:26:53.13,Default,,0000,0000,0000,,
Dialogue: 0,0:26:53.13,0:26:56.74,Default,,0000,0000,0000,,Can somebody tell me why the\Ndot product between t and itself
Dialogue: 0,0:26:56.74,0:27:00.85,Default,,0000,0000,0000,,is the square of a length of t?
Dialogue: 0,0:27:00.85,0:27:04.93,Default,,0000,0000,0000,,What's the definition\Nof the dot product?
Dialogue: 0,0:27:04.93,0:27:08.29,Default,,0000,0000,0000,,Magnitude of the first\Nvector, times the magnitude
Dialogue: 0,0:27:08.29,0:27:11.34,Default,,0000,0000,0000,,of the second vector--\Nthere i am already--
Dialogue: 0,0:27:11.34,0:27:15.41,Default,,0000,0000,0000,,times the cosine of the\Nangle between the two vectors
Dialogue: 0,0:27:15.41,0:27:17.48,Default,,0000,0000,0000,,Duh, that's 0.
Dialogue: 0,0:27:17.48,0:27:20.50,Default,,0000,0000,0000,,So cosine of 0 is 1, I'm done.
Dialogue: 0,0:27:20.50,0:27:21.39,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:27:21.39,0:27:26.58,Default,,0000,0000,0000,,Now, I have a vector function\Ntimes a vector function--
Dialogue: 0,0:27:26.58,0:27:31.27,Default,,0000,0000,0000,,this is crazy, right-- equals 1.
Dialogue: 0,0:27:31.27,0:27:34.25,Default,,0000,0000,0000,,I'm going to go ahead\Nand differentiate.
Dialogue: 0,0:27:34.25,0:27:37.69,Default,,0000,0000,0000,,Keep in mind that\Nthis is a product.
Dialogue: 0,0:27:37.69,0:27:39.77,Default,,0000,0000,0000,,What's the product?
Dialogue: 0,0:27:39.77,0:27:42.33,Default,,0000,0000,0000,,One of my professors,\Ncolleagues,
Dialogue: 0,0:27:42.33,0:27:44.98,Default,,0000,0000,0000,,was telling me, now,\Nlet's be serious.
Dialogue: 0,0:27:44.98,0:27:49.37,Default,,0000,0000,0000,,In five years, how many\Nof your engineering majors
Dialogue: 0,0:27:49.37,0:27:51.25,Default,,0000,0000,0000,,will remember the product?
Dialogue: 0,0:27:51.25,0:27:53.18,Default,,0000,0000,0000,,I really was\Nthinking about this.
Dialogue: 0,0:27:53.18,0:27:56.68,Default,,0000,0000,0000,,I hope everybody, if\Nthey were my students,
Dialogue: 0,0:27:56.68,0:27:59.18,Default,,0000,0000,0000,,because we are going to\Nhave enough practice.
Dialogue: 0,0:27:59.18,0:28:01.78,Default,,0000,0000,0000,,So the prime rule in\NCalc 1 said that if you
Dialogue: 0,0:28:01.78,0:28:05.12,Default,,0000,0000,0000,,have f of t times g of\Nt, you have a product.
Dialogue: 0,0:28:05.12,0:28:08.32,Default,,0000,0000,0000,,You prime that product,\Nand never write
Dialogue: 0,0:28:08.32,0:28:12.53,Default,,0000,0000,0000,,f prime times g prime unless you\Nwant me to call you around 2 AM
Dialogue: 0,0:28:12.53,0:28:15.26,Default,,0000,0000,0000,,to say you should never do that.
Dialogue: 0,0:28:15.26,0:28:20.10,Default,,0000,0000,0000,,
Dialogue: 0,0:28:20.10,0:28:23.76,Default,,0000,0000,0000,,So how does the\Nproduct rule work?
Dialogue: 0,0:28:23.76,0:28:27.50,Default,,0000,0000,0000,,The first one prime\Ntimes the second unprime
Dialogue: 0,0:28:27.50,0:28:32.31,Default,,0000,0000,0000,,plus the first one unprime\Ntimes the second prime.
Dialogue: 0,0:28:32.31,0:28:34.50,Default,,0000,0000,0000,,My students know\Nthe product rule.
Dialogue: 0,0:28:34.50,0:28:37.48,Default,,0000,0000,0000,,I don't care if the rest\Nof the world doesn't.
Dialogue: 0,0:28:37.48,0:28:40.07,Default,,0000,0000,0000,,I don't care about any\Ncommunity college who
Dialogue: 0,0:28:40.07,0:28:42.59,Default,,0000,0000,0000,,would say, I don't want the\Nproduct rule to be known,
Dialogue: 0,0:28:42.59,0:28:44.76,Default,,0000,0000,0000,,you can differentiate\Nwith a calculator.
Dialogue: 0,0:28:44.76,0:28:46.00,Default,,0000,0000,0000,,That's a no, no, no.
Dialogue: 0,0:28:46.00,0:28:50.11,Default,,0000,0000,0000,,You don't know calculus if you\Ndon't know the product rule.
Dialogue: 0,0:28:50.11,0:28:53.48,Default,,0000,0000,0000,,So the product rule is\Na blessing from God.
Dialogue: 0,0:28:53.48,0:28:58.02,Default,,0000,0000,0000,,It helps everywhere in physics,\Nin mechanics, in engineering.
Dialogue: 0,0:28:58.02,0:29:00.99,Default,,0000,0000,0000,,It really helps in\Ndifferential geometry
Dialogue: 0,0:29:00.99,0:29:03.76,Default,,0000,0000,0000,,with the directional\Nderivative, the Lie derivative.
Dialogue: 0,0:29:03.76,0:29:07.91,Default,,0000,0000,0000,,It helps you understand all\Nthe upper level mathematics.
Dialogue: 0,0:29:07.91,0:29:11.70,Default,,0000,0000,0000,,Now here you have t prime,\Nthe first prime times
Dialogue: 0,0:29:11.70,0:29:16.30,Default,,0000,0000,0000,,the second unprime, plus the\Nfirst unprime times the second
Dialogue: 0,0:29:16.30,0:29:17.39,Default,,0000,0000,0000,,prime.
Dialogue: 0,0:29:17.39,0:29:21.04,Default,,0000,0000,0000,,It's the same as for\Nregular scalar functions.
Dialogue: 0,0:29:21.04,0:29:23.75,Default,,0000,0000,0000,,What's the derivative of 1?
Dialogue: 0,0:29:23.75,0:29:24.25,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:29:24.25,0:29:25.61,Default,,0000,0000,0000,,PROFESSOR: 0.
Dialogue: 0,0:29:25.61,0:29:26.86,Default,,0000,0000,0000,,Look at this guy!
Dialogue: 0,0:29:26.86,0:29:29.27,Default,,0000,0000,0000,,Doesn't he look funny?
Dialogue: 0,0:29:29.27,0:29:32.63,Default,,0000,0000,0000,,It is the dot product community.
Dialogue: 0,0:29:32.63,0:29:34.26,Default,,0000,0000,0000,,Yes it is, by definition.
Dialogue: 0,0:29:34.26,0:29:40.12,Default,,0000,0000,0000,,So you have twice T\Ntimes T prime equals 0.
Dialogue: 0,0:29:40.12,0:29:44.05,Default,,0000,0000,0000,,This 2 is-- stinking\Nguy, let's divide by 2.
Dialogue: 0,0:29:44.05,0:29:45.08,Default,,0000,0000,0000,,Forget about that.
Dialogue: 0,0:29:45.08,0:29:46.59,Default,,0000,0000,0000,,What does this say?
Dialogue: 0,0:29:46.59,0:29:53.84,Default,,0000,0000,0000,,The dot product of T times--\NI mean by T prime is 0.
Dialogue: 0,0:29:53.84,0:29:57.22,Default,,0000,0000,0000,,When are two vectors\Ngiving you dot product 0?
Dialogue: 0,0:29:57.22,0:29:58.72,Default,,0000,0000,0000,,STUDENT: When they're\Nperpendicular.
Dialogue: 0,0:29:58.72,0:29:59.55,Default,,0000,0000,0000,,
Dialogue: 0,0:29:59.55,0:30:01.90,Default,,0000,0000,0000,,PROFESSOR: So if both\Nof them are non-zero,
Dialogue: 0,0:30:01.90,0:30:03.37,Default,,0000,0000,0000,,they have to be like that.
Dialogue: 0,0:30:03.37,0:30:06.33,Default,,0000,0000,0000,,They have to be like this,\Nperpendicular, right?
Dialogue: 0,0:30:06.33,0:30:11.76,Default,,0000,0000,0000,,So it follows that t has to\Nbe perpendicular to T prime.
Dialogue: 0,0:30:11.76,0:30:15.55,Default,,0000,0000,0000,,And now, that's why n\Nis perpendicular to t.
Dialogue: 0,0:30:15.55,0:30:19.37,Default,,0000,0000,0000,,But, because n is\Ncollinear to t prime.
Dialogue: 0,0:30:19.37,0:30:20.10,Default,,0000,0000,0000,,Hello.
Dialogue: 0,0:30:20.10,0:30:22.50,Default,,0000,0000,0000,,n is collinear to t prime.
Dialogue: 0,0:30:22.50,0:30:25.66,Default,,0000,0000,0000,,So this is t prime.
Dialogue: 0,0:30:25.66,0:30:27.84,Default,,0000,0000,0000,,Is t prime unitary?
Dialogue: 0,0:30:27.84,0:30:29.42,Default,,0000,0000,0000,,I'm going to measure it.
Dialogue: 0,0:30:29.42,0:30:30.72,Default,,0000,0000,0000,,No it's not.
Dialogue: 0,0:30:30.72,0:30:31.62,Default,,0000,0000,0000,,t prime.
Dialogue: 0,0:30:31.62,0:30:33.65,Default,,0000,0000,0000,,So if I want to\Nmake it unitary, I'm
Dialogue: 0,0:30:33.65,0:30:36.43,Default,,0000,0000,0000,,going to chop my-- no,\NI'm not going to chop.
Dialogue: 0,0:30:36.43,0:30:40.16,Default,,0000,0000,0000,,I just take it, t prime,\Nand divide by its magnitude.
Dialogue: 0,0:30:40.16,0:30:43.45,Default,,0000,0000,0000,,Then I'm going to get that\Nvector n, which is unitary.
Dialogue: 0,0:30:43.45,0:30:47.79,Default,,0000,0000,0000,,So from here it follows that t\Nand n are indeed perpendicular,
Dialogue: 0,0:30:47.79,0:30:52.67,Default,,0000,0000,0000,,and your colleague over there\Nsaid, hey, it has to be normal.
Dialogue: 0,0:30:52.67,0:30:55.14,Default,,0000,0000,0000,,That's perpendicular\Nto t, but which one?
Dialogue: 0,0:30:55.14,0:30:58.18,Default,,0000,0000,0000,,A special one, because\NI have many normals.
Dialogue: 0,0:30:58.18,0:31:02.31,Default,,0000,0000,0000,,Now, this special one is\Neasy to find like that.
Dialogue: 0,0:31:02.31,0:31:05.73,Default,,0000,0000,0000,,Where shall I put here--\NI'll draw him very nicely.
Dialogue: 0,0:31:05.73,0:31:08.88,Default,,0000,0000,0000,,
Dialogue: 0,0:31:08.88,0:31:10.00,Default,,0000,0000,0000,,I'll draw him.
Dialogue: 0,0:31:10.00,0:31:13.41,Default,,0000,0000,0000,,Now you guys have to\Nimagine-- am I drawing
Dialogue: 0,0:31:13.41,0:31:14.58,Default,,0000,0000,0000,,well enough for you?
Dialogue: 0,0:31:14.58,0:31:15.90,Default,,0000,0000,0000,,I don't even know.
Dialogue: 0,0:31:15.90,0:31:17.78,Default,,0000,0000,0000,,t and n should be perpendicular.
Dialogue: 0,0:31:17.78,0:31:21.96,Default,,0000,0000,0000,,Can you imagine them having that\N90 degree angle between them?
Dialogue: 0,0:31:21.96,0:31:22.46,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:31:22.46,0:31:26.71,Default,,0000,0000,0000,,Now there is a magic one that\Nyou don't even have to define.
Dialogue: 0,0:31:26.71,0:31:28.67,Default,,0000,0000,0000,,And yes sir?
Dialogue: 0,0:31:28.67,0:31:31.18,Default,,0000,0000,0000,,STUDENT: In this\Nthing, can [INAUDIBLE]
Dialogue: 0,0:31:31.18,0:31:34.41,Default,,0000,0000,0000,,this T vector [INAUDIBLE]\Nwritten by the definition
Dialogue: 0,0:31:34.41,0:31:36.17,Default,,0000,0000,0000,,thing?
Dialogue: 0,0:31:36.17,0:31:37.83,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,0:31:37.83,0:31:39.37,Default,,0000,0000,0000,,STUDENT: N vector\Ntimes the magnitude
Dialogue: 0,0:31:39.37,0:31:42.06,Default,,0000,0000,0000,,of t vector derivative?
Dialogue: 0,0:31:42.06,0:31:47.05,Default,,0000,0000,0000,,PROFESSOR: So\Ntechnically you have
Dialogue: 0,0:31:47.05,0:31:50.76,Default,,0000,0000,0000,,t prime would be the\Nmagnitude of t prime times n.
Dialogue: 0,0:31:50.76,0:31:51.64,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,0:31:51.64,0:31:54.44,Default,,0000,0000,0000,,PROFESSOR: But keep in mind\Nthat sometimes is tricky,
Dialogue: 0,0:31:54.44,0:31:57.31,Default,,0000,0000,0000,,because this is, in\Ngeneral, not a constant.
Dialogue: 0,0:31:57.31,0:31:59.47,Default,,0000,0000,0000,,Always keep it in mind,\Nit's not a constant.
Dialogue: 0,0:31:59.47,0:32:02.00,Default,,0000,0000,0000,,We'll have some examples later.
Dialogue: 0,0:32:02.00,0:32:04.69,Default,,0000,0000,0000,,There is a magic\Nguy called binormal.
Dialogue: 0,0:32:04.69,0:32:09.71,Default,,0000,0000,0000,,That binormal is the\Nnormal to both t and n.
Dialogue: 0,0:32:09.71,0:32:12.43,Default,,0000,0000,0000,,And he's defined as\Nt plus n because it's
Dialogue: 0,0:32:12.43,0:32:14.44,Default,,0000,0000,0000,,normal to both of them.
Dialogue: 0,0:32:14.44,0:32:18.37,Default,,0000,0000,0000,,So I'm going to write this\Nb vector is t cross n.
Dialogue: 0,0:32:18.37,0:32:22.00,Default,,0000,0000,0000,,Now I'm asking you to draw it.
Dialogue: 0,0:32:22.00,0:32:23.71,Default,,0000,0000,0000,,Can anybody come to\Nthe board and draw it
Dialogue: 0,0:32:23.71,0:32:26.96,Default,,0000,0000,0000,,for 0.01 extra credit?
Dialogue: 0,0:32:26.96,0:32:29.64,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:32:29.64,0:32:30.66,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:32:30.66,0:32:35.00,Default,,0000,0000,0000,,PROFESSOR: Draw that on the\Npicture like t and n, t and n,
Dialogue: 0,0:32:35.00,0:32:38.47,Default,,0000,0000,0000,,t is the-- who the heck\Nis t? t is the red one,
Dialogue: 0,0:32:38.47,0:32:40.86,Default,,0000,0000,0000,,and blue is the n.
Dialogue: 0,0:32:40.86,0:32:42.93,Default,,0000,0000,0000,,So does it go down or up?
Dialogue: 0,0:32:42.93,0:32:46.46,Default,,0000,0000,0000,,We should be perpendicular\Nto both of them.
Dialogue: 0,0:32:46.46,0:32:49.34,Default,,0000,0000,0000,,Is b unitary or not?
Dialogue: 0,0:32:49.34,0:32:51.82,Default,,0000,0000,0000,,If you have two unit vectors,\Nwill the cross product
Dialogue: 0,0:32:51.82,0:32:53.06,Default,,0000,0000,0000,,be a unit vector?
Dialogue: 0,0:32:53.06,0:32:56.37,Default,,0000,0000,0000,,
Dialogue: 0,0:32:56.37,0:33:00.20,Default,,0000,0000,0000,,Only if the two vectors\Nare perpendicular,
Dialogue: 0,0:33:00.20,0:33:04.68,Default,,0000,0000,0000,,it is going to be, right?
Dialogue: 0,0:33:04.68,0:33:12.42,Default,,0000,0000,0000,,So you have-- well, I\Nthink it goes that--
Dialogue: 0,0:33:12.42,0:33:13.76,Default,,0000,0000,0000,,in which direction does it go?
Dialogue: 0,0:33:13.76,0:33:14.62,Default,,0000,0000,0000,,Because
Dialogue: 0,0:33:14.62,0:33:16.36,Default,,0000,0000,0000,,STUDENT: It should\Nnot be how we have it.
Dialogue: 0,0:33:16.36,0:33:17.28,Default,,0000,0000,0000,,PROFESSOR: No, no, no.
Dialogue: 0,0:33:17.28,0:33:18.37,Default,,0000,0000,0000,,Because this is--
Dialogue: 0,0:33:18.37,0:33:19.27,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:33:19.27,0:33:19.82,Default,,0000,0000,0000,,I'm using--
Dialogue: 0,0:33:19.82,0:33:22.94,Default,,0000,0000,0000,,PROFESSOR: So t\Ngoes over n, so I'm
Dialogue: 0,0:33:22.94,0:33:27.47,Default,,0000,0000,0000,,going to try-- it is\Nlike that, sort of.
Dialogue: 0,0:33:27.47,0:33:28.71,Default,,0000,0000,0000,,STUDENT: Into the chord?
Dialogue: 0,0:33:28.71,0:33:31.45,Default,,0000,0000,0000,,PROFESSOR: So again, it's\Nnot very clear because
Dialogue: 0,0:33:31.45,0:33:33.86,Default,,0000,0000,0000,,of my stinking art, here.
Dialogue: 0,0:33:33.86,0:33:36.00,Default,,0000,0000,0000,,It's really not nice art.
Dialogue: 0,0:33:36.00,0:33:39.83,Default,,0000,0000,0000,,t, and this is n.
Dialogue: 0,0:33:39.83,0:33:43.54,Default,,0000,0000,0000,,And if I go t going over n.
Dialogue: 0,0:33:43.54,0:33:47.63,Default,,0000,0000,0000,,T going over n goes up or down?
Dialogue: 0,0:33:47.63,0:33:48.21,Default,,0000,0000,0000,,STUDENT: Down.
Dialogue: 0,0:33:48.21,0:33:49.08,Default,,0000,0000,0000,,PROFESSOR: Goes down.
Dialogue: 0,0:33:49.08,0:33:52.31,Default,,0000,0000,0000,,So it's going to look\Nmore like this, feet.
Dialogue: 0,0:33:52.31,0:33:54.67,Default,,0000,0000,0000,,Now guys, when we--\Nthank you so much.
Dialogue: 0,0:33:54.67,0:33:57.97,Default,,0000,0000,0000,,So you've like a\N0.01 extra credit.
Dialogue: 0,0:33:57.97,0:34:00.61,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:34:00.61,0:34:03.40,Default,,0000,0000,0000,,Tangent, normal, and\Nbinormal form a corner.
Dialogue: 0,0:34:03.40,0:34:04.32,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:34:04.32,0:34:07.46,Default,,0000,0000,0000,,STUDENT: Is rt-- rt is\Nthe function at the--
Dialogue: 0,0:34:07.46,0:34:09.16,Default,,0000,0000,0000,,for the flag that's flying?
Dialogue: 0,0:34:09.16,0:34:12.12,Default,,0000,0000,0000,,PROFESSOR: The r of t\Nis the position vector
Dialogue: 0,0:34:12.12,0:34:15.03,Default,,0000,0000,0000,,of the flag that was\Nflying that he was drunk.
Dialogue: 0,0:34:15.03,0:34:21.35,Default,,0000,0000,0000,,STUDENT: Why wasn't the\Nderivative of it perpendicular?
Dialogue: 0,0:34:21.35,0:34:24.42,Default,,0000,0000,0000,,Why isn't t perpendicular to rt?
Dialogue: 0,0:34:24.42,0:34:26.20,Default,,0000,0000,0000,,PROFESSOR: If--\Nwell, good question.
Dialogue: 0,0:34:26.20,0:34:28.77,Default,,0000,0000,0000,,
Dialogue: 0,0:34:28.77,0:34:30.60,Default,,0000,0000,0000,,We'll talk about it.
Dialogue: 0,0:34:30.60,0:34:34.69,Default,,0000,0000,0000,,If the length of r\Nwould be a constant,
Dialogue: 0,0:34:34.69,0:34:38.69,Default,,0000,0000,0000,,can we prove that r and r\Nprime are perpendicular?
Dialogue: 0,0:34:38.69,0:34:40.98,Default,,0000,0000,0000,,Let's do that as\Nanother exercise.
Dialogue: 0,0:34:40.98,0:34:42.88,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:34:42.88,0:34:46.19,Default,,0000,0000,0000,,So tnb looks like a corner.
Dialogue: 0,0:34:46.19,0:34:51.81,Default,,0000,0000,0000,,Look at the corner that the\Nvideo cannot see over there.
Dialogue: 0,0:34:51.81,0:34:53.66,Default,,0000,0000,0000,,TN and B are mutually octagonal.
Dialogue: 0,0:34:53.66,0:34:56.30,Default,,0000,0000,0000,,
Dialogue: 0,0:34:56.30,0:34:57.72,Default,,0000,0000,0000,,I'm going to draw them.
Dialogue: 0,0:34:57.72,0:35:00.81,Default,,0000,0000,0000,,This is an arbitrary\Npoint on a curve,
Dialogue: 0,0:35:00.81,0:35:04.41,Default,,0000,0000,0000,,and this is t, which is\Nalways tangent to the curve,
Dialogue: 0,0:35:04.41,0:35:05.61,Default,,0000,0000,0000,,and this is n.
Dialogue: 0,0:35:05.61,0:35:08.49,Default,,0000,0000,0000,,Let's say that's the\Nunit principle normal.
Dialogue: 0,0:35:08.49,0:35:11.11,Default,,0000,0000,0000,,And t cross n will\Ngo, again, down.
Dialogue: 0,0:35:11.11,0:35:11.65,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:35:11.65,0:35:14.56,Default,,0000,0000,0000,,I have an obsession\Nwith me going down.
Dialogue: 0,0:35:14.56,0:35:16.48,Default,,0000,0000,0000,,This is called the\NFrenet Trihedron.
Dialogue: 0,0:35:16.48,0:35:20.22,Default,,0000,0000,0000,,
Dialogue: 0,0:35:20.22,0:35:23.00,Default,,0000,0000,0000,,And I have a proposal\Nfor a problem
Dialogue: 0,0:35:23.00,0:35:34.95,Default,,0000,0000,0000,,that maybe I should give\Nmy students in the future.
Dialogue: 0,0:35:34.95,0:35:46.45,Default,,0000,0000,0000,,Show that for a circle,\Nplaying in space, I don't know.
Dialogue: 0,0:35:46.45,0:36:07.21,Default,,0000,0000,0000,,The position vector and the\Nvelocity vector are always how?
Dialogue: 0,0:36:07.21,0:36:07.71,Default,,0000,0000,0000,,Friends.
Dialogue: 0,0:36:07.71,0:36:09.14,Default,,0000,0000,0000,,Let's say friends.
Dialogue: 0,0:36:09.14,0:36:12.47,Default,,0000,0000,0000,,No, come on, I'm kidding.
Dialogue: 0,0:36:12.47,0:36:13.27,Default,,0000,0000,0000,,How are they?
Dialogue: 0,0:36:13.27,0:36:15.39,Default,,0000,0000,0000,,STUDENT: Perpendicular.
Dialogue: 0,0:36:15.39,0:36:16.64,Default,,0000,0000,0000,,PROFESSOR: How do you do that?
Dialogue: 0,0:36:16.64,0:36:18.36,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:36:18.36,0:36:20.88,Default,,0000,0000,0000,,We should be smart\Nenough to do that, right?
Dialogue: 0,0:36:20.88,0:36:21.98,Default,,0000,0000,0000,,I have a circle.
Dialogue: 0,0:36:21.98,0:36:26.38,Default,,0000,0000,0000,,That circle has what-- what\Nis the property of a circle?
Dialogue: 0,0:36:26.38,0:36:29.88,Default,,0000,0000,0000,,Euclid defined that-- this is\None of the axioms of Euclid.
Dialogue: 0,0:36:29.88,0:36:32.43,Default,,0000,0000,0000,,Does anybody know which axiom?
Dialogue: 0,0:36:32.43,0:36:35.79,Default,,0000,0000,0000,,That there exists\Nsuch a set of points
Dialogue: 0,0:36:35.79,0:36:38.66,Default,,0000,0000,0000,,that are all at the same\Ndistance from a given point
Dialogue: 0,0:36:38.66,0:36:40.63,Default,,0000,0000,0000,,called center.
Dialogue: 0,0:36:40.63,0:36:42.85,Default,,0000,0000,0000,,So that is a circle, right?
Dialogue: 0,0:36:42.85,0:36:44.27,Default,,0000,0000,0000,,That's what Mr. Euclid said.
Dialogue: 0,0:36:44.27,0:36:45.40,Default,,0000,0000,0000,,He was a genius.
Dialogue: 0,0:36:45.40,0:36:51.98,Default,,0000,0000,0000,,So no matter where I put that\Ncircle, I can take r of t
Dialogue: 0,0:36:51.98,0:36:55.29,Default,,0000,0000,0000,,in magnitude measured\Nfrom the origin
Dialogue: 0,0:36:55.29,0:36:57.19,Default,,0000,0000,0000,,from the center of the circle.
Dialogue: 0,0:36:57.19,0:37:01.20,Default,,0000,0000,0000,,Keep in mind, always the\Ncenter of the circle.
Dialogue: 0,0:37:01.20,0:37:06.00,Default,,0000,0000,0000,,I put it at the origin of the\Nspace-- origin of the universe.
Dialogue: 0,0:37:06.00,0:37:08.42,Default,,0000,0000,0000,,No, origin of the\Nspace, actually.
Dialogue: 0,0:37:08.42,0:37:12.84,Default,,0000,0000,0000,,R of T magnitude\Nwould be a constant.
Dialogue: 0,0:37:12.84,0:37:13.100,Default,,0000,0000,0000,,Give me a constant, guys.
Dialogue: 0,0:37:13.100,0:37:14.50,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:37:14.50,0:37:16.48,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:37:16.48,0:37:18.46,Default,,0000,0000,0000,,Let me draw.
Dialogue: 0,0:37:18.46,0:37:20.27,Default,,0000,0000,0000,,I want to draw in plane, OK?
Dialogue: 0,0:37:20.27,0:37:25.98,Default,,0000,0000,0000,,Because I'm getting tired.\Nx y, and this is r of t,
Dialogue: 0,0:37:25.98,0:37:30.21,Default,,0000,0000,0000,,and the magnitude of this r of\Nt is the radius of the circle.
Dialogue: 0,0:37:30.21,0:37:32.54,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:37:32.54,0:37:36.68,Default,,0000,0000,0000,,So let's say, this is\Nthe radius of the circle.
Dialogue: 0,0:37:36.68,0:37:40.88,Default,,0000,0000,0000,,
Dialogue: 0,0:37:40.88,0:37:43.93,Default,,0000,0000,0000,,How in the world do I\Nprove the same idea?
Dialogue: 0,0:37:43.93,0:37:47.98,Default,,0000,0000,0000,,Who helps me prove\Nthat r is always
Dialogue: 0,0:37:47.98,0:37:50.61,Default,,0000,0000,0000,,perpendicular to r prime?
Dialogue: 0,0:37:50.61,0:37:53.86,Default,,0000,0000,0000,,Which way do you want to move,\Ncounterclockwise or clockwise?
Dialogue: 0,0:37:53.86,0:37:54.94,Default,,0000,0000,0000,,STUDENT: Counterclockwise.
Dialogue: 0,0:37:54.94,0:37:56.11,Default,,0000,0000,0000,,PROFESSOR: Counterclockwise.
Dialogue: 0,0:37:56.11,0:37:58.16,Default,,0000,0000,0000,,Because if you are\Na real scientist,
Dialogue: 0,0:37:58.16,0:38:00.11,Default,,0000,0000,0000,,I'm proud of you guys.
Dialogue: 0,0:38:00.11,0:38:01.95,Default,,0000,0000,0000,,It's clear from the\Npicture that r prime
Dialogue: 0,0:38:01.95,0:38:04.79,Default,,0000,0000,0000,,would be perpendicular to r.
Dialogue: 0,0:38:04.79,0:38:05.82,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:38:05.82,0:38:07.59,Default,,0000,0000,0000,,How am I going to do that?
Dialogue: 0,0:38:07.59,0:38:11.51,Default,,0000,0000,0000,,Now, mimic everything I--\Ndon't look at your notes,
Dialogue: 0,0:38:11.51,0:38:17.26,Default,,0000,0000,0000,,and try to tell me how\NI show that quickly.
Dialogue: 0,0:38:17.26,0:38:18.29,Default,,0000,0000,0000,,What am I going to do?
Dialogue: 0,0:38:18.29,0:38:23.38,Default,,0000,0000,0000,,So all I know, all\Nthat gave me was r of t
Dialogue: 0,0:38:23.38,0:38:27.68,Default,,0000,0000,0000,,equals k in magnitude constant.
Dialogue: 0,0:38:27.68,0:38:30.53,Default,,0000,0000,0000,,For every t, this same constant.
Dialogue: 0,0:38:30.53,0:38:31.51,Default,,0000,0000,0000,,What's next?
Dialogue: 0,0:38:31.51,0:38:34.92,Default,,0000,0000,0000,,What do I want to do next?
Dialogue: 0,0:38:34.92,0:38:36.31,Default,,0000,0000,0000,,STUDENT: Square it?
Dialogue: 0,0:38:36.31,0:38:38.39,Default,,0000,0000,0000,,PROFESSOR: Square\Nit, differentiate it.
Dialogue: 0,0:38:38.39,0:38:40.45,Default,,0000,0000,0000,,I can also go ahead\Nand differentiate it
Dialogue: 0,0:38:40.45,0:38:41.100,Default,,0000,0000,0000,,without squaring\Nit, but that's going
Dialogue: 0,0:38:41.100,0:38:47.35,Default,,0000,0000,0000,,to be a little bit of more pain.
Dialogue: 0,0:38:47.35,0:38:52.17,Default,,0000,0000,0000,,So square it, differentiate it.
Dialogue: 0,0:38:52.17,0:38:53.39,Default,,0000,0000,0000,,I'm too lazy.
Dialogue: 0,0:38:53.39,0:38:56.14,Default,,0000,0000,0000,,When I differentiate,\Nwhat am I going to get?
Dialogue: 0,0:38:56.14,0:39:05.32,Default,,0000,0000,0000,,From the product rule, twice\Nr dot r primed of t equals 0.
Dialogue: 0,0:39:05.32,0:39:06.87,Default,,0000,0000,0000,,Well, I'm done.
Dialogue: 0,0:39:06.87,0:39:12.00,Default,,0000,0000,0000,,Because it means that for\Nevery t that radius-- not
Dialogue: 0,0:39:12.00,0:39:13.27,Default,,0000,0000,0000,,the radius, guys, I'm sorry.
Dialogue: 0,0:39:13.27,0:39:16.87,Default,,0000,0000,0000,,The position vector will be\Nperpendicular to the velocity
Dialogue: 0,0:39:16.87,0:39:17.56,Default,,0000,0000,0000,,vector.
Dialogue: 0,0:39:17.56,0:39:22.18,Default,,0000,0000,0000,,Now, if I draw the\Ntrajectory of my drunken flag
Dialogue: 0,0:39:22.18,0:39:24.66,Default,,0000,0000,0000,,this [INAUDIBLE]\Nis not true, right?
Dialogue: 0,0:39:24.66,0:39:27.43,Default,,0000,0000,0000,,This is crazy.
Dialogue: 0,0:39:27.43,0:39:29.92,Default,,0000,0000,0000,,Of course this is r,\Nand this is r prime,
Dialogue: 0,0:39:29.92,0:39:35.59,Default,,0000,0000,0000,,and there is an arbitrary\Nangle between r and r prime.
Dialogue: 0,0:39:35.59,0:39:38.26,Default,,0000,0000,0000,,The good thing is that\Nthe arbitrary angle always
Dialogue: 0,0:39:38.26,0:39:41.32,Default,,0000,0000,0000,,exists, and is\Ncontinuous as a function.
Dialogue: 0,0:39:41.32,0:39:43.45,Default,,0000,0000,0000,,I never have that\Nangle disappear.
Dialogue: 0,0:39:43.45,0:39:46.90,Default,,0000,0000,0000,,That's way I want that\Nprime never to become 0.
Dialogue: 0,0:39:46.90,0:39:49.33,Default,,0000,0000,0000,,Because if the bag was\Nstopping its motion,
Dialogue: 0,0:39:49.33,0:39:54.02,Default,,0000,0000,0000,,goodbye angle, goodbye\Nanalysis, right?
Dialogue: 0,0:39:54.02,0:39:54.73,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:39:54.73,0:39:55.54,Default,,0000,0000,0000,,Very nice.
Dialogue: 0,0:39:55.54,0:39:57.47,Default,,0000,0000,0000,,So don't give me more ideas.
Dialogue: 0,0:39:57.47,0:39:59.94,Default,,0000,0000,0000,,You smart people, if\Nyou give me more ideas,
Dialogue: 0,0:39:59.94,0:40:02.55,Default,,0000,0000,0000,,I'm going to come up with\Nall sorts of problems.
Dialogue: 0,0:40:02.55,0:40:05.00,Default,,0000,0000,0000,,And this is actually one\Nof the first problems
Dialogue: 0,0:40:05.00,0:40:08.98,Default,,0000,0000,0000,,you learn in a graduate\Nlevel geometry class.
Dialogue: 0,0:40:08.98,0:40:13.93,Default,,0000,0000,0000,,
Dialogue: 0,0:40:13.93,0:40:16.67,Default,,0000,0000,0000,,Let me give you another\Npiece of information
Dialogue: 0,0:40:16.67,0:40:20.24,Default,,0000,0000,0000,,that you're going\Nto love, which could
Dialogue: 0,0:40:20.24,0:40:22.39,Default,,0000,0000,0000,,be one of those\Ntypes of combined
Dialogue: 0,0:40:22.39,0:40:25.30,Default,,0000,0000,0000,,problems on a final\Nexam or midterm,
Dialogue: 0,0:40:25.30,0:40:29.90,Default,,0000,0000,0000,,A, B, C, D, E. The\Ncurvature of a curve
Dialogue: 0,0:40:29.90,0:40:33.93,Default,,0000,0000,0000,,is a measure of how\Nthe curve will bend.
Dialogue: 0,0:40:33.93,0:40:35.72,Default,,0000,0000,0000,,Say what?
Dialogue: 0,0:40:35.72,0:40:45.97,Default,,0000,0000,0000,,The curvature of a\Ncurve is a measure
Dialogue: 0,0:40:45.97,0:40:48.54,Default,,0000,0000,0000,,of the bending of that curve.
Dialogue: 0,0:40:48.54,0:40:58.80,Default,,0000,0000,0000,,
Dialogue: 0,0:40:58.80,0:41:04.38,Default,,0000,0000,0000,,By definition, you have\Nto take it like that.
Dialogue: 0,0:41:04.38,0:41:20.73,Default,,0000,0000,0000,,If the curve is parameterized\Nin arc length-- somebody
Dialogue: 0,0:41:20.73,0:41:23.07,Default,,0000,0000,0000,,remind me what that is.
Dialogue: 0,0:41:23.07,0:41:24.96,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:41:24.96,0:41:32.44,Default,,0000,0000,0000,,That is r of s such\Nthat-- what does it mean,
Dialogue: 0,0:41:32.44,0:41:33.64,Default,,0000,0000,0000,,parameterizing arc length--
Dialogue: 0,0:41:33.64,0:41:34.56,Default,,0000,0000,0000,,STUDENT: r prime of s.
Dialogue: 0,0:41:34.56,0:41:37.32,Default,,0000,0000,0000,,PROFESSOR: r primed of\Ns in magnitude is 1.
Dialogue: 0,0:41:37.32,0:41:37.91,Default,,0000,0000,0000,,The speed 1.
Dialogue: 0,0:41:37.91,0:41:39.20,Default,,0000,0000,0000,,It's a speed 1 curve.
Dialogue: 0,0:41:39.20,0:41:42.51,Default,,0000,0000,0000,,
Dialogue: 0,0:41:42.51,0:42:00.69,Default,,0000,0000,0000,,Then, the curvature of this\Ncurve is defined as k of s
Dialogue: 0,0:42:00.69,0:42:06.32,Default,,0000,0000,0000,,equals the magnitude of\Nthe acceleration vector
Dialogue: 0,0:42:06.32,0:42:09.46,Default,,0000,0000,0000,,will respect the S.\NSay what, Magdalena?
Dialogue: 0,0:42:09.46,0:42:12.89,Default,,0000,0000,0000,,I can also write\Nit magnitude of d--
Dialogue: 0,0:42:12.89,0:42:17.07,Default,,0000,0000,0000,,oh my gosh, second derivative\Nwith respect s of r.
Dialogue: 0,0:42:17.07,0:42:20.34,Default,,0000,0000,0000,,I'll do it right now. d2r ds2.
Dialogue: 0,0:42:20.34,0:42:21.91,Default,,0000,0000,0000,,And I know you get\Na headache when
Dialogue: 0,0:42:21.91,0:42:25.94,Default,,0000,0000,0000,,I solve, when I write that,\Nbecause you are not used to it.
Dialogue: 0,0:42:25.94,0:42:33.45,Default,,0000,0000,0000,,A quick and beautiful example\Nthat can be on the homework,
Dialogue: 0,0:42:33.45,0:42:39.38,Default,,0000,0000,0000,,and would also be on the\Nexam, maybe on all the exams,
Dialogue: 0,0:42:39.38,0:42:41.86,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:42:41.86,0:42:48.14,Default,,0000,0000,0000,,Compute the curvature of a\Ncircle of radius a Say what?
Dialogue: 0,0:42:48.14,0:43:02.67,Default,,0000,0000,0000,,Compute the curvature of a\Ncircle of radius a And you say,
Dialogue: 0,0:43:02.67,0:43:03.62,Default,,0000,0000,0000,,wait a minute.
Dialogue: 0,0:43:03.62,0:43:07.04,Default,,0000,0000,0000,,For a circle of\Nradius a in plane--
Dialogue: 0,0:43:07.04,0:43:08.99,Default,,0000,0000,0000,,why can I assume it's in plane?
Dialogue: 0,0:43:08.99,0:43:12.59,Default,,0000,0000,0000,,Because if the circle\Nis a planar curve,
Dialogue: 0,0:43:12.59,0:43:15.86,Default,,0000,0000,0000,,I can always assume\Nit to be in plane.
Dialogue: 0,0:43:15.86,0:43:19.03,Default,,0000,0000,0000,,And it has radius a I\Ncan find infinitely many
Dialogue: 0,0:43:19.03,0:43:20.16,Default,,0000,0000,0000,,parameterizations.
Dialogue: 0,0:43:20.16,0:43:23.10,Default,,0000,0000,0000,,So what, am I crazy?
Dialogue: 0,0:43:23.10,0:43:25.44,Default,,0000,0000,0000,,Well, yes, I am, but\Nthat's another story.
Dialogue: 0,0:43:25.44,0:43:28.33,Default,,0000,0000,0000,,Now, if I want to\Nparameterize, I
Dialogue: 0,0:43:28.33,0:43:31.17,Default,,0000,0000,0000,,have to parameterize\Nin arc length.
Dialogue: 0,0:43:31.17,0:43:34.02,Default,,0000,0000,0000,,If I do anything else,\Nthat means I'm stupid.
Dialogue: 0,0:43:34.02,0:43:38.96,Default,,0000,0000,0000,,So, r of s will be what?
Dialogue: 0,0:43:38.96,0:43:41.39,Default,,0000,0000,0000,,Can somebody tell me\Nhow I parameterize
Dialogue: 0,0:43:41.39,0:43:46.28,Default,,0000,0000,0000,,a curve in arc length\Nfor a-- what is this guy?
Dialogue: 0,0:43:46.28,0:43:49.21,Default,,0000,0000,0000,,A circle of radius a.
Dialogue: 0,0:43:49.21,0:43:50.52,Default,,0000,0000,0000,,Yeah, I cannot do it.
Dialogue: 0,0:43:50.52,0:43:52.47,Default,,0000,0000,0000,,I'm not smart enough.
Dialogue: 0,0:43:52.47,0:43:59.39,Default,,0000,0000,0000,,So I'll say R of T will be\Na cosine t, a sine t and 0.
Dialogue: 0,0:43:59.39,0:44:03.15,Default,,0000,0000,0000,,And here I stop, because\NI had a headache.
Dialogue: 0,0:44:03.15,0:44:08.51,Default,,0000,0000,0000,,t is from 0 to 2 pi, and\NI think this a is making
Dialogue: 0,0:44:08.51,0:44:14.98,Default,,0000,0000,0000,,my life miserable,\Nbecause it's telling me,
Dialogue: 0,0:44:14.98,0:44:17.41,Default,,0000,0000,0000,,you don't have\Nspeed 1, Magdalena.
Dialogue: 0,0:44:17.41,0:44:19.20,Default,,0000,0000,0000,,Drive to Amarillo\Nand back, you're
Dialogue: 0,0:44:19.20,0:44:22.05,Default,,0000,0000,0000,,not going to get speed 1.
Dialogue: 0,0:44:22.05,0:44:23.32,Default,,0000,0000,0000,,Why don't I have speed 1?
Dialogue: 0,0:44:23.32,0:44:23.95,Default,,0000,0000,0000,,Think about it.
Dialogue: 0,0:44:23.95,0:44:24.93,Default,,0000,0000,0000,,Bear with me.
Dialogue: 0,0:44:24.93,0:44:28.55,Default,,0000,0000,0000,,Minus a sine t equals sine t, 0.
Dialogue: 0,0:44:28.55,0:44:29.42,Default,,0000,0000,0000,,Bad.
Dialogue: 0,0:44:29.42,0:44:31.15,Default,,0000,0000,0000,,What is the speed?
Dialogue: 0,0:44:31.15,0:44:32.75,Default,,0000,0000,0000,,a.
Dialogue: 0,0:44:32.75,0:44:35.57,Default,,0000,0000,0000,,If you do the math,\Nthe speed will be a.
Dialogue: 0,0:44:35.57,0:44:39.65,Default,,0000,0000,0000,,So length of our\Nprime of t will be a.
Dialogue: 0,0:44:39.65,0:44:40.76,Default,,0000,0000,0000,,Somebody help me.
Dialogue: 0,0:44:40.76,0:44:42.30,Default,,0000,0000,0000,,Get me out of trouble.
Dialogue: 0,0:44:42.30,0:44:43.22,Default,,0000,0000,0000,,Who is this?
Dialogue: 0,0:44:43.22,0:44:45.40,Default,,0000,0000,0000,,I want to do it in arc length.
Dialogue: 0,0:44:45.40,0:44:48.25,Default,,0000,0000,0000,,Otherwise, how can\NI do the curvature?
Dialogue: 0,0:44:48.25,0:44:50.79,Default,,0000,0000,0000,,So somebody tell\Nme how to get to s.
Dialogue: 0,0:44:50.79,0:44:52.27,Default,,0000,0000,0000,,What the heck is that?
Dialogue: 0,0:44:52.27,0:44:58.61,Default,,0000,0000,0000,,s of t is integral from\N0 to t of-- who tells me?
Dialogue: 0,0:44:58.61,0:44:59.84,Default,,0000,0000,0000,,The speed, right?
Dialogue: 0,0:44:59.84,0:45:04.39,Default,,0000,0000,0000,,Was it not the displacement,\Nthe arc length traveled along,
Dialogue: 0,0:45:04.39,0:45:08.04,Default,,0000,0000,0000,,and the curve is integral\Nin time of the speed.
Dialogue: 0,0:45:08.04,0:45:11.95,Default,,0000,0000,0000,,
Dialogue: 0,0:45:11.95,0:45:13.03,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:45:13.03,0:45:16.42,Default,,0000,0000,0000,,So I have-- what is that?
Dialogue: 0,0:45:16.42,0:45:17.56,Default,,0000,0000,0000,,Speed is?
Dialogue: 0,0:45:17.56,0:45:18.36,Default,,0000,0000,0000,,STUDENT: Um--
Dialogue: 0,0:45:18.36,0:45:18.98,Default,,0000,0000,0000,,PROFESSOR: a.
Dialogue: 0,0:45:18.98,0:45:22.85,Default,,0000,0000,0000,,So a time t, am I right,\Nguys? s is a times t.
Dialogue: 0,0:45:22.85,0:45:24.65,Default,,0000,0000,0000,,So what do I have to do?
Dialogue: 0,0:45:24.65,0:45:30.95,Default,,0000,0000,0000,,Take Mr. t, shake his hand,\Nand replace him with s over a.
Dialogue: 0,0:45:30.95,0:45:31.81,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:45:31.81,0:45:39.74,Default,,0000,0000,0000,,So instead of r of t, I'll say--\Nwhat other letters do I have?
Dialogue: 0,0:45:39.74,0:45:40.48,Default,,0000,0000,0000,,Not r.
Dialogue: 0,0:45:40.48,0:45:41.44,Default,,0000,0000,0000,,Rho of s.
Dialogue: 0,0:45:41.44,0:45:42.40,Default,,0000,0000,0000,,I love rho.
Dialogue: 0,0:45:42.40,0:45:44.42,Default,,0000,0000,0000,,Rho is the Greek [INAUDIBLE].
Dialogue: 0,0:45:44.42,0:45:46.35,Default,,0000,0000,0000,,Is this finally an arc length?
Dialogue: 0,0:45:46.35,0:45:51.00,Default,,0000,0000,0000,,Cosine of-- what\Nis t, guys, again?
Dialogue: 0,0:45:51.00,0:45:53.14,Default,,0000,0000,0000,,s over a.
Dialogue: 0,0:45:53.14,0:45:58.22,Default,,0000,0000,0000,,s over a, a sine\Ns over a, and 0.
Dialogue: 0,0:45:58.22,0:46:01.71,Default,,0000,0000,0000,,This is the parameterization\Nin arc length.
Dialogue: 0,0:46:01.71,0:46:08.47,Default,,0000,0000,0000,,This is an arc length\Nparameterization of the circle.
Dialogue: 0,0:46:08.47,0:46:11.47,Default,,0000,0000,0000,,And then what is this\Ndefinition of curvature?
Dialogue: 0,0:46:11.47,0:46:13.76,Default,,0000,0000,0000,,It's here.
Dialogue: 0,0:46:13.76,0:46:16.93,Default,,0000,0000,0000,,Do that rho once, twice.
Dialogue: 0,0:46:16.93,0:46:19.51,Default,,0000,0000,0000,,Prime it twice,\Nand do the length.
Dialogue: 0,0:46:19.51,0:46:20.76,Default,,0000,0000,0000,,So rho prime.
Dialogue: 0,0:46:20.76,0:46:25.06,Default,,0000,0000,0000,,Oh my God is it hard.
Dialogue: 0,0:46:25.06,0:46:29.42,Default,,0000,0000,0000,,a times minus sine of s over a.
Dialogue: 0,0:46:29.42,0:46:30.27,Default,,0000,0000,0000,,Am I done, though?
Dialogue: 0,0:46:30.27,0:46:30.96,Default,,0000,0000,0000,,Chain rule.
Dialogue: 0,0:46:30.96,0:46:32.00,Default,,0000,0000,0000,,Pay attention, Magdalena.
Dialogue: 0,0:46:32.00,0:46:34.07,Default,,0000,0000,0000,,Don't screwed up with this one.
Dialogue: 0,0:46:34.07,0:46:36.37,Default,,0000,0000,0000,,1 over a.
Dialogue: 0,0:46:36.37,0:46:37.73,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:46:37.73,0:46:38.58,Default,,0000,0000,0000,,Next.
Dialogue: 0,0:46:38.58,0:46:41.72,Default,,0000,0000,0000,,a cosine of s over a.
Dialogue: 0,0:46:41.72,0:46:42.67,Default,,0000,0000,0000,,Chain rule.
Dialogue: 0,0:46:42.67,0:46:44.30,Default,,0000,0000,0000,,Don't forget,\Nmultiply by 1 over a.
Dialogue: 0,0:46:44.30,0:46:46.86,Default,,0000,0000,0000,,OK, that makes my life easier.
Dialogue: 0,0:46:46.86,0:46:47.82,Default,,0000,0000,0000,,We simplify.
Dialogue: 0,0:46:47.82,0:46:51.59,Default,,0000,0000,0000,,Thank God a simplifies\Nhere, a simplifies there,
Dialogue: 0,0:46:51.59,0:46:54.18,Default,,0000,0000,0000,,so that is that derivative.
Dialogue: 0,0:46:54.18,0:46:55.87,Default,,0000,0000,0000,,What's the second derivative?
Dialogue: 0,0:46:55.87,0:47:01.22,Default,,0000,0000,0000,,Rho double prime of s will\Nbe-- somebody help me, OK?
Dialogue: 0,0:47:01.22,0:47:02.72,Default,,0000,0000,0000,,Because this is a\Nlot of derivation.
Dialogue: 0,0:47:02.72,0:47:03.47,Default,,0000,0000,0000,,STUDENT: --cosine--
Dialogue: 0,0:47:03.47,0:47:04.55,Default,,0000,0000,0000,,PROFESSOR: Thank you, sir.
Dialogue: 0,0:47:04.55,0:47:06.81,Default,,0000,0000,0000,,Minus cosine of s over a.
Dialogue: 0,0:47:06.81,0:47:07.81,Default,,0000,0000,0000,,STUDENT: Times 1 over a.
Dialogue: 0,0:47:07.81,0:47:13.33,Default,,0000,0000,0000,,PROFESSOR: Times 1 over a,\Ncomma, minus sine of s over a.
Dialogue: 0,0:47:13.33,0:47:15.60,Default,,0000,0000,0000,,That's all I have left\Nin my life, right?
Dialogue: 0,0:47:15.60,0:47:19.86,Default,,0000,0000,0000,,Minus sine of s over a times\N1 over a from the chain rule.
Dialogue: 0,0:47:19.86,0:47:22.69,Default,,0000,0000,0000,,I have to pay attention and see.
Dialogue: 0,0:47:22.69,0:47:24.35,Default,,0000,0000,0000,,What's the magnitude of this?
Dialogue: 0,0:47:24.35,0:47:28.64,Default,,0000,0000,0000,,The magnitude of this of this\Nanimal will be the curvature.
Dialogue: 0,0:47:28.64,0:47:30.06,Default,,0000,0000,0000,,Oh, my God.
Dialogue: 0,0:47:30.06,0:47:32.32,Default,,0000,0000,0000,,So what is k?
Dialogue: 0,0:47:32.32,0:47:35.12,Default,,0000,0000,0000,,k of s will be--\Ncould somebody tell me
Dialogue: 0,0:47:35.12,0:47:39.61,Default,,0000,0000,0000,,what magnitude I get after I\Nsquare all these individuals,
Dialogue: 0,0:47:39.61,0:47:42.91,Default,,0000,0000,0000,,sum them up, and take\Nthe square root of them?
Dialogue: 0,0:47:42.91,0:47:44.36,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:47:44.36,0:47:49.55,Default,,0000,0000,0000,,PROFESSOR: Square root\Nof 1 over 1 squared.
Dialogue: 0,0:47:49.55,0:47:50.74,Default,,0000,0000,0000,,And I get 1 over a.
Dialogue: 0,0:47:50.74,0:47:53.24,Default,,0000,0000,0000,,You are too fast for\Nme, you teach me that.
Dialogue: 0,0:47:53.24,0:47:54.11,Default,,0000,0000,0000,,No, I'm just kidding.
Dialogue: 0,0:47:54.11,0:47:56.29,Default,,0000,0000,0000,,I knew it was 1 over a.
Dialogue: 0,0:47:56.29,0:47:59.32,Default,,0000,0000,0000,,Now, how did\Nengineers know that?
Dialogue: 0,0:47:59.32,0:48:02.17,Default,,0000,0000,0000,,Actually, for hundreds of years,\Nmathematicians, engineers,
Dialogue: 0,0:48:02.17,0:48:04.46,Default,,0000,0000,0000,,and physicists knew that.
Dialogue: 0,0:48:04.46,0:48:07.47,Default,,0000,0000,0000,,And that's the last thing\NI want to teach you today.
Dialogue: 0,0:48:07.47,0:48:09.91,Default,,0000,0000,0000,,We have two circles.
Dialogue: 0,0:48:09.91,0:48:17.39,Default,,0000,0000,0000,,This is of, let's say, radius\N1/2, and this is radius 2.
Dialogue: 0,0:48:17.39,0:48:21.03,Default,,0000,0000,0000,,The engineer, mathematician,\Nphysicist, whoever they are,
Dialogue: 0,0:48:21.03,0:48:26.46,Default,,0000,0000,0000,,they knew that the curvature\Nis inverse proportional
Dialogue: 0,0:48:26.46,0:48:28.25,Default,,0000,0000,0000,,to the radius.
Dialogue: 0,0:48:28.25,0:48:30.44,Default,,0000,0000,0000,,That radius is 1/2.
Dialogue: 0,0:48:30.44,0:48:33.52,Default,,0000,0000,0000,,The curvature will\Nbe 2 in this case.
Dialogue: 0,0:48:33.52,0:48:38.07,Default,,0000,0000,0000,,The radius is 2, the\Ncurvature will be 1/2.
Dialogue: 0,0:48:38.07,0:48:41.89,Default,,0000,0000,0000,,Does that make sense, this\Ninverse proportionality?
Dialogue: 0,0:48:41.89,0:48:45.66,Default,,0000,0000,0000,,The bigger the radius,\Nthe lesser the curvature,
Dialogue: 0,0:48:45.66,0:48:47.67,Default,,0000,0000,0000,,that less bent you are.
Dialogue: 0,0:48:47.67,0:48:49.60,Default,,0000,0000,0000,,The more fat-- well, OK.
Dialogue: 0,0:48:49.60,0:48:53.08,Default,,0000,0000,0000,,I'm not going to say anything\Npolitically incorrect.
Dialogue: 0,0:48:53.08,0:48:58.54,Default,,0000,0000,0000,,So this is really curved because\Nthe radius is really small.
Dialogue: 0,0:48:58.54,0:49:02.62,Default,,0000,0000,0000,,This less curved,\Nalmost-- at infinity,
Dialogue: 0,0:49:02.62,0:49:05.55,Default,,0000,0000,0000,,this curvature\Nbecomes 0, because
Dialogue: 0,0:49:05.55,0:49:09.08,Default,,0000,0000,0000,,at infinity, that radius\Nexplodes to plus infinity bag
Dialogue: 0,0:49:09.08,0:49:10.09,Default,,0000,0000,0000,,theory.
Dialogue: 0,0:49:10.09,0:49:13.60,Default,,0000,0000,0000,,Then you have 1 over\Ninfinity will be 0,
Dialogue: 0,0:49:13.60,0:49:19.06,Default,,0000,0000,0000,,and that will be the curvature\Nof a circle of infinite radius.
Dialogue: 0,0:49:19.06,0:49:20.42,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:49:20.42,0:49:22.53,Default,,0000,0000,0000,,So we learned something today.
Dialogue: 0,0:49:22.53,0:49:25.49,Default,,0000,0000,0000,,We learned about the\Ncurvature of a circle, which
Dialogue: 0,0:49:25.49,0:49:26.03,Default,,0000,0000,0000,,is something.
Dialogue: 0,0:49:26.03,0:49:30.93,Default,,0000,0000,0000,,But this is the same\Nway for any curve.
Dialogue: 0,0:49:30.93,0:49:31.84,Default,,0000,0000,0000,,You reparameterize.
Dialogue: 0,0:49:31.84,0:49:34.46,Default,,0000,0000,0000,,Now you understand why you need\Nto reparameterize in arc length
Dialogue: 0,0:49:34.46,0:49:35.88,Default,,0000,0000,0000,,s.
Dialogue: 0,0:49:35.88,0:49:37.85,Default,,0000,0000,0000,,You take the acceleration\Nin arc length.
Dialogue: 0,0:49:37.85,0:49:38.77,Default,,0000,0000,0000,,You get the magnitude.
Dialogue: 0,0:49:38.77,0:49:41.52,Default,,0000,0000,0000,,That measures how\Nbent the curve is.
Dialogue: 0,0:49:41.52,0:49:47.21,Default,,0000,0000,0000,,Next time, you're going to\Ndo how bent the helix is.
Dialogue: 0,0:49:47.21,0:49:47.75,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:49:47.75,0:49:49.44,Default,,0000,0000,0000,,At every point.
Dialogue: 0,0:49:49.44,0:49:51.24,Default,,0000,0000,0000,,Enjoy your WeBWorK homework.
Dialogue: 0,0:49:51.24,0:49:54.94,Default,,0000,0000,0000,,Ask me anytime, and\Nask me also Thursday.
Dialogue: 0,0:49:54.94,0:49:58.71,Default,,0000,0000,0000,,Do not have a block about\Nyour homework questions.
Dialogue: 0,0:49:58.71,0:50:05.12,Default,,0000,0000,0000,,You can ask me anytime\Nby email, or in person.
Dialogue: 0,0:50:05.12,0:50:10.49,Default,,0000,0000,0000,,