[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.51,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.51,0:00:04.89,Default,,0000,0000,0000,,I love this model.
Dialogue: 0,0:00:04.89,0:00:06.33,Default,,0000,0000,0000,,Again, thank you, Casey.
Dialogue: 0,0:00:06.33,0:00:09.86,Default,,0000,0000,0000,,I'm not going to take\Nany credit for that.
Dialogue: 0,0:00:09.86,0:00:11.63,Default,,0000,0000,0000,,So if you want to\Nimagine the stool
Dialogue: 0,0:00:11.63,0:00:16.72,Default,,0000,0000,0000,,I was talking about as\Na bamboo object, that
Dialogue: 0,0:00:16.72,0:00:21.11,Default,,0000,0000,0000,,is about the same thing,\Nat the same scale, compared
Dialogue: 0,0:00:21.11,0:00:28.13,Default,,0000,0000,0000,,to the diameter and the height,\Nscaled or dialated five times.
Dialogue: 0,0:00:28.13,0:00:30.45,Default,,0000,0000,0000,,Uniform, no alterations.
Dialogue: 0,0:00:30.45,0:00:35.04,Default,,0000,0000,0000,,And one can sit on it,\N[? and circle, ?] to sit on it.
Dialogue: 0,0:00:35.04,0:00:39.71,Default,,0000,0000,0000,,Now, as you see this is\Na doubly ruled surface.
Dialogue: 0,0:00:39.71,0:00:41.35,Default,,0000,0000,0000,,And you say, oh wait a minute.
Dialogue: 0,0:00:41.35,0:00:45.45,Default,,0000,0000,0000,,You said rule surface, why all\Nof a sudden, why doubly ruled
Dialogue: 0,0:00:45.45,0:00:46.21,Default,,0000,0000,0000,,surface?
Dialogue: 0,0:00:46.21,0:00:51.52,Default,,0000,0000,0000,,Because it is a surface\Nthat is ruled and generated
Dialogue: 0,0:00:51.52,0:00:58.15,Default,,0000,0000,0000,,by two different one\Nparameter families.
Dialogue: 0,0:00:58.15,0:00:59.69,Default,,0000,0000,0000,,Each of them has a\Ncertain parameter
Dialogue: 0,0:00:59.69,0:01:02.60,Default,,0000,0000,0000,,and that gives them continuity.
Dialogue: 0,0:01:02.60,0:01:04.32,Default,,0000,0000,0000,,So you have two\Nfamilies of lines.
Dialogue: 0,0:01:04.32,0:01:06.86,Default,,0000,0000,0000,,
Dialogue: 0,0:01:06.86,0:01:09.100,Default,,0000,0000,0000,,One family is in this direction.
Dialogue: 0,0:01:09.100,0:01:11.33,Default,,0000,0000,0000,,Do you see it?
Dialogue: 0,0:01:11.33,0:01:14.85,Default,,0000,0000,0000,,So these lines-- this\Nline is in motion.
Dialogue: 0,0:01:14.85,0:01:16.94,Default,,0000,0000,0000,,It moves to the right, to\Nthe right, to the right,
Dialogue: 0,0:01:16.94,0:01:19.97,Default,,0000,0000,0000,,and it generated.
Dialogue: 0,0:01:19.97,0:01:22.54,Default,,0000,0000,0000,,And the other family\Nof lines is this one
Dialogue: 0,0:01:22.54,0:01:24.63,Default,,0000,0000,0000,,in the other direction.
Dialogue: 0,0:01:24.63,0:01:28.23,Default,,0000,0000,0000,,You have a continuity\Nparameter for each of them.
Dialogue: 0,0:01:28.23,0:01:33.99,Default,,0000,0000,0000,,So you have to imagine\Nsome real parameter going
Dialogue: 0,0:01:33.99,0:01:36.75,Default,,0000,0000,0000,,along the entire\N[? infinite real ?] axis.
Dialogue: 0,0:01:36.75,0:01:40.46,Default,,0000,0000,0000,,Or along a circle which would\Nbe about the same thing.
Dialogue: 0,0:01:40.46,0:01:46.16,Default,,0000,0000,0000,,But in any case, you have\Na one parameter family
Dialogue: 0,0:01:46.16,0:01:49.13,Default,,0000,0000,0000,,and another one\Nparameter family.
Dialogue: 0,0:01:49.13,0:01:52.30,Default,,0000,0000,0000,,Both of them are\Ntogether generating
Dialogue: 0,0:01:52.30,0:01:55.98,Default,,0000,0000,0000,,this beautiful\None-sheeted hyperboloid.
Dialogue: 0,0:01:55.98,0:02:00.89,Default,,0000,0000,0000,,It's incredible because you\Nsee where these sort of round,
Dialogue: 0,0:02:00.89,0:02:07.08,Default,,0000,0000,0000,,but if you go towards the\Nends, it's topologically
Dialogue: 0,0:02:07.08,0:02:08.45,Default,,0000,0000,0000,,a cylinder or a tube.
Dialogue: 0,0:02:08.45,0:02:15.37,Default,,0000,0000,0000,,But if you look towards\Nthe end, the two ends
Dialogue: 0,0:02:15.37,0:02:18.81,Default,,0000,0000,0000,,will look more straight.
Dialogue: 0,0:02:18.81,0:02:23.79,Default,,0000,0000,0000,,And you will see the\Nstraight lines more clearly.
Dialogue: 0,0:02:23.79,0:02:27.68,Default,,0000,0000,0000,,So imagine that you\Nhave a continuation
Dialogue: 0,0:02:27.68,0:02:31.21,Default,,0000,0000,0000,,to infinity in this direction,\Nand in the other direction.
Dialogue: 0,0:02:31.21,0:02:36.11,Default,,0000,0000,0000,,And this actually should be an\Ninfinite surface in your model.
Dialogue: 0,0:02:36.11,0:02:38.55,Default,,0000,0000,0000,,You're just cutting it\Nbetween two z planes,
Dialogue: 0,0:02:38.55,0:02:41.100,Default,,0000,0000,0000,,so you have a patch of a\None-sheeted hyperboloid.
Dialogue: 0,0:02:41.100,0:02:43.37,Default,,0000,0000,0000,,Yeah, the one-sheeted\Nhyperboloid
Dialogue: 0,0:02:43.37,0:02:47.22,Default,,0000,0000,0000,,that we wrote last time,\Ndo you guys remember
Dialogue: 0,0:02:47.22,0:02:49.99,Default,,0000,0000,0000,,x squared over a\Nsquared plus y squared
Dialogue: 0,0:02:49.99,0:02:53.37,Default,,0000,0000,0000,,over b squared minus z squared?\Nz should be this [INAUDIBLE].
Dialogue: 0,0:02:53.37,0:02:56.43,Default,,0000,0000,0000,,Minus z squared over\Nc squared minus 1
Dialogue: 0,0:02:56.43,0:03:00.95,Default,,0000,0000,0000,,equals 0 is an\Ninfinite surface area.
Dialogue: 0,0:03:00.95,0:03:04.100,Default,,0000,0000,0000,,At both ends you keep going.
Dialogue: 0,0:03:04.100,0:03:06.48,Default,,0000,0000,0000,,Very beautiful.
Dialogue: 0,0:03:06.48,0:03:08.38,Default,,0000,0000,0000,,Thank you so much.
Dialogue: 0,0:03:08.38,0:03:09.37,Default,,0000,0000,0000,,I appreciate.
Dialogue: 0,0:03:09.37,0:03:12.24,Default,,0000,0000,0000,,And keep the brownies.
Dialogue: 0,0:03:12.24,0:03:14.03,Default,,0000,0000,0000,,No, then I have to pay more.
Dialogue: 0,0:03:14.03,0:03:16.70,Default,,0000,0000,0000,,Than I have to pay money.
Dialogue: 0,0:03:16.70,0:03:18.94,Default,,0000,0000,0000,,STUDENT: It's made\Nout of [INAUDIBLE].
Dialogue: 0,0:03:18.94,0:03:20.32,Default,,0000,0000,0000,,PROFESSOR: When\Nis your birthday?
Dialogue: 0,0:03:20.32,0:03:23.39,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,0:03:23.39,0:03:24.00,Default,,0000,0000,0000,,Really?
Dialogue: 0,0:03:24.00,0:03:24.14,Default,,0000,0000,0000,,When is it?
Dialogue: 0,0:03:24.14,0:03:25.01,Default,,0000,0000,0000,,STUDENT: February 29.
Dialogue: 0,0:03:25.01,0:03:27.98,Default,,0000,0000,0000,,PROFESSOR: Oh, it's coming.
Dialogue: 0,0:03:27.98,0:03:28.81,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:03:28.81,0:03:32.67,Default,,0000,0000,0000,,
Dialogue: 0,0:03:32.67,0:03:35.56,Default,,0000,0000,0000,,STUDENT: It's coming\Nin a year, too.
Dialogue: 0,0:03:35.56,0:03:38.96,Default,,0000,0000,0000,,PROFESSOR: That was a smart one.
Dialogue: 0,0:03:38.96,0:03:41.12,Default,,0000,0000,0000,,Anyway, I'll remember that.
Dialogue: 0,0:03:41.12,0:03:42.87,Default,,0000,0000,0000,,I appreciate the gift very much.
Dialogue: 0,0:03:42.87,0:03:45.47,Default,,0000,0000,0000,,And I will cherish\Nit and I'll use it
Dialogue: 0,0:03:45.47,0:03:46.94,Default,,0000,0000,0000,,with both my\Nundergraduate students
Dialogue: 0,0:03:46.94,0:03:50.38,Default,,0000,0000,0000,,and my graduate students who\Nare just learning about-- some
Dialogue: 0,0:03:50.38,0:03:54.10,Default,,0000,0000,0000,,of them don't know the\None-sheeted hyperboloid model,
Dialogue: 0,0:03:54.10,0:03:57.13,Default,,0000,0000,0000,,but they will learn about it.
Dialogue: 0,0:03:57.13,0:04:00.08,Default,,0000,0000,0000,,Coming back to our lesson.
Dialogue: 0,0:04:00.08,0:04:05.02,Default,,0000,0000,0000,,I announced Section 10.1.
Dialogue: 0,0:04:05.02,0:04:06.78,Default,,0000,0000,0000,,Say goodbye to\Nquadrant for a while.
Dialogue: 0,0:04:06.78,0:04:09.64,Default,,0000,0000,0000,,I know you love them,\Nbut they will be there
Dialogue: 0,0:04:09.64,0:04:11.45,Default,,0000,0000,0000,,for you in Chapter 11.
Dialogue: 0,0:04:11.45,0:04:13.10,Default,,0000,0000,0000,,They will wait for you.
Dialogue: 0,0:04:13.10,0:04:20.58,Default,,0000,0000,0000,,Now, let's go to Section\N10.1 of Chapter 10.
Dialogue: 0,0:04:20.58,0:04:23.08,Default,,0000,0000,0000,,Chapter 10 is a\Nbeautiful chapter.
Dialogue: 0,0:04:23.08,0:04:26.58,Default,,0000,0000,0000,,As you know very well,\NI announced last time,
Dialogue: 0,0:04:26.58,0:04:29.17,Default,,0000,0000,0000,,it is about\Nvector-valued functions.
Dialogue: 0,0:04:29.17,0:04:40.80,Default,,0000,0000,0000,,
Dialogue: 0,0:04:40.80,0:04:43.22,Default,,0000,0000,0000,,And you say, oh\Nmy god, I've never
Dialogue: 0,0:04:43.22,0:04:45.68,Default,,0000,0000,0000,,heard about vector-valued\Nfunctions before.
Dialogue: 0,0:04:45.68,0:04:48.62,Default,,0000,0000,0000,,You deal with them every day.
Dialogue: 0,0:04:48.62,0:04:51.57,Default,,0000,0000,0000,,Every time you move,\Nyou are dealing
Dialogue: 0,0:04:51.57,0:04:54.79,Default,,0000,0000,0000,,with a vector-valued\Nfunction, which
Dialogue: 0,0:04:54.79,0:05:01.64,Default,,0000,0000,0000,,is the displacement, which\Ntakes values in a subset in R3.
Dialogue: 0,0:05:01.64,0:05:06.82,Default,,0000,0000,0000,,So let's try and see what\Nyou should understand
Dialogue: 0,0:05:06.82,0:05:10.02,Default,,0000,0000,0000,,when you start Section 10.1.
Dialogue: 0,0:05:10.02,0:05:14.38,Default,,0000,0000,0000,,Because the book is pretty\Ngood, not that I'm a co-author.
Dialogue: 0,0:05:14.38,0:05:18.88,Default,,0000,0000,0000,,But it was meant to be really\Nwritten for the students
Dialogue: 0,0:05:18.88,0:05:22.51,Default,,0000,0000,0000,,and explain concepts\Nreally well.
Dialogue: 0,0:05:22.51,0:05:26.38,Default,,0000,0000,0000,,How many of you took physics?
Dialogue: 0,0:05:26.38,0:05:29.02,Default,,0000,0000,0000,,OK, quite a lot of\Nyou took physics.
Dialogue: 0,0:05:29.02,0:05:33.84,Default,,0000,0000,0000,,Now, one of my students\Nin a previous honors class
Dialogue: 0,0:05:33.84,0:05:38.48,Default,,0000,0000,0000,,told me he enjoyed my\Nclass greatly in general.
Dialogue: 0,0:05:38.48,0:05:41.48,Default,,0000,0000,0000,,The most [INAUDIBLE] thing\Nhe had from my class, he
Dialogue: 0,0:05:41.48,0:05:45.75,Default,,0000,0000,0000,,learned from my class was the\Nmotion of the drunken bug.
Dialogue: 0,0:05:45.75,0:05:47.88,Default,,0000,0000,0000,,And I said, did I say that?
Dialogue: 0,0:05:47.88,0:05:49.69,Default,,0000,0000,0000,,Absolutely, you said that.
Dialogue: 0,0:05:49.69,0:05:54.41,Default,,0000,0000,0000,,So apparently I had\Nstarted one of my lessons
Dialogue: 0,0:05:54.41,0:05:59.83,Default,,0000,0000,0000,,with imagine you have a fly\Nwho went into your coffee mug.
Dialogue: 0,0:05:59.83,0:06:00.50,Default,,0000,0000,0000,,I think I did.
Dialogue: 0,0:06:00.50,0:06:03.52,Default,,0000,0000,0000,,He reproduced the whole\Nthing the way I said it.
Dialogue: 0,0:06:03.52,0:06:05.63,Default,,0000,0000,0000,,It was quite spontaneous.
Dialogue: 0,0:06:05.63,0:06:10.43,Default,,0000,0000,0000,,So imagine your coffee mug had\Nsome Baileys Irish Creme in it.
Dialogue: 0,0:06:10.43,0:06:15.83,Default,,0000,0000,0000,,And the fly was really\Nhappy after she got up.
Dialogue: 0,0:06:15.83,0:06:17.89,Default,,0000,0000,0000,,She managed to get up.
Dialogue: 0,0:06:17.89,0:06:21.26,Default,,0000,0000,0000,,And the trajectory of the\Nfly was something more
Dialogue: 0,0:06:21.26,0:06:23.33,Default,,0000,0000,0000,,like a helix.
Dialogue: 0,0:06:23.33,0:06:25.79,Default,,0000,0000,0000,,And this is how I actually\Nintroduced the helix
Dialogue: 0,0:06:25.79,0:06:27.28,Default,,0000,0000,0000,,in my classroom.
Dialogue: 0,0:06:27.28,0:06:29.83,Default,,0000,0000,0000,,And I thought, OK,\Nis that unusual?
Dialogue: 0,0:06:29.83,0:06:30.33,Default,,0000,0000,0000,,Very.
Dialogue: 0,0:06:30.33,0:06:33.20,Default,,0000,0000,0000,,And I said, but that's\Nan honors class.
Dialogue: 0,0:06:33.20,0:06:36.08,Default,,0000,0000,0000,,Everything is supposed\Nto be unusual, right?
Dialogue: 0,0:06:36.08,0:06:50.19,Default,,0000,0000,0000,,So let's think about the\Nposition vector or some sort
Dialogue: 0,0:06:50.19,0:06:53.63,Default,,0000,0000,0000,,of vector-valued function that\Nyou're familiar with already
Dialogue: 0,0:06:53.63,0:06:55.41,Default,,0000,0000,0000,,from physics.
Dialogue: 0,0:06:55.41,0:06:57.49,Default,,0000,0000,0000,,He is one of your best friends.
Dialogue: 0,0:06:57.49,0:07:01.18,Default,,0000,0000,0000,,You have a function r of t.
Dialogue: 0,0:07:01.18,0:07:10.12,Default,,0000,0000,0000,,And I will point out that r is\Npractically the position vector
Dialogue: 0,0:07:10.12,0:07:16.00,Default,,0000,0000,0000,,measure that time t, or\Nobserved at time t in R3.
Dialogue: 0,0:07:16.00,0:07:19.15,Default,,0000,0000,0000,,So he takes values in R3.
Dialogue: 0,0:07:19.15,0:07:20.07,Default,,0000,0000,0000,,How?
Dialogue: 0,0:07:20.07,0:07:23.93,Default,,0000,0000,0000,,As the mathematician, because\NI like to write mathematically
Dialogue: 0,0:07:23.93,0:07:26.74,Default,,0000,0000,0000,,all the notion I\Nhave, r is defined
Dialogue: 0,0:07:26.74,0:07:34.47,Default,,0000,0000,0000,,on I was a sub-interval\Nof R with values in R3.
Dialogue: 0,0:07:34.47,0:07:38.70,Default,,0000,0000,0000,,And he asked me, my student\Nsaid, what is this I?
Dialogue: 0,0:07:38.70,0:07:40.85,Default,,0000,0000,0000,,Well, this I could\Nbe any interval,
Dialogue: 0,0:07:40.85,0:07:43.29,Default,,0000,0000,0000,,but let's assume for\Nthe time being it's
Dialogue: 0,0:07:43.29,0:07:46.41,Default,,0000,0000,0000,,just an open\Ninterval of the type
Dialogue: 0,0:07:46.41,0:07:52.18,Default,,0000,0000,0000,,a, b, where a and b are\Nreal numbers, a less than b.
Dialogue: 0,0:07:52.18,0:07:58.31,Default,,0000,0000,0000,,So this is practically the time\Nfor my bug from the moment,
Dialogue: 0,0:07:58.31,0:08:03.36,Default,,0000,0000,0000,,let's say a equals 0 when\Nshe or he starts flying up,
Dialogue: 0,0:08:03.36,0:08:06.92,Default,,0000,0000,0000,,until the moment she\Ncompletely freaks
Dialogue: 0,0:08:06.92,0:08:11.74,Default,,0000,0000,0000,,out or drops from the\Nmaximum point she reached.
Dialogue: 0,0:08:11.74,0:08:13.40,Default,,0000,0000,0000,,And she eventually dies.
Dialogue: 0,0:08:13.40,0:08:15.34,Default,,0000,0000,0000,,Or maybe she doesn't die.
Dialogue: 0,0:08:15.34,0:08:19.28,Default,,0000,0000,0000,,Maybe she's just drunk and she\Nwill wake up after a while.
Dialogue: 0,0:08:19.28,0:08:24.40,Default,,0000,0000,0000,,OK, so what do I mean by\Nthis displacement vector?
Dialogue: 0,0:08:24.40,0:08:25.50,Default,,0000,0000,0000,,I mean, a function--
Dialogue: 0,0:08:25.50,0:08:26.48,Default,,0000,0000,0000,,STUDENT: Is that Tc?
Dialogue: 0,0:08:26.48,0:08:27.48,Default,,0000,0000,0000,,Do you have [INAUDIBLE]?
Dialogue: 0,0:08:27.48,0:08:28.77,Default,,0000,0000,0000,,PROFESSOR: This is r, little r.
Dialogue: 0,0:08:28.77,0:08:30.51,Default,,0000,0000,0000,,STUDENT: I know, but the Tc.
Dialogue: 0,0:08:30.51,0:08:32.01,Default,,0000,0000,0000,,PROFESSOR: Tc?
Dialogue: 0,0:08:32.01,0:08:33.43,Default,,0000,0000,0000,,STUDENT: Or is that an I?
Dialogue: 0,0:08:33.43,0:08:34.01,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,0:08:34.01,0:08:37.46,Default,,0000,0000,0000,,This is I interval,\Nwhich is the same as a,
Dialogue: 0,0:08:37.46,0:08:41.72,Default,,0000,0000,0000,,b open interval, like\Nfrom 2 to 7, included.
Dialogue: 0,0:08:41.72,0:08:46.18,Default,,0000,0000,0000,,This is inclusion\N[INAUDIBLE] included in R.
Dialogue: 0,0:08:46.18,0:08:51.52,Default,,0000,0000,0000,,So I mean R is the real number\Nset and a, b is my interval.
Dialogue: 0,0:08:51.52,0:08:55.01,Default,,0000,0000,0000,,
Dialogue: 0,0:08:55.01,0:08:58.46,Default,,0000,0000,0000,,OK, so r of t is\Ngoing to be what?
Dialogue: 0,0:08:58.46,0:09:01.68,Default,,0000,0000,0000,,x of t, y of t, z of t.
Dialogue: 0,0:09:01.68,0:09:05.18,Default,,0000,0000,0000,,The book tells you, hey, guys--\Nit doesn't say hey, guys,
Dialogue: 0,0:09:05.18,0:09:09.80,Default,,0000,0000,0000,,but it's quite informal-- if\Nyou live in Rn, if your image is
Dialogue: 0,0:09:09.80,0:09:12.72,Default,,0000,0000,0000,,in Rn, instead of x\Nof t, y of t, z of t,
Dialogue: 0,0:09:12.72,0:09:19.03,Default,,0000,0000,0000,,you are going to get something\Nlike x1 of t, y1 of t.
Dialogue: 0,0:09:19.03,0:09:23.56,Default,,0000,0000,0000,,x1 of t, x2 of t,\Nx3 of t, et cetera.
Dialogue: 0,0:09:23.56,0:09:26.48,Default,,0000,0000,0000,,What do we assume about R?
Dialogue: 0,0:09:26.48,0:09:28.59,Default,,0000,0000,0000,,We have to assume\Nsomething about it, right?
Dialogue: 0,0:09:28.59,0:09:30.13,Default,,0000,0000,0000,,STUDENT: It's a\Nfunction [INAUDIBLE].
Dialogue: 0,0:09:30.13,0:09:33.34,Default,,0000,0000,0000,,PROFESSOR: It's a function\Nthat is differentiable
Dialogue: 0,0:09:33.34,0:09:36.63,Default,,0000,0000,0000,,most of the times, right?
Dialogue: 0,0:09:36.63,0:09:37.90,Default,,0000,0000,0000,,What does it mean smooth?
Dialogue: 0,0:09:37.90,0:09:42.91,Default,,0000,0000,0000,,I saw that your books\Nbefore college level
Dialogue: 0,0:09:42.91,0:09:44.30,Default,,0000,0000,0000,,never mention smooth.
Dialogue: 0,0:09:44.30,0:09:48.55,Default,,0000,0000,0000,,A smooth function is a\Nfunction that is differentiable
Dialogue: 0,0:09:48.55,0:09:51.40,Default,,0000,0000,0000,,and whose first\Nderivative is continuous.
Dialogue: 0,0:09:51.40,0:09:55.71,Default,,0000,0000,0000,,Some mathematicians even assume\Nthat you have c infinity, which
Dialogue: 0,0:09:55.71,0:10:00.32,Default,,0000,0000,0000,,means you have a function that's\Ninfinitely many differentiable.
Dialogue: 0,0:10:00.32,0:10:02.57,Default,,0000,0000,0000,,So you have first derivative,\Nsecond derivative, third
Dialogue: 0,0:10:02.57,0:10:03.78,Default,,0000,0000,0000,,derivative, fifth derivative.
Dialogue: 0,0:10:03.78,0:10:05.06,Default,,0000,0000,0000,,Somebody stop me.
Dialogue: 0,0:10:05.06,0:10:09.24,Default,,0000,0000,0000,,All the derivatives exist\Nand they are all continuous.
Dialogue: 0,0:10:09.24,0:10:13.27,Default,,0000,0000,0000,,By smooth, I will\Nassume c1 in this case.
Dialogue: 0,0:10:13.27,0:10:16.00,Default,,0000,0000,0000,,I know it's not accurate,\Nbut let's assume c1.
Dialogue: 0,0:10:16.00,0:10:18.74,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:10:18.74,0:10:23.66,Default,,0000,0000,0000,,Differentiable function whose\Nderivative is continuous.
Dialogue: 0,0:10:23.66,0:10:30.94,Default,,0000,0000,0000,,
Dialogue: 0,0:10:30.94,0:10:35.22,Default,,0000,0000,0000,,And I will assume\None more thing.
Dialogue: 0,0:10:35.22,0:10:37.36,Default,,0000,0000,0000,,That is not enough for me.
Dialogue: 0,0:10:37.36,0:10:41.90,Default,,0000,0000,0000,,I will also assume that\Nr prime of t in this case
Dialogue: 0,0:10:41.90,0:10:49.70,Default,,0000,0000,0000,,is different from 0 for\Nevery t in the interval I.
Dialogue: 0,0:10:49.70,0:10:53.78,Default,,0000,0000,0000,,Could somebody tell me in\Neveryday words what that means?
Dialogue: 0,0:10:53.78,0:10:55.74,Default,,0000,0000,0000,,We call that regular function.
Dialogue: 0,0:10:55.74,0:10:56.24,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:10:56.24,0:11:00.24,Default,,0000,0000,0000,,
Dialogue: 0,0:11:00.24,0:11:01.77,Default,,0000,0000,0000,,You have a brownie [INAUDIBLE].
Dialogue: 0,0:11:01.77,0:11:03.39,Default,,0000,0000,0000,,I have no brownies with me.
Dialogue: 0,0:11:03.39,0:11:05.55,Default,,0000,0000,0000,,But if you answer, so what--
Dialogue: 0,0:11:05.55,0:11:08.22,Default,,0000,0000,0000,,STUDENT: So that means you've\Ngot no relative mins or maxes,
Dialogue: 0,0:11:08.22,0:11:11.40,Default,,0000,0000,0000,,and you never-- the\Nobject never stops moving.
Dialogue: 0,0:11:11.40,0:11:15.25,Default,,0000,0000,0000,,PROFESSOR: Well, actually,\Nyou can have relative mins
Dialogue: 0,0:11:15.25,0:11:18.47,Default,,0000,0000,0000,,and maxes in some way.
Dialogue: 0,0:11:18.47,0:11:22.63,Default,,0000,0000,0000,,I'm talking about something\Nlike that, r prime.
Dialogue: 0,0:11:22.63,0:11:28.33,Default,,0000,0000,0000,,
Dialogue: 0,0:11:28.33,0:11:29.85,Default,,0000,0000,0000,,This is r of t.
Dialogue: 0,0:11:29.85,0:11:33.38,Default,,0000,0000,0000,,And r prime of t\Nis the derivative.
Dialogue: 0,0:11:33.38,0:11:34.98,Default,,0000,0000,0000,,It's never going to stop.
Dialogue: 0,0:11:34.98,0:11:36.00,Default,,0000,0000,0000,,The velocity.
Dialogue: 0,0:11:36.00,0:11:38.03,Default,,0000,0000,0000,,I'm talking about this\Npiece of information.
Dialogue: 0,0:11:38.03,0:11:42.49,Default,,0000,0000,0000,,Velocity [INAUDIBLE] 0 means\Nthat drunken bug between time
Dialogue: 0,0:11:42.49,0:11:45.83,Default,,0000,0000,0000,,a and time b never stops.
Dialogue: 0,0:11:45.83,0:11:50.62,Default,,0000,0000,0000,,He stops at the end, but the end\Nis b, is outside [INAUDIBLE].
Dialogue: 0,0:11:50.62,0:11:54.74,Default,,0000,0000,0000,,So he stops at b and he falls.
Dialogue: 0,0:11:54.74,0:11:56.11,Default,,0000,0000,0000,,So I don't stop.
Dialogue: 0,0:11:56.11,0:11:59.08,Default,,0000,0000,0000,,I move on from time a to time b.
Dialogue: 0,0:11:59.08,0:12:01.39,Default,,0000,0000,0000,,I don't stop at all.
Dialogue: 0,0:12:01.39,0:12:02.83,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:12:02.83,0:12:06.63,Default,,0000,0000,0000,,STUDENT: Wouldn't the derivative\Nof that line at some point
Dialogue: 0,0:12:06.63,0:12:08.03,Default,,0000,0000,0000,,equal 0 where it flattens out?
Dialogue: 0,0:12:08.03,0:12:10.89,Default,,0000,0000,0000,,PROFESSOR: Let me\Ndraw very well.
Dialogue: 0,0:12:10.89,0:12:14.60,Default,,0000,0000,0000,,So at time r of t, this\Nis the position vector.
Dialogue: 0,0:12:14.60,0:12:16.55,Default,,0000,0000,0000,,What is the derivative?
Dialogue: 0,0:12:16.55,0:12:19.58,Default,,0000,0000,0000,,The derivative represents\Nthe velocity vector.
Dialogue: 0,0:12:19.58,0:12:24.25,Default,,0000,0000,0000,,A beautiful thing about the\Nvelocity vector r prime of t
Dialogue: 0,0:12:24.25,0:12:26.91,Default,,0000,0000,0000,,is that it has a\Nbeautiful property.
Dialogue: 0,0:12:26.91,0:12:30.13,Default,,0000,0000,0000,,It's always tangent\Nto the trajectory.
Dialogue: 0,0:12:30.13,0:12:32.61,Default,,0000,0000,0000,,So at every point\Nyou're going to have
Dialogue: 0,0:12:32.61,0:12:36.47,Default,,0000,0000,0000,,a velocity vector that is\Ntangent to the trajectory.
Dialogue: 0,0:12:36.47,0:12:37.95,Default,,0000,0000,0000,,[INAUDIBLE] in physics.
Dialogue: 0,0:12:37.95,0:12:41.90,Default,,0000,0000,0000,,This r prime of t\Nshould never become 0.
Dialogue: 0,0:12:41.90,0:12:46.59,Default,,0000,0000,0000,,So you will never have a\Npoint instead of a segment
Dialogue: 0,0:12:46.59,0:12:51.07,Default,,0000,0000,0000,,when it comes to r prime.
Dialogue: 0,0:12:51.07,0:12:52.00,Default,,0000,0000,0000,,So you don't stop.
Dialogue: 0,0:12:52.00,0:12:58.56,Default,,0000,0000,0000,,
Dialogue: 0,0:12:58.56,0:13:00.06,Default,,0000,0000,0000,,You are going to\Nsay, wait a minute?
Dialogue: 0,0:13:00.06,0:13:04.45,Default,,0000,0000,0000,,But are you always going to\Nconsider curves, regular curves
Dialogue: 0,0:13:04.45,0:13:06.22,Default,,0000,0000,0000,,in space?
Dialogue: 0,0:13:06.22,0:13:10.49,Default,,0000,0000,0000,,Regular curves in space.
Dialogue: 0,0:13:10.49,0:13:15.72,Default,,0000,0000,0000,,And by space, I know you guys\Nmean the Euclidean three space.
Dialogue: 0,0:13:15.72,0:13:20.46,Default,,0000,0000,0000,,Actually, many times I will\Nconsider curves in plane.
Dialogue: 0,0:13:20.46,0:13:22.84,Default,,0000,0000,0000,,And the plane is\Npart of the space.
Dialogue: 0,0:13:22.84,0:13:25.68,Default,,0000,0000,0000,,And you say, give us an example.
Dialogue: 0,0:13:25.68,0:13:28.21,Default,,0000,0000,0000,,I will give you an\Nexample right now.
Dialogue: 0,0:13:28.21,0:13:30.38,Default,,0000,0000,0000,,You're going to laugh\Nhow simple that is.
Dialogue: 0,0:13:30.38,0:13:33.18,Default,,0000,0000,0000,,
Dialogue: 0,0:13:33.18,0:13:37.24,Default,,0000,0000,0000,,Now, I have another bug\Nwho is really happy,
Dialogue: 0,0:13:37.24,0:13:39.88,Default,,0000,0000,0000,,but it's not drunk at all.
Dialogue: 0,0:13:39.88,0:13:46.58,Default,,0000,0000,0000,,And this bug knows how to\Ncircle around a certain point
Dialogue: 0,0:13:46.58,0:13:49.31,Default,,0000,0000,0000,,at the same speed.
Dialogue: 0,0:13:49.31,0:13:51.80,Default,,0000,0000,0000,,So very organized bug.
Dialogue: 0,0:13:51.80,0:13:52.50,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:13:52.50,0:13:55.01,Default,,0000,0000,0000,,STUDENT: Where did\Nyou get c prime?
Dialogue: 0,0:13:55.01,0:13:55.68,Default,,0000,0000,0000,,PROFESSOR: What?
Dialogue: 0,0:13:55.68,0:14:01.39,Default,,0000,0000,0000,,STUDENT: You have c prime is\Ndifferentiable, is [INAUDIBLE].
Dialogue: 0,0:14:01.39,0:14:02.07,Default,,0000,0000,0000,,PROFESSOR: c1.
Dialogue: 0,0:14:02.07,0:14:03.00,Default,,0000,0000,0000,,STUDENT: c1.
Dialogue: 0,0:14:03.00,0:14:03.75,Default,,0000,0000,0000,,PROFESSOR: OK. c1.
Dialogue: 0,0:14:03.75,0:14:09.25,Default,,0000,0000,0000,,This is the notation for any\Nfunction that is differentiable
Dialogue: 0,0:14:09.25,0:14:11.86,Default,,0000,0000,0000,,and whose derivative\Nis continuous.
Dialogue: 0,0:14:11.86,0:14:16.85,Default,,0000,0000,0000,,So again, give an\Nexample of a c1 function.
Dialogue: 0,0:14:16.85,0:14:18.33,Default,,0000,0000,0000,,STUDENT: x squared.
Dialogue: 0,0:14:18.33,0:14:19.22,Default,,0000,0000,0000,,PROFESSOR: Yeah.
Dialogue: 0,0:14:19.22,0:14:20.67,Default,,0000,0000,0000,,On some real interval.
Dialogue: 0,0:14:20.67,0:14:27.30,Default,,0000,0000,0000,,How about absolute value\Nof x over the real line?
Dialogue: 0,0:14:27.30,0:14:29.51,Default,,0000,0000,0000,,What's the problem with that?
Dialogue: 0,0:14:29.51,0:14:30.94,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:14:30.94,0:14:33.28,Default,,0000,0000,0000,,PROFESSOR: It's not\Ndifferentiable at 0.
Dialogue: 0,0:14:33.28,0:14:36.64,Default,,0000,0000,0000,,OK, so we'll talk a little\Nbit later about smoothness.
Dialogue: 0,0:14:36.64,0:14:39.85,Default,,0000,0000,0000,,It's a little bit\Ndelicate as a notion.
Dialogue: 0,0:14:39.85,0:14:42.69,Default,,0000,0000,0000,,It's really beautiful\Non the other side.
Dialogue: 0,0:14:42.69,0:14:49.83,Default,,0000,0000,0000,,Let's find the nice picture\Ntrajectory for the bug.
Dialogue: 0,0:14:49.83,0:14:51.46,Default,,0000,0000,0000,,This is a ladybug.
Dialogue: 0,0:14:51.46,0:14:54.17,Default,,0000,0000,0000,,I cannot draw her, anyway.
Dialogue: 0,0:14:54.17,0:14:56.87,Default,,0000,0000,0000,,She is moving along this circle.
Dialogue: 0,0:14:56.87,0:15:00.56,Default,,0000,0000,0000,,And I'll give you\Nthe law of motion.
Dialogue: 0,0:15:00.56,0:15:06.96,Default,,0000,0000,0000,,And that reminds me of a\Nstudent who told me, what
Dialogue: 0,0:15:06.96,0:15:08.63,Default,,0000,0000,0000,,do I care about law of motion?
Dialogue: 0,0:15:08.63,0:15:10.79,Default,,0000,0000,0000,,He never had me as a\Nteacher, obviously.
Dialogue: 0,0:15:10.79,0:15:14.08,Default,,0000,0000,0000,,But he was telling me,\Nwell, after I graduated,
Dialogue: 0,0:15:14.08,0:15:18.46,Default,,0000,0000,0000,,I always thought, what do I\Ncare about the law of motion?
Dialogue: 0,0:15:18.46,0:15:20.65,Default,,0000,0000,0000,,I mean, I took calculus.
Dialogue: 0,0:15:20.65,0:15:24.24,Default,,0000,0000,0000,,Everything was about\Nthe law of motion.
Dialogue: 0,0:15:24.24,0:15:27.34,Default,,0000,0000,0000,,I'm sorry, you should care\Nabout the law of motion.
Dialogue: 0,0:15:27.34,0:15:30.28,Default,,0000,0000,0000,,Once you're not there anymore,\Nabsolutely you don't care.
Dialogue: 0,0:15:30.28,0:15:33.08,Default,,0000,0000,0000,,But why do you want to\N[INAUDIBLE] doing calculus?
Dialogue: 0,0:15:33.08,0:15:34.70,Default,,0000,0000,0000,,When you bring\N[INAUDIBLE] to calculus,
Dialogue: 0,0:15:34.70,0:15:37.46,Default,,0000,0000,0000,,when you walk into\Ncalculus, it's law of motion
Dialogue: 0,0:15:37.46,0:15:39.99,Default,,0000,0000,0000,,everywhere whether\Nyou like it or not.
Dialogue: 0,0:15:39.99,0:15:49.06,Default,,0000,0000,0000,,So let's try cosine t\Nsine t and z to b 1.
Dialogue: 0,0:15:49.06,0:15:52.24,Default,,0000,0000,0000,,Let's make it 1 to\Nmake your life easier.
Dialogue: 0,0:15:52.24,0:15:54.47,Default,,0000,0000,0000,,What kind of curve\Nis this and why am I
Dialogue: 0,0:15:54.47,0:15:58.29,Default,,0000,0000,0000,,claiming that the ladybug\Nfollowing this curve
Dialogue: 0,0:15:58.29,0:16:00.62,Default,,0000,0000,0000,,is moving at a constant speed?
Dialogue: 0,0:16:00.62,0:16:01.49,Default,,0000,0000,0000,,Oh my god.
Dialogue: 0,0:16:01.49,0:16:02.62,Default,,0000,0000,0000,,Go ahead, Alexander.
Dialogue: 0,0:16:02.62,0:16:03.66,Default,,0000,0000,0000,,STUDENT: That's a circle.
Dialogue: 0,0:16:03.66,0:16:05.12,Default,,0000,0000,0000,,PROFESSOR: That's the circle.
Dialogue: 0,0:16:05.12,0:16:06.52,Default,,0000,0000,0000,,It's more than a circle.
Dialogue: 0,0:16:06.52,0:16:07.92,Default,,0000,0000,0000,,It's a parametrized circle.
Dialogue: 0,0:16:07.92,0:16:10.49,Default,,0000,0000,0000,,It's a vector-valued function.
Dialogue: 0,0:16:10.49,0:16:15.46,Default,,0000,0000,0000,,Now, like every mathematician\NI should specify the domain.
Dialogue: 0,0:16:15.46,0:16:18.14,Default,,0000,0000,0000,,I am just winding\Naround one time,
Dialogue: 0,0:16:18.14,0:16:20.68,Default,,0000,0000,0000,,and I stop where I started.
Dialogue: 0,0:16:20.68,0:16:24.43,Default,,0000,0000,0000,,So I better be smart and\Nrealize time is not infinity.
Dialogue: 0,0:16:24.43,0:16:25.54,Default,,0000,0000,0000,,It could be.
Dialogue: 0,0:16:25.54,0:16:28.13,Default,,0000,0000,0000,,I'm wrapping around the\Ncircle infinitely many times.
Dialogue: 0,0:16:28.13,0:16:30.32,Default,,0000,0000,0000,,They do that in\Ntopology actually when
Dialogue: 0,0:16:30.32,0:16:34.31,Default,,0000,0000,0000,,you're going to be--\Nseniors takes topology.
Dialogue: 0,0:16:34.31,0:16:38.42,Default,,0000,0000,0000,,But I'm not going around\Nin circles only one time.
Dialogue: 0,0:16:38.42,0:16:41.06,Default,,0000,0000,0000,,So my time will\Nstart at 0 when I
Dialogue: 0,0:16:41.06,0:16:44.82,Default,,0000,0000,0000,,start my motion and\Nend at 2 pi seconds
Dialogue: 0,0:16:44.82,0:16:47.93,Default,,0000,0000,0000,,if the time is in seconds
Dialogue: 0,0:16:47.93,0:16:52.10,Default,,0000,0000,0000,,So I say r is defined\Non the interval I which
Dialogue: 0,0:16:52.10,0:16:53.67,Default,,0000,0000,0000,,is-- say it again, Magdalena.
Dialogue: 0,0:16:53.67,0:16:55.21,Default,,0000,0000,0000,,You just said it.
Dialogue: 0,0:16:55.21,0:16:56.11,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:16:56.11,0:16:58.07,Default,,0000,0000,0000,,PROFESSOR: 0 to pi.
Dialogue: 0,0:16:58.07,0:17:01.05,Default,,0000,0000,0000,,If you want to take\N0 together, fine.
Dialogue: 0,0:17:01.05,0:17:05.92,Default,,0000,0000,0000,,But for consistency, let's\Ntake it like before, 0 to 2 pi.
Dialogue: 0,0:17:05.92,0:17:07.78,Default,,0000,0000,0000,,I'm actually\Nexcluding the origin.
Dialogue: 0,0:17:07.78,0:17:10.55,Default,,0000,0000,0000,,
Dialogue: 0,0:17:10.55,0:17:12.20,Default,,0000,0000,0000,,And with values in R3.
Dialogue: 0,0:17:12.20,0:17:17.62,Default,,0000,0000,0000,,Although, this is a [? plane ?]\Ncurve, z will be constant.
Dialogue: 0,0:17:17.62,0:17:19.87,Default,,0000,0000,0000,,Do I care about that very much?
Dialogue: 0,0:17:19.87,0:17:21.97,Default,,0000,0000,0000,,You will see the beauty of it.
Dialogue: 0,0:17:21.97,0:17:25.77,Default,,0000,0000,0000,,I have the velocity vector\Nbeing really pretty.
Dialogue: 0,0:17:25.77,0:17:28.08,Default,,0000,0000,0000,,What is the velocity vector?
Dialogue: 0,0:17:28.08,0:17:30.18,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:17:30.18,0:17:31.61,Default,,0000,0000,0000,,PROFESSOR: Negative sign t.
Dialogue: 0,0:17:31.61,0:17:32.48,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:17:32.48,0:17:33.35,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:17:33.35,0:17:35.86,Default,,0000,0000,0000,,PROFESSOR: Cosine t.
Dialogue: 0,0:17:35.86,0:17:37.20,Default,,0000,0000,0000,,And 0, finally.
Dialogue: 0,0:17:37.20,0:17:40.93,Default,,0000,0000,0000,,Because as you saw\Nvery well in the book,
Dialogue: 0,0:17:40.93,0:17:43.94,Default,,0000,0000,0000,,the way we compute\Nthe velocity vector
Dialogue: 0,0:17:43.94,0:17:47.31,Default,,0000,0000,0000,,is by taking x of\Nt, y of t, z of t
Dialogue: 0,0:17:47.31,0:17:50.13,Default,,0000,0000,0000,,and differentiating\Nthem in terms of time.
Dialogue: 0,0:17:50.13,0:17:54.48,Default,,0000,0000,0000,,
Dialogue: 0,0:17:54.48,0:17:55.00,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:17:55.00,0:17:58.26,Default,,0000,0000,0000,,Is this a regular function?
Dialogue: 0,0:17:58.26,0:18:02.52,Default,,0000,0000,0000,,As the bug moves between\Ntime 0 and time equals 2 pi,
Dialogue: 0,0:18:02.52,0:18:06.59,Default,,0000,0000,0000,,is the bug ever going to\Nstop between these times?
Dialogue: 0,0:18:06.59,0:18:07.43,Default,,0000,0000,0000,,STUDENT: No.
Dialogue: 0,0:18:07.43,0:18:08.01,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,0:18:08.01,0:18:08.68,Default,,0000,0000,0000,,How do you know?
Dialogue: 0,0:18:08.68,0:18:10.51,Default,,0000,0000,0000,,You guys are faster\Nthan me, right?
Dialogue: 0,0:18:10.51,0:18:11.27,Default,,0000,0000,0000,,What did you do?
Dialogue: 0,0:18:11.27,0:18:12.81,Default,,0000,0000,0000,,You did the speed.
Dialogue: 0,0:18:12.81,0:18:14.21,Default,,0000,0000,0000,,What's the relationship?
Dialogue: 0,0:18:14.21,0:18:16.54,Default,,0000,0000,0000,,What's the difference\Nbetween velocity and speed?
Dialogue: 0,0:18:16.54,0:18:18.58,Default,,0000,0000,0000,,STUDENT: Speed is the\Nabsolute value [INAUDIBLE].
Dialogue: 0,0:18:18.58,0:18:19.28,Default,,0000,0000,0000,,PROFESSOR: Wonderful.
Dialogue: 0,0:18:19.28,0:18:20.08,Default,,0000,0000,0000,,This is very good.
Dialogue: 0,0:18:20.08,0:18:21.95,Default,,0000,0000,0000,,You should tell everybody\Nthat because people
Dialogue: 0,0:18:21.95,0:18:23.74,Default,,0000,0000,0000,,confuse that left and right.
Dialogue: 0,0:18:23.74,0:18:26.79,Default,,0000,0000,0000,,So the velocity is\Na vector, like you
Dialogue: 0,0:18:26.79,0:18:28.18,Default,,0000,0000,0000,,learned in engineering.
Dialogue: 0,0:18:28.18,0:18:29.96,Default,,0000,0000,0000,,You learned in physics.
Dialogue: 0,0:18:29.96,0:18:31.09,Default,,0000,0000,0000,,Velocity is a vector.
Dialogue: 0,0:18:31.09,0:18:32.08,Default,,0000,0000,0000,,It changes direction.
Dialogue: 0,0:18:32.08,0:18:33.96,Default,,0000,0000,0000,,I'm going to Amarillo this way.
Dialogue: 0,0:18:33.96,0:18:34.90,Default,,0000,0000,0000,,I'm driving.
Dialogue: 0,0:18:34.90,0:18:37.34,Default,,0000,0000,0000,,The velocity will be a\Nvector pointing this way.
Dialogue: 0,0:18:37.34,0:18:40.79,Default,,0000,0000,0000,,As I come back, will\Npoint the opposite way.
Dialogue: 0,0:18:40.79,0:18:43.82,Default,,0000,0000,0000,,The speed will be a\Nscalar, not a vector.
Dialogue: 0,0:18:43.82,0:18:46.46,Default,,0000,0000,0000,,It's a magnitude of\Na velocity vector.
Dialogue: 0,0:18:46.46,0:18:48.06,Default,,0000,0000,0000,,So say it again, Magdalena.
Dialogue: 0,0:18:48.06,0:18:49.24,Default,,0000,0000,0000,,What is the speed?
Dialogue: 0,0:18:49.24,0:18:55.54,Default,,0000,0000,0000,,The speed is the magnitude\Nof the velocity vector.
Dialogue: 0,0:18:55.54,0:18:58.58,Default,,0000,0000,0000,,It's a scalar.
Dialogue: 0,0:18:58.58,0:19:00.78,Default,,0000,0000,0000,,Speed.
Dialogue: 0,0:19:00.78,0:19:02.82,Default,,0000,0000,0000,,Speed.
Dialogue: 0,0:19:02.82,0:19:06.04,Default,,0000,0000,0000,,I heard that before in\Ncars, in the movie Cars.
Dialogue: 0,0:19:06.04,0:19:10.70,Default,,0000,0000,0000,,Anyway, r prime of t magnitude.
Dialogue: 0,0:19:10.70,0:19:12.44,Default,,0000,0000,0000,,In magnitude.
Dialogue: 0,0:19:12.44,0:19:17.36,Default,,0000,0000,0000,,Remember, there is a big\Ndifference between the velocity
Dialogue: 0,0:19:17.36,0:19:19.10,Default,,0000,0000,0000,,as the notion.
Dialogue: 0,0:19:19.10,0:19:22.77,Default,,0000,0000,0000,,Velocity is a vector.
Dialogue: 0,0:19:22.77,0:19:25.57,Default,,0000,0000,0000,,The speed is a\Nmagnitude, is a scalar.
Dialogue: 0,0:19:25.57,0:19:27.80,Default,,0000,0000,0000,,I'm going to go\Nahead and erase that
Dialogue: 0,0:19:27.80,0:19:33.19,Default,,0000,0000,0000,,and I'm going to ask\Nyou what the speed is
Dialogue: 0,0:19:33.19,0:19:36.20,Default,,0000,0000,0000,,for my fellow over here.
Dialogue: 0,0:19:36.20,0:19:40.61,Default,,0000,0000,0000,,What is the speed\Nof a trajectory
Dialogue: 0,0:19:40.61,0:19:46.35,Default,,0000,0000,0000,,of the bug who is sober and\Nmoves at the constant speed?
Dialogue: 0,0:19:46.35,0:19:46.85,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:19:46.85,0:19:49.21,Default,,0000,0000,0000,,As I already told\Nyou, it's constant.
Dialogue: 0,0:19:49.21,0:19:50.39,Default,,0000,0000,0000,,What is that constant?
Dialogue: 0,0:19:50.39,0:19:53.80,Default,,0000,0000,0000,,
Dialogue: 0,0:19:53.80,0:19:56.59,Default,,0000,0000,0000,,What's the constant speed\NI was talking about?
Dialogue: 0,0:19:56.59,0:19:58.79,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:19:58.79,0:20:01.84,Default,,0000,0000,0000,,PROFESSOR: I say the\Nmagnitude of that.
Dialogue: 0,0:20:01.84,0:20:04.36,Default,,0000,0000,0000,,I'm too lazy to write it down.
Dialogue: 0,0:20:04.36,0:20:06.22,Default,,0000,0000,0000,,It's a Tuesday, almost morning.
Dialogue: 0,0:20:06.22,0:20:10.24,Default,,0000,0000,0000,,So I go square root\Nof minus I squared
Dialogue: 0,0:20:10.24,0:20:12.05,Default,,0000,0000,0000,,plus cosine squared plus 0.
Dialogue: 0,0:20:12.05,0:20:13.78,Default,,0000,0000,0000,,I don't need to write that down.
Dialogue: 0,0:20:13.78,0:20:15.27,Default,,0000,0000,0000,,You write it down.
Dialogue: 0,0:20:15.27,0:20:16.58,Default,,0000,0000,0000,,And how much is that?
Dialogue: 0,0:20:16.58,0:20:17.46,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:20:17.46,0:20:18.17,Default,,0000,0000,0000,,PROFESSOR: 1.
Dialogue: 0,0:20:18.17,0:20:24.60,Default,,0000,0000,0000,,So I love this curve because\Nin mathematician slang,
Dialogue: 0,0:20:24.60,0:20:28.80,Default,,0000,0000,0000,,especially in [? a geometer's ?]\Nslang-- and my area
Dialogue: 0,0:20:28.80,0:20:30.24,Default,,0000,0000,0000,,is differential geometry.
Dialogue: 0,0:20:30.24,0:20:35.25,Default,,0000,0000,0000,,So in a way, I do calculus in\NR3 every day on a daily basis.
Dialogue: 0,0:20:35.25,0:20:37.03,Default,,0000,0000,0000,,So I have what?
Dialogue: 0,0:20:37.03,0:20:42.57,Default,,0000,0000,0000,,This is a special kind of curve.
Dialogue: 0,0:20:42.57,0:20:46.38,Default,,0000,0000,0000,,It's a curve parameterized\Nin arc length.
Dialogue: 0,0:20:46.38,0:20:57.84,Default,,0000,0000,0000,,So definition, we say\Nthat a curve in R3,
Dialogue: 0,0:20:57.84,0:21:10.13,Default,,0000,0000,0000,,or Rn, well anyway, is\Nparameterized in arc length.
Dialogue: 0,0:21:10.13,0:21:12.94,Default,,0000,0000,0000,,
Dialogue: 0,0:21:12.94,0:21:13.44,Default,,0000,0000,0000,,When?
Dialogue: 0,0:21:13.44,0:21:14.45,Default,,0000,0000,0000,,Say it again, Magdalena.
Dialogue: 0,0:21:14.45,0:21:32.27,Default,,0000,0000,0000,,Whenever, if and only if,\Nits speed is constantly 1.
Dialogue: 0,0:21:32.27,0:21:36.15,Default,,0000,0000,0000,,
Dialogue: 0,0:21:36.15,0:21:40.63,Default,,0000,0000,0000,,So this is an example\Nwhere the speed is 1.
Dialogue: 0,0:21:40.63,0:21:45.55,Default,,0000,0000,0000,,In such cases, we avoid\Nthe notation with t.
Dialogue: 0,0:21:45.55,0:21:46.47,Default,,0000,0000,0000,,You say, oh my god.
Dialogue: 0,0:21:46.47,0:21:47.46,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:21:47.46,0:21:50.10,Default,,0000,0000,0000,,When the curve is\Nparameterized in arc length,
Dialogue: 0,0:21:50.10,0:21:54.97,Default,,0000,0000,0000,,from now on the we\Nwill actually try
Dialogue: 0,0:21:54.97,0:21:58.92,Default,,0000,0000,0000,,to use s whatever we\Nknow it's an arc length.
Dialogue: 0,0:21:58.92,0:22:00.85,Default,,0000,0000,0000,,We use s instead of t.
Dialogue: 0,0:22:00.85,0:22:05.21,Default,,0000,0000,0000,,So I'm sorry for the people\Nwho cannot change that,
Dialogue: 0,0:22:05.21,0:22:08.70,Default,,0000,0000,0000,,but you should all be\Nable t change that.
Dialogue: 0,0:22:08.70,0:22:12.55,Default,,0000,0000,0000,,So everything will be\Nin s because we just
Dialogue: 0,0:22:12.55,0:22:15.45,Default,,0000,0000,0000,,discovered\N[? Discovery Channel, ?] we
Dialogue: 0,0:22:15.45,0:22:19.40,Default,,0000,0000,0000,,just discovered that speed is 1.
Dialogue: 0,0:22:19.40,0:22:24.30,Default,,0000,0000,0000,,So there is something\Nspecial about this s.
Dialogue: 0,0:22:24.30,0:22:29.38,Default,,0000,0000,0000,,
Dialogue: 0,0:22:29.38,0:22:32.72,Default,,0000,0000,0000,,In this example-- oh, you\Ncan rewrite the whole example
Dialogue: 0,0:22:32.72,0:22:37.37,Default,,0000,0000,0000,,if you want in s so you don't\Nhave to smudge the paper.
Dialogue: 0,0:22:37.37,0:22:38.83,Default,,0000,0000,0000,,OK, it's beautiful.
Dialogue: 0,0:22:38.83,0:22:41.42,Default,,0000,0000,0000,,So I am already arc length.
Dialogue: 0,0:22:41.42,0:22:43.95,Default,,0000,0000,0000,,And in that case, I'm going\Nto call my time parameter
Dialogue: 0,0:22:43.95,0:22:46.19,Default,,0000,0000,0000,,little s. s comes from special.
Dialogue: 0,0:22:46.19,0:22:48.46,Default,,0000,0000,0000,,No, s comes from\Nspeed [INAUDIBLE].
Dialogue: 0,0:22:48.46,0:22:51.59,Default,,0000,0000,0000,,STUDENT: So you use s\Nwhen it's [INAUDIBLE]?
Dialogue: 0,0:22:51.59,0:22:57.14,Default,,0000,0000,0000,,PROFESSOR: We use s whenever the\Nspeed of that curve will be 1.
Dialogue: 0,0:22:57.14,0:22:58.14,Default,,0000,0000,0000,,STUDENT: So [INAUDIBLE].
Dialogue: 0,0:22:58.14,0:23:00.47,Default,,0000,0000,0000,,PROFESSOR: And we call that\Narc length parameterization.
Dialogue: 0,0:23:00.47,0:23:02.97,Default,,0000,0000,0000,,
Dialogue: 0,0:23:02.97,0:23:06.24,Default,,0000,0000,0000,,I'm moving into the duration\Nof your final thoughts.
Dialogue: 0,0:23:06.24,0:23:07.96,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:23:07.96,0:23:09.50,Default,,0000,0000,0000,,STUDENT: When we\Nget the question, so
Dialogue: 0,0:23:09.50,0:23:10.63,Default,,0000,0000,0000,,before solving [INAUDIBLE].
Dialogue: 0,0:23:10.63,0:23:13.14,Default,,0000,0000,0000,,
Dialogue: 0,0:23:13.14,0:23:14.46,Default,,0000,0000,0000,,PROFESSOR: We don't know.
Dialogue: 0,0:23:14.46,0:23:17.92,Default,,0000,0000,0000,,That's why it was our\Ndiscovery that, hey, at the end
Dialogue: 0,0:23:17.92,0:23:22.43,Default,,0000,0000,0000,,it is an arc length, so I better\Nchange [INAUDIBLE] t into s
Dialogue: 0,0:23:22.43,0:23:26.92,Default,,0000,0000,0000,,because that will help me in\Nthe future remember to do that.
Dialogue: 0,0:23:26.92,0:23:30.38,Default,,0000,0000,0000,,Every time I have arc length,\Nthat it means speed 1.
Dialogue: 0,0:23:30.38,0:23:33.39,Default,,0000,0000,0000,,I will call it s instead of y.
Dialogue: 0,0:23:33.39,0:23:34.92,Default,,0000,0000,0000,,There is a reason for that.
Dialogue: 0,0:23:34.92,0:23:36.55,Default,,0000,0000,0000,,I'm going to erase\Nthe definition
Dialogue: 0,0:23:36.55,0:23:42.86,Default,,0000,0000,0000,,and I'm going to give\Nyou the-- more or less,
Dialogue: 0,0:23:42.86,0:23:45.58,Default,,0000,0000,0000,,the explanation that my\Nphysics professor gave me.
Dialogue: 0,0:23:45.58,0:23:50.17,Default,,0000,0000,0000,,Because as a freshman,\Nmy mathematics professor
Dialogue: 0,0:23:50.17,0:23:54.45,Default,,0000,0000,0000,,in that area, in geometry,\Nwas not very, very active.
Dialogue: 0,0:23:54.45,0:23:57.33,Default,,0000,0000,0000,,But practically, what my physics\Nprofessor told me is that,
Dialogue: 0,0:23:57.33,0:24:05.90,Default,,0000,0000,0000,,hey, I would like to have\Nsome sort of a uniform tangent
Dialogue: 0,0:24:05.90,0:24:09.83,Default,,0000,0000,0000,,vector, something that is\Nstandardized to be in speed 1.
Dialogue: 0,0:24:09.83,0:24:15.86,Default,,0000,0000,0000,,So I would like that tangent\Nvector to be important to us.
Dialogue: 0,0:24:15.86,0:24:19.84,Default,,0000,0000,0000,,And if r is an\Narc length, then r
Dialogue: 0,0:24:19.84,0:24:24.11,Default,,0000,0000,0000,,prime would be that unit\Nvector that I'm talking about.
Dialogue: 0,0:24:24.11,0:24:30.79,Default,,0000,0000,0000,,So he introduced for any r of\Nt, which is x of t, y of t,
Dialogue: 0,0:24:30.79,0:24:32.12,Default,,0000,0000,0000,,z of t.
Dialogue: 0,0:24:32.12,0:24:36.58,Default,,0000,0000,0000,,My physics professor introduced\Nthe following terminology.
Dialogue: 0,0:24:36.58,0:24:42.87,Default,,0000,0000,0000,,The tangent unit vector\Nfor a regular curve--
Dialogue: 0,0:24:42.87,0:24:46.63,Default,,0000,0000,0000,,he was very well-organized\NI might add about him--
Dialogue: 0,0:24:46.63,0:24:52.67,Default,,0000,0000,0000,,is by definition r\Nprime of t as a vector
Dialogue: 0,0:24:52.67,0:24:54.38,Default,,0000,0000,0000,,divided by the\Nspeed of the vector.
Dialogue: 0,0:24:54.38,0:24:56.25,Default,,0000,0000,0000,,So what is he doing?
Dialogue: 0,0:24:56.25,0:24:58.54,Default,,0000,0000,0000,,He is unitarizing the velocity.
Dialogue: 0,0:24:58.54,0:25:00.13,Default,,0000,0000,0000,,Say it again, Magdalena.
Dialogue: 0,0:25:00.13,0:25:03.21,Default,,0000,0000,0000,,He has unitarized\Nthe velocity in order
Dialogue: 0,0:25:03.21,0:25:08.50,Default,,0000,0000,0000,,to make research more consistent\Nfrom the viewpoint of Frenet
Dialogue: 0,0:25:08.50,0:25:10.04,Default,,0000,0000,0000,,frame.
Dialogue: 0,0:25:10.04,0:25:12.52,Default,,0000,0000,0000,,So in Frenet frame, you\Nwill see-- you probably
Dialogue: 0,0:25:12.52,0:25:14.31,Default,,0000,0000,0000,,learned about the\NFrenet frame if you
Dialogue: 0,0:25:14.31,0:25:18.26,Default,,0000,0000,0000,,are a mechanics major, or some\Nsolid mechanics or physics
Dialogue: 0,0:25:18.26,0:25:19.13,Default,,0000,0000,0000,,major.
Dialogue: 0,0:25:19.13,0:25:22.60,Default,,0000,0000,0000,,The Frenet frame is\Nan orthogonal frame
Dialogue: 0,0:25:22.60,0:25:28.75,Default,,0000,0000,0000,,moving along a line in time\Nwhere the three components are
Dialogue: 0,0:25:28.75,0:25:33.20,Default,,0000,0000,0000,,t, and the principal normal\Nvector, and b the [INAUDIBLE].
Dialogue: 0,0:25:33.20,0:25:35.51,Default,,0000,0000,0000,,We only know of the\Nfirst of them, which
Dialogue: 0,0:25:35.51,0:25:38.65,Default,,0000,0000,0000,,is T, which is a unit vector.
Dialogue: 0,0:25:38.65,0:25:40.32,Default,,0000,0000,0000,,Say it again who it was.
Dialogue: 0,0:25:40.32,0:25:44.98,Default,,0000,0000,0000,,It was the velocity vector\Ndivided by its magnitude.
Dialogue: 0,0:25:44.98,0:25:47.33,Default,,0000,0000,0000,,So the velocity vector could\Nbe any wild, crazy vector
Dialogue: 0,0:25:47.33,0:25:54.56,Default,,0000,0000,0000,,that's tangent to the trajectory\Nat the point where you are.
Dialogue: 0,0:25:54.56,0:25:58.12,Default,,0000,0000,0000,,His magnitude varies from\None point to the other.
Dialogue: 0,0:25:58.12,0:25:59.91,Default,,0000,0000,0000,,He's absolutely crazy.
Dialogue: 0,0:25:59.91,0:26:01.43,Default,,0000,0000,0000,,He or she, the velocity vector.
Dialogue: 0,0:26:01.43,0:26:02.29,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:26:02.29,0:26:03.16,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:26:03.16,0:26:06.85,Default,,0000,0000,0000,,
Dialogue: 0,0:26:06.85,0:26:07.52,Default,,0000,0000,0000,,PROFESSOR: Here?
Dialogue: 0,0:26:07.52,0:26:08.33,Default,,0000,0000,0000,,Here?
Dialogue: 0,0:26:08.33,0:26:09.42,Default,,0000,0000,0000,,STUDENT: Yeah, down there.
Dialogue: 0,0:26:09.42,0:26:10.62,Default,,0000,0000,0000,,PROFESSOR: D-E-F, definition.
Dialogue: 0,0:26:10.62,0:26:12.82,Default,,0000,0000,0000,,That's how a mathematician\Ndefines things.
Dialogue: 0,0:26:12.82,0:26:18.34,Default,,0000,0000,0000,,So to define you write def\Non top of an equality sign
Dialogue: 0,0:26:18.34,0:26:20.96,Default,,0000,0000,0000,,or double dot equal.
Dialogue: 0,0:26:20.96,0:26:23.63,Default,,0000,0000,0000,,That's a formal way a\Nmathematician introduces
Dialogue: 0,0:26:23.63,0:26:24.78,Default,,0000,0000,0000,,a definition.
Dialogue: 0,0:26:24.78,0:26:27.60,Default,,0000,0000,0000,,Well, he was a physicist,\Nbut he does math.
Dialogue: 0,0:26:27.60,0:26:29.33,Default,,0000,0000,0000,,So what do we do?
Dialogue: 0,0:26:29.33,0:26:32.50,Default,,0000,0000,0000,,We say all the blue\Nguys that are not equal,
Dialogue: 0,0:26:32.50,0:26:34.58,Default,,0000,0000,0000,,divide yourselves\Nby your magnitude.
Dialogue: 0,0:26:34.58,0:26:39.72,Default,,0000,0000,0000,,And I'm going to have\Nthe T here is next one,
Dialogue: 0,0:26:39.72,0:26:42.75,Default,,0000,0000,0000,,the T here is next one,\Nthe T here is next one.
Dialogue: 0,0:26:42.75,0:26:43.58,Default,,0000,0000,0000,,They are all equal.
Dialogue: 0,0:26:43.58,0:26:51.20,Default,,0000,0000,0000,,So that T changes direction, but\Nits magnitude will always be 1.
Dialogue: 0,0:26:51.20,0:26:51.70,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:26:51.70,0:26:55.00,Default,,0000,0000,0000,,Know that the magnitude--\Nthat's what unit vector means,
Dialogue: 0,0:26:55.00,0:26:58.20,Default,,0000,0000,0000,,the magnitude is 1.
Dialogue: 0,0:26:58.20,0:27:00.71,Default,,0000,0000,0000,,Why am I so happy about that?
Dialogue: 0,0:27:00.71,0:27:03.94,Default,,0000,0000,0000,,Well let me tell\Nyou that we can have
Dialogue: 0,0:27:03.94,0:27:07.48,Default,,0000,0000,0000,,another parametrization\Nand another parametrization
Dialogue: 0,0:27:07.48,0:27:11.12,Default,,0000,0000,0000,,and another parametrization\Nof the same curve.
Dialogue: 0,0:27:11.12,0:27:12.26,Default,,0000,0000,0000,,Say what?
Dialogue: 0,0:27:12.26,0:27:14.82,Default,,0000,0000,0000,,The parametrization of\Na curve is not unique?
Dialogue: 0,0:27:14.82,0:27:15.72,Default,,0000,0000,0000,,No.
Dialogue: 0,0:27:15.72,0:27:18.81,Default,,0000,0000,0000,,There are infinitely\Nmany parametrizations
Dialogue: 0,0:27:18.81,0:27:21.93,Default,,0000,0000,0000,,for a physical curve.
Dialogue: 0,0:27:21.93,0:27:34.16,Default,,0000,0000,0000,,There are infinitely\Nmany parametrizations
Dialogue: 0,0:27:34.16,0:27:39.73,Default,,0000,0000,0000,,for an even physical curve.
Dialogue: 0,0:27:39.73,0:27:43.06,Default,,0000,0000,0000,,
Dialogue: 0,0:27:43.06,0:27:45.03,Default,,0000,0000,0000,,Like [INAUDIBLE]\Nthe regular one?
Dialogue: 0,0:27:45.03,0:27:47.50,Default,,0000,0000,0000,,Well let me give you\Nanother example that
Dialogue: 0,0:27:47.50,0:27:51.48,Default,,0000,0000,0000,,says that this is\Ncurrently R of T
Dialogue: 0,0:27:51.48,0:27:58.29,Default,,0000,0000,0000,,equals cosine 5T sine 5T and 1.
Dialogue: 0,0:27:58.29,0:27:59.06,Default,,0000,0000,0000,,Why 1?
Dialogue: 0,0:27:59.06,0:28:02.56,Default,,0000,0000,0000,,I still want to have\Nthe same physical curve.
Dialogue: 0,0:28:02.56,0:28:03.69,Default,,0000,0000,0000,,What's different, guys?
Dialogue: 0,0:28:03.69,0:28:06.74,Default,,0000,0000,0000,,Look at that and then\Nsay oh OK, is this
Dialogue: 0,0:28:06.74,0:28:11.89,Default,,0000,0000,0000,,the same curve as\Na physical curve?
Dialogue: 0,0:28:11.89,0:28:13.41,Default,,0000,0000,0000,,What's different in this case?
Dialogue: 0,0:28:13.41,0:28:14.80,Default,,0000,0000,0000,,I'm still here.
Dialogue: 0,0:28:14.80,0:28:16.62,Default,,0000,0000,0000,,It's still the\N[? red ?] physical curve
Dialogue: 0,0:28:16.62,0:28:18.22,Default,,0000,0000,0000,,I'm moving along.
Dialogue: 0,0:28:18.22,0:28:18.97,Default,,0000,0000,0000,,What is different?
Dialogue: 0,0:28:18.97,0:28:19.89,Default,,0000,0000,0000,,STUDENT: The velocity.
Dialogue: 0,0:28:19.89,0:28:20.96,Default,,0000,0000,0000,,PROFESSOR: The velocity.
Dialogue: 0,0:28:20.96,0:28:23.88,Default,,0000,0000,0000,,The velocity and\Nactually the speed.
Dialogue: 0,0:28:23.88,0:28:29.38,Default,,0000,0000,0000,,I'm moving faster or slower, I\Ndon't know, we have to decide.
Dialogue: 0,0:28:29.38,0:28:34.23,Default,,0000,0000,0000,,Now how do I realize\Nhow many times
Dialogue: 0,0:28:34.23,0:28:36.38,Default,,0000,0000,0000,,I'm moving along this curve?
Dialogue: 0,0:28:36.38,0:28:39.74,Default,,0000,0000,0000,,I can be smart and say\Nhey, I'm not stupid.
Dialogue: 0,0:28:39.74,0:28:43.42,Default,,0000,0000,0000,,I know how to move only one\Ntime and stop where I started.
Dialogue: 0,0:28:43.42,0:28:47.04,Default,,0000,0000,0000,,So if I start with\Nmy T in the interval
Dialogue: 0,0:28:47.04,0:28:52.63,Default,,0000,0000,0000,,zero-- I start at\Nzero, where do I stop?
Dialogue: 0,0:28:52.63,0:28:54.46,Default,,0000,0000,0000,,I can hear your brain buzzing.
Dialogue: 0,0:28:54.46,0:28:55.33,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:28:55.33,0:28:57.90,Default,,0000,0000,0000,,PROFESSOR: 2pi over 5.
Dialogue: 0,0:28:57.90,0:28:58.84,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:28:58.84,0:29:00.26,Default,,0000,0000,0000,,Excellent answer.
Dialogue: 0,0:29:00.26,0:29:03.35,Default,,0000,0000,0000,,STUDENT: Because when you\Nplug it in, it's [INAUDIBLE].
Dialogue: 0,0:29:03.35,0:29:05.15,Default,,0000,0000,0000,,PROFESSOR: 5 times 2pi over 5.
Dialogue: 0,0:29:05.15,0:29:06.13,Default,,0000,0000,0000,,That's where I stop.
Dialogue: 0,0:29:06.13,0:29:08.36,Default,,0000,0000,0000,,So this is not the same\Ninterval as before.
Dialogue: 0,0:29:08.36,0:29:09.68,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:29:09.68,0:29:16.65,Default,,0000,0000,0000,,This is a new guy, which\Nis called J. Oh, all right.
Dialogue: 0,0:29:16.65,0:29:19.46,Default,,0000,0000,0000,,So there is a\Nrelationship between the T
Dialogue: 0,0:29:19.46,0:29:23.48,Default,,0000,0000,0000,,and the S. That's why I\Nuse different notations.
Dialogue: 0,0:29:23.48,0:29:26.74,Default,,0000,0000,0000,,And I wish my teachers\Nstarted it just
Dialogue: 0,0:29:26.74,0:29:29.79,Default,,0000,0000,0000,,like that when I took math\Nanalysis as a freshman,
Dialogue: 0,0:29:29.79,0:29:30.63,Default,,0000,0000,0000,,or calculus.
Dialogue: 0,0:29:30.63,0:29:32.33,Default,,0000,0000,0000,,That's calculus.
Dialogue: 0,0:29:32.33,0:29:35.70,Default,,0000,0000,0000,,Because what they started\Nwith was a diagram.
Dialogue: 0,0:29:35.70,0:29:37.30,Default,,0000,0000,0000,,What kind of diagram?
Dialogue: 0,0:29:37.30,0:29:41.84,Default,,0000,0000,0000,,Say OK, the\Nparametrizations are both
Dialogue: 0,0:29:41.84,0:29:45.06,Default,,0000,0000,0000,,starting from\Ndifferent intervals.
Dialogue: 0,0:29:45.06,0:29:47.55,Default,,0000,0000,0000,,And first I have\Nthe parametrization
Dialogue: 0,0:29:47.55,0:29:50.29,Default,,0000,0000,0000,,from I going to our 3.
Dialogue: 0,0:29:50.29,0:29:53.35,Default,,0000,0000,0000,,And that's called-- how\Ndid we baptize that?
Dialogue: 0,0:29:53.35,0:29:57.64,Default,,0000,0000,0000,,R. And the other\None, from J to R3,
Dialogue: 0,0:29:57.64,0:30:01.79,Default,,0000,0000,0000,,we call that big R.\NThey're both vectors.
Dialogue: 0,0:30:01.79,0:30:05.17,Default,,0000,0000,0000,,And hey guys, we\Nshould have some sort
Dialogue: 0,0:30:05.17,0:30:08.72,Default,,0000,0000,0000,,of correspondence\Nfunctions between I
Dialogue: 0,0:30:08.72,0:30:14.14,Default,,0000,0000,0000,,and J that are both 1 to 1, and\Nthey are 1 being [INAUDIBLE]
Dialogue: 0,0:30:14.14,0:30:16.42,Default,,0000,0000,0000,,the other.
Dialogue: 0,0:30:16.42,0:30:18.31,Default,,0000,0000,0000,,I swear to God,\Nwhen they started
Dialogue: 0,0:30:18.31,0:30:21.00,Default,,0000,0000,0000,,with this theoretical\Nmodel, I didn't understand
Dialogue: 0,0:30:21.00,0:30:23.19,Default,,0000,0000,0000,,the motivation at all.
Dialogue: 0,0:30:23.19,0:30:25.22,Default,,0000,0000,0000,,At all.
Dialogue: 0,0:30:25.22,0:30:27.62,Default,,0000,0000,0000,,Now with an example,\NI can get you
Dialogue: 0,0:30:27.62,0:30:30.89,Default,,0000,0000,0000,,closer to the motivation\Nof such a diagram.
Dialogue: 0,0:30:30.89,0:30:34.70,Default,,0000,0000,0000,,So where does our\Nprimary S live?
Dialogue: 0,0:30:34.70,0:30:39.13,Default,,0000,0000,0000,,S lives in I, and\NT lives in J. So I
Dialogue: 0,0:30:39.13,0:30:42.79,Default,,0000,0000,0000,,have to have a correspondence\Nthat takes S to T or T to S.
Dialogue: 0,0:30:42.79,0:30:46.35,Default,,0000,0000,0000,,STUDENT: Wait I\Nthought since R of T
Dialogue: 0,0:30:46.35,0:30:48.48,Default,,0000,0000,0000,,is also pretty much\N[INAUDIBLE] that we should also
Dialogue: 0,0:30:48.48,0:30:49.59,Default,,0000,0000,0000,,use S [INAUDIBLE].
Dialogue: 0,0:30:49.59,0:30:53.04,Default,,0000,0000,0000,,PROFESSOR: It's very--\Nactually it's very easy.
Dialogue: 0,0:30:53.04,0:30:55.88,Default,,0000,0000,0000,,This is 5T.
Dialogue: 0,0:30:55.88,0:31:02.25,Default,,0000,0000,0000,,And we cannot use S\Ninstead of this T,
Dialogue: 0,0:31:02.25,0:31:05.08,Default,,0000,0000,0000,,because if we use S\Ninstead of this T,
Dialogue: 0,0:31:05.08,0:31:07.69,Default,,0000,0000,0000,,and we compute the\Nspeed, we get 5.
Dialogue: 0,0:31:07.69,0:31:10.94,Default,,0000,0000,0000,,So it cannot be called S.\NThis is very important.
Dialogue: 0,0:31:10.94,0:31:15.14,Default,,0000,0000,0000,,So T is not an arc\Nlength parameter.
Dialogue: 0,0:31:15.14,0:31:18.17,Default,,0000,0000,0000,,I wonder what the speed\Nwill be for this guy.
Dialogue: 0,0:31:18.17,0:31:20.47,Default,,0000,0000,0000,,So who wants to\Ncompute R prime of T?
Dialogue: 0,0:31:20.47,0:31:22.58,Default,,0000,0000,0000,,Nobody, but I'll force you to.
Dialogue: 0,0:31:22.58,0:31:26.52,Default,,0000,0000,0000,,And the magnitude of that\Nwill be god knows what.
Dialogue: 0,0:31:26.52,0:31:27.74,Default,,0000,0000,0000,,I claim it's 5.
Dialogue: 0,0:31:27.74,0:31:30.11,Default,,0000,0000,0000,,Maybe I'm wrong.
Dialogue: 0,0:31:30.11,0:31:31.27,Default,,0000,0000,0000,,I did this in my head.
Dialogue: 0,0:31:31.27,0:31:33.15,Default,,0000,0000,0000,,I have to do it on paper, right.
Dialogue: 0,0:31:33.15,0:31:35.31,Default,,0000,0000,0000,,So I have what?
Dialogue: 0,0:31:35.31,0:31:38.73,Default,,0000,0000,0000,,I have to differentiate\Ncomponent-wise.
Dialogue: 0,0:31:38.73,0:31:42.25,Default,,0000,0000,0000,,And I have [INAUDIBLE] that,\Nbecause I'm running out of gas.
Dialogue: 0,0:31:42.25,0:31:43.03,Default,,0000,0000,0000,,STUDENT: Minus 5--
Dialogue: 0,0:31:43.03,0:31:45.59,Default,,0000,0000,0000,,PROFESSOR: Minus 5, very good.
Dialogue: 0,0:31:45.59,0:31:47.72,Default,,0000,0000,0000,,Sine of 5T.
Dialogue: 0,0:31:47.72,0:31:49.32,Default,,0000,0000,0000,,What have we applied?
Dialogue: 0,0:31:49.32,0:31:51.51,Default,,0000,0000,0000,,In case you don't\Nknow that, out.
Dialogue: 0,0:31:51.51,0:31:52.77,Default,,0000,0000,0000,,That was Calc 1.
Dialogue: 0,0:31:52.77,0:31:53.59,Default,,0000,0000,0000,,Chain rule.
Dialogue: 0,0:31:53.59,0:31:55.42,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:31:55.42,0:32:00.08,Default,,0000,0000,0000,,So 5 times cosine 5T.
Dialogue: 0,0:32:00.08,0:32:03.72,Default,,0000,0000,0000,,And finally, 1\Nprime, which is 0.
Dialogue: 0,0:32:03.72,0:32:09.85,Default,,0000,0000,0000,,Now let's be brave and\Nwrite the whole thing down.
Dialogue: 0,0:32:09.85,0:32:13.15,Default,,0000,0000,0000,,I know I'm lazy today, but I'm\Ngoing to have to do something.
Dialogue: 0,0:32:13.15,0:32:13.65,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:32:13.65,0:32:18.39,Default,,0000,0000,0000,,So I'll say minus 5\Nsine 5T is all squared.
Dialogue: 0,0:32:18.39,0:32:20.88,Default,,0000,0000,0000,,Let me take it and square it.
Dialogue: 0,0:32:20.88,0:32:23.89,Default,,0000,0000,0000,,Because I see one\Nface is confused.
Dialogue: 0,0:32:23.89,0:32:26.84,Default,,0000,0000,0000,,And since one face\Nis confused, it
Dialogue: 0,0:32:26.84,0:32:29.97,Default,,0000,0000,0000,,doesn't matter that the\Nothers are not confused.
Dialogue: 0,0:32:29.97,0:32:31.06,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:32:31.06,0:32:36.28,Default,,0000,0000,0000,,So I have square root of this\Nplus square of [INAUDIBLE] plus
Dialogue: 0,0:32:36.28,0:32:38.54,Default,,0000,0000,0000,,[INAUDIBLE] computing\Nthe magnitude.
Dialogue: 0,0:32:38.54,0:32:39.75,Default,,0000,0000,0000,,What do I get out of here?
Dialogue: 0,0:32:39.75,0:32:40.48,Default,,0000,0000,0000,,STUDENT: Five.
Dialogue: 0,0:32:40.48,0:32:41.17,Default,,0000,0000,0000,,PROFESSOR: Five.
Dialogue: 0,0:32:41.17,0:32:41.78,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:32:41.78,0:32:45.43,Default,,0000,0000,0000,,This is 5 sine squared\Nplus 5 cosine squared.
Dialogue: 0,0:32:45.43,0:32:49.67,Default,,0000,0000,0000,,Now yes, then I have 5 times 1.
Dialogue: 0,0:32:49.67,0:32:54.55,Default,,0000,0000,0000,,So I have square root\Nof 25 here will be 5.
Dialogue: 0,0:32:54.55,0:32:55.70,Default,,0000,0000,0000,,What is 5?
Dialogue: 0,0:32:55.70,0:33:03.48,Default,,0000,0000,0000,,5 is the speed of the [? bug ?]\Nalong the same physical curve
Dialogue: 0,0:33:03.48,0:33:04.64,Default,,0000,0000,0000,,the other way around.
Dialogue: 0,0:33:04.64,0:33:06.99,Default,,0000,0000,0000,,The second time around.
Dialogue: 0,0:33:06.99,0:33:10.21,Default,,0000,0000,0000,,Now can you tell me the\Nrelationship between T and S?
Dialogue: 0,0:33:10.21,0:33:12.58,Default,,0000,0000,0000,,They are related.
Dialogue: 0,0:33:12.58,0:33:19.22,Default,,0000,0000,0000,,They are like if you're my\Nuncle, then I'm your niece.
Dialogue: 0,0:33:19.22,0:33:21.29,Default,,0000,0000,0000,,It's the same way.
Dialogue: 0,0:33:21.29,0:33:23.02,Default,,0000,0000,0000,,It depends where you look at.
Dialogue: 0,0:33:23.02,0:33:26.04,Default,,0000,0000,0000,,T is a function of S,\Nand S is a function of T.
Dialogue: 0,0:33:26.04,0:33:32.22,Default,,0000,0000,0000,,So it has to be a 1 to 1\Ncorrespondence between the two.
Dialogue: 0,0:33:32.22,0:33:38.24,Default,,0000,0000,0000,,Now any ideas of how I what\Nto compute the-- how do I
Dialogue: 0,0:33:38.24,0:33:43.18,Default,,0000,0000,0000,,want to write the\Nrelationship between them.
Dialogue: 0,0:33:43.18,0:33:46.40,Default,,0000,0000,0000,,Well, S is a\Nfunction of T, right?
Dialogue: 0,0:33:46.40,0:33:50.53,Default,,0000,0000,0000,,I just don't know what\Nfunction of T that is.
Dialogue: 0,0:33:50.53,0:33:52.45,Default,,0000,0000,0000,,And I wish my professor\Nhad started like that,
Dialogue: 0,0:33:52.45,0:33:54.71,Default,,0000,0000,0000,,but he started\Nwith this diagram.
Dialogue: 0,0:33:54.71,0:33:58.89,Default,,0000,0000,0000,,So simply here you\Nhave S equals S of T,
Dialogue: 0,0:33:58.89,0:34:01.38,Default,,0000,0000,0000,,and here you have\NT equals T of S,
Dialogue: 0,0:34:01.38,0:34:03.17,Default,,0000,0000,0000,,the inverse of that function.
Dialogue: 0,0:34:03.17,0:34:05.82,Default,,0000,0000,0000,,And when you-- when\Nsomebody starts that
Dialogue: 0,0:34:05.82,0:34:09.56,Default,,0000,0000,0000,,without an example as a\Ngeneral diagram philosophy,
Dialogue: 0,0:34:09.56,0:34:12.05,Default,,0000,0000,0000,,then it's really, really tough.
Dialogue: 0,0:34:12.05,0:34:13.05,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:34:13.05,0:34:16.05,Default,,0000,0000,0000,,So I'd like to know\Nwho S of T-- how
Dialogue: 0,0:34:16.05,0:34:19.53,Default,,0000,0000,0000,,in the world do I want\Nto define that S of T.
Dialogue: 0,0:34:19.53,0:34:25.57,Default,,0000,0000,0000,,He spoonfed us S of T. I don't\Nwant to spoonfeed you anything.
Dialogue: 0,0:34:25.57,0:34:27.73,Default,,0000,0000,0000,,Because this is\Nhonors class, and you
Dialogue: 0,0:34:27.73,0:34:30.93,Default,,0000,0000,0000,,should be able to figure\Nthis out yourselves.
Dialogue: 0,0:34:30.93,0:34:35.84,Default,,0000,0000,0000,,So who is big R of T?
Dialogue: 0,0:34:35.84,0:34:42.20,Default,,0000,0000,0000,,Big R of T should\Nbe, what, should
Dialogue: 0,0:34:42.20,0:34:44.82,Default,,0000,0000,0000,,be the same thing in\Nthe end as R of S.
Dialogue: 0,0:34:44.82,0:34:56.69,Default,,0000,0000,0000,,But I should say maybe it's\NR of function T of S, right?
Dialogue: 0,0:34:56.69,0:34:59.66,Default,,0000,0000,0000,,Which is the same\Nthing as R of S. So
Dialogue: 0,0:34:59.66,0:35:05.82,Default,,0000,0000,0000,,what should be the\Nrelationship between T and S?
Dialogue: 0,0:35:05.82,0:35:11.28,Default,,0000,0000,0000,,We have to call them-- one of\Nthem should be T equals T of S.
Dialogue: 0,0:35:11.28,0:35:12.52,Default,,0000,0000,0000,,How about this function?
Dialogue: 0,0:35:12.52,0:35:15.59,Default,,0000,0000,0000,,Give it a Greek name,\Nwhat do you want.
Dialogue: 0,0:35:15.59,0:35:16.12,Default,,0000,0000,0000,,Alpha?
Dialogue: 0,0:35:16.12,0:35:16.67,Default,,0000,0000,0000,,Beta?
Dialogue: 0,0:35:16.67,0:35:16.80,Default,,0000,0000,0000,,What?
Dialogue: 0,0:35:16.80,0:35:17.68,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:35:17.68,0:35:19.25,Default,,0000,0000,0000,,PROFESSOR: Alpha?
Dialogue: 0,0:35:19.25,0:35:19.79,Default,,0000,0000,0000,,Beta?
Dialogue: 0,0:35:19.79,0:35:20.29,Default,,0000,0000,0000,,Alpha?
Dialogue: 0,0:35:20.29,0:35:21.70,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:35:21.70,0:35:25.52,Default,,0000,0000,0000,,So S going to T, alpha.
Dialogue: 0,0:35:25.52,0:35:27.27,Default,,0000,0000,0000,,And this is going\Nto be alpha inverse.
Dialogue: 0,0:35:27.27,0:35:30.64,Default,,0000,0000,0000,,
Dialogue: 0,0:35:30.64,0:35:32.09,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:35:32.09,0:35:37.30,Default,,0000,0000,0000,,So T equals alpha of S.\NIt's more elegant to call it
Dialogue: 0,0:35:37.30,0:35:44.98,Default,,0000,0000,0000,,like that than T of S. T\Nequals alpha of S. Alpha of S.
Dialogue: 0,0:35:44.98,0:35:49.08,Default,,0000,0000,0000,,So from this thing,\NI realize that I
Dialogue: 0,0:35:49.08,0:35:54.30,Default,,0000,0000,0000,,get that R composed with\Nalpha equals R. Say what?
Dialogue: 0,0:35:54.30,0:35:54.80,Default,,0000,0000,0000,,Magdalena?
Dialogue: 0,0:35:54.80,0:35:57.17,Default,,0000,0000,0000,,Yeah, yeah, that\Nwas pre-calculus.
Dialogue: 0,0:35:57.17,0:36:01.11,Default,,0000,0000,0000,,R composed with alpha\Nequals little r.
Dialogue: 0,0:36:01.11,0:36:09.14,Default,,0000,0000,0000,,So how do I get a little r\Nby composing R with alpha?
Dialogue: 0,0:36:09.14,0:36:12.09,Default,,0000,0000,0000,,How do we say that?
Dialogue: 0,0:36:12.09,0:36:17.49,Default,,0000,0000,0000,,Alpha followed by R.\NR composed with alpha.
Dialogue: 0,0:36:17.49,0:36:22.03,Default,,0000,0000,0000,,R of alpha of S equals\NR of S. Say it again.
Dialogue: 0,0:36:22.03,0:36:30.77,Default,,0000,0000,0000,,R of alpha of S, which is T--\Nthis T is alpha of S-- equals
Dialogue: 0,0:36:30.77,0:36:31.42,Default,,0000,0000,0000,,R.
Dialogue: 0,0:36:31.42,0:36:39.19,Default,,0000,0000,0000,,This is the composition\Nthat we learned in pre-calc.
Dialogue: 0,0:36:39.19,0:36:40.92,Default,,0000,0000,0000,,Who can find me the\Ndefinition of S?
Dialogue: 0,0:36:40.92,0:36:44.37,Default,,0000,0000,0000,,Because this may be\Na little bit hard.
Dialogue: 0,0:36:44.37,0:36:46.58,Default,,0000,0000,0000,,This may be a little bit hard.
Dialogue: 0,0:36:46.58,0:36:48.90,Default,,0000,0000,0000,,STUDENT: S [INAUDIBLE].
Dialogue: 0,0:36:48.90,0:36:52.43,Default,,0000,0000,0000,,PROFESSOR: Eh, yeah,\Nlet me write it down.
Dialogue: 0,0:36:52.43,0:36:56.87,Default,,0000,0000,0000,,I want to find out\Nwhat S of T is.
Dialogue: 0,0:36:56.87,0:36:59.94,Default,,0000,0000,0000,,
Dialogue: 0,0:36:59.94,0:37:11.30,Default,,0000,0000,0000,,Equals what in terms of the\Nfunction R of T. The one
Dialogue: 0,0:37:11.30,0:37:13.79,Default,,0000,0000,0000,,that's given here.
Dialogue: 0,0:37:13.79,0:37:14.78,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:37:14.78,0:37:22.76,Default,,0000,0000,0000,,
Dialogue: 0,0:37:22.76,0:37:26.06,Default,,0000,0000,0000,,Let's try some sort\Nof chain rule, right?
Dialogue: 0,0:37:26.06,0:37:28.70,Default,,0000,0000,0000,,So what do I know I have?
Dialogue: 0,0:37:28.70,0:37:29.82,Default,,0000,0000,0000,,I have that.
Dialogue: 0,0:37:29.82,0:37:32.74,Default,,0000,0000,0000,,Look at that.
Dialogue: 0,0:37:32.74,0:37:39.07,Default,,0000,0000,0000,,R prime of S, which\Nis the velocity of-- I
Dialogue: 0,0:37:39.07,0:37:43.84,Default,,0000,0000,0000,,erased it-- the velocity of R\Nwith respect to the arc length
Dialogue: 0,0:37:43.84,0:37:46.56,Default,,0000,0000,0000,,parameter is going to be what?
Dialogue: 0,0:37:46.56,0:37:52.09,Default,,0000,0000,0000,,R of alpha of S prime\Nwith respect to S, right?
Dialogue: 0,0:37:52.09,0:37:53.84,Default,,0000,0000,0000,,So I should put DDS.
Dialogue: 0,0:37:53.84,0:37:55.43,Default,,0000,0000,0000,,Well I'm a little bit lazy.
Dialogue: 0,0:37:55.43,0:37:58.19,Default,,0000,0000,0000,,Let's do it again.
Dialogue: 0,0:37:58.19,0:38:06.07,Default,,0000,0000,0000,,DDS, R of alpha of S.
Dialogue: 0,0:38:06.07,0:38:07.93,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:38:07.93,0:38:11.06,Default,,0000,0000,0000,,And what do I have in this case?
Dialogue: 0,0:38:11.06,0:38:18.56,Default,,0000,0000,0000,,Well, I have R prime of-- who is\Nalpha of S. T, [INAUDIBLE] of T
Dialogue: 0,0:38:18.56,0:38:27.04,Default,,0000,0000,0000,,and alpha of S times\NR prime of alpha
Dialogue: 0,0:38:27.04,0:38:30.40,Default,,0000,0000,0000,,of S times the prime outside.
Dialogue: 0,0:38:30.40,0:38:32.30,Default,,0000,0000,0000,,How do we prime\Nin the chain rule?
Dialogue: 0,0:38:32.30,0:38:35.22,Default,,0000,0000,0000,,From the outside to the\Ninside, one at a time.
Dialogue: 0,0:38:35.22,0:38:38.76,Default,,0000,0000,0000,,So I differentiated the\Nouter shell, R prime,
Dialogue: 0,0:38:38.76,0:38:39.91,Default,,0000,0000,0000,,and then times what?
Dialogue: 0,0:38:39.91,0:38:41.39,Default,,0000,0000,0000,,Chain rule, guys.
Dialogue: 0,0:38:41.39,0:38:44.89,Default,,0000,0000,0000,,Alpha prime of S. Very good.
Dialogue: 0,0:38:44.89,0:38:50.49,Default,,0000,0000,0000,,Alpha prime of S.
Dialogue: 0,0:38:50.49,0:38:51.10,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:38:51.10,0:38:54.75,Default,,0000,0000,0000,,So I would like\Nto understand how
Dialogue: 0,0:38:54.75,0:39:02.64,Default,,0000,0000,0000,,I want to compute-- how I want\Nto define S of T. If I take
Dialogue: 0,0:39:02.64,0:39:06.59,Default,,0000,0000,0000,,this in absolute value, R\Nprime of S in absolute value
Dialogue: 0,0:39:06.59,0:39:11.99,Default,,0000,0000,0000,,equals R prime of T in absolute\Nvalue times alpha prime of S
Dialogue: 0,0:39:11.99,0:39:14.56,Default,,0000,0000,0000,,in absolute value.
Dialogue: 0,0:39:14.56,0:39:15.14,Default,,0000,0000,0000,,What do I get?
Dialogue: 0,0:39:15.14,0:39:20.51,Default,,0000,0000,0000,,
Dialogue: 0,0:39:20.51,0:39:22.41,Default,,0000,0000,0000,,Who is R prime of S?
Dialogue: 0,0:39:22.41,0:39:26.16,Default,,0000,0000,0000,,This is my original\Nfunction in arc length,
Dialogue: 0,0:39:26.16,0:39:28.66,Default,,0000,0000,0000,,and that's the\Nspeed in arc length.
Dialogue: 0,0:39:28.66,0:39:30.98,Default,,0000,0000,0000,,What was the speed\Nin arc length?
Dialogue: 0,0:39:30.98,0:39:31.82,Default,,0000,0000,0000,,STUDENT: One.
Dialogue: 0,0:39:31.82,0:39:33.90,Default,,0000,0000,0000,,PROFESSOR: One.
Dialogue: 0,0:39:33.90,0:39:37.08,Default,,0000,0000,0000,,And what is the speed\Nin not in arc length?
Dialogue: 0,0:39:37.08,0:39:38.47,Default,,0000,0000,0000,,STUDENT: Five.
Dialogue: 0,0:39:38.47,0:39:41.81,Default,,0000,0000,0000,,PROFESSOR: In that case,\Nthis is going to be five.
Dialogue: 0,0:39:41.81,0:39:46.32,Default,,0000,0000,0000,,And so what is this\Nalpha prime of S guy?
Dialogue: 0,0:39:46.32,0:39:47.20,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:39:47.20,0:39:51.02,Default,,0000,0000,0000,,PROFESSOR: It's going to be 1/5.
Dialogue: 0,0:39:51.02,0:39:52.44,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:39:52.44,0:39:52.96,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:39:52.96,0:39:56.12,Default,,0000,0000,0000,,Actually alpha of S,\Nwho is that going to be?
Dialogue: 0,0:39:56.12,0:40:03.90,Default,,0000,0000,0000,,Alpha of S.
Dialogue: 0,0:40:03.90,0:40:06.61,Default,,0000,0000,0000,,Do you notice the\Ncorrespondence?
Dialogue: 0,0:40:06.61,0:40:12.07,Default,,0000,0000,0000,,We simply have to re-define\Nthis as S. That's how it goes.
Dialogue: 0,0:40:12.07,0:40:14.63,Default,,0000,0000,0000,,That five times\Nis nothing but S.
Dialogue: 0,0:40:14.63,0:40:17.01,Default,,0000,0000,0000,,STUDENT: How did you\Nget the [INAUDIBLE]?
Dialogue: 0,0:40:17.01,0:40:21.45,Default,,0000,0000,0000,,PROFESSOR: Because 1\Nequals 5 times what?
Dialogue: 0,0:40:21.45,0:40:26.20,Default,,0000,0000,0000,,1, which is arc length\Nspeed, equals 5 times what?
Dialogue: 0,0:40:26.20,0:40:26.70,Default,,0000,0000,0000,,1/5.
Dialogue: 0,0:40:26.70,0:40:27.60,Default,,0000,0000,0000,,STUDENT: Yeah, but then\Nwhere'd you get the 1?
Dialogue: 0,0:40:27.60,0:40:29.06,Default,,0000,0000,0000,,PROFESSOR: That's\None way to do it.
Dialogue: 0,0:40:29.06,0:40:32.29,Default,,0000,0000,0000,,Oh, this is by definition,\Nbecause little r means
Dialogue: 0,0:40:32.29,0:40:35.60,Default,,0000,0000,0000,,curve in arc length, and little\Ns is the arc length parameter.
Dialogue: 0,0:40:35.60,0:40:39.17,Default,,0000,0000,0000,,By definition, that\Nmeans you get speed 1.
Dialogue: 0,0:40:39.17,0:40:40.83,Default,,0000,0000,0000,,This was our assumption.
Dialogue: 0,0:40:40.83,0:40:44.14,Default,,0000,0000,0000,,So we could've gotten\Nthat much faster saying
Dialogue: 0,0:40:44.14,0:40:46.22,Default,,0000,0000,0000,,oh, well, forget\Nabout this diagram
Dialogue: 0,0:40:46.22,0:40:48.75,Default,,0000,0000,0000,,that you introduced-- and\Nit's also in the book.
Dialogue: 0,0:40:48.75,0:40:52.96,Default,,0000,0000,0000,,Simply take 5T to BS, 5T to BS.
Dialogue: 0,0:40:52.96,0:40:56.32,Default,,0000,0000,0000,,Then I get my old\Nfriend, the curve.
Dialogue: 0,0:40:56.32,0:40:59.20,Default,,0000,0000,0000,,The arc length\Nparameter is the curve.
Dialogue: 0,0:40:59.20,0:41:04.52,Default,,0000,0000,0000,,So this is the same as cosine\Nof S, sine of S, and 1.
Dialogue: 0,0:41:04.52,0:41:07.65,Default,,0000,0000,0000,,So what is the correspondence\Nbetween S and T?
Dialogue: 0,0:41:07.65,0:41:10.59,Default,,0000,0000,0000,,
Dialogue: 0,0:41:10.59,0:41:14.93,Default,,0000,0000,0000,,Since S is 5T in\Nthis example, I'll
Dialogue: 0,0:41:14.93,0:41:16.40,Default,,0000,0000,0000,,put it-- where shall I put it.
Dialogue: 0,0:41:16.40,0:41:19.81,Default,,0000,0000,0000,,I'll put it here.
Dialogue: 0,0:41:19.81,0:41:22.64,Default,,0000,0000,0000,,S is 5T.
Dialogue: 0,0:41:22.64,0:41:24.78,Default,,0000,0000,0000,,I'll say S of T is 5T.
Dialogue: 0,0:41:24.78,0:41:28.09,Default,,0000,0000,0000,,
Dialogue: 0,0:41:28.09,0:41:32.24,Default,,0000,0000,0000,,and T of S, what\Nis T in terms of S?
Dialogue: 0,0:41:32.24,0:41:37.05,Default,,0000,0000,0000,,T in terms of S is S over 5.
Dialogue: 0,0:41:37.05,0:41:39.90,Default,,0000,0000,0000,,So instead of T of\NS, we call this alpha
Dialogue: 0,0:41:39.90,0:41:47.80,Default,,0000,0000,0000,,of S. So the correspondence\Nbetween S and T, what is T?
Dialogue: 0,0:41:47.80,0:41:51.97,Default,,0000,0000,0000,,T is exactly S over\N5 in this example.
Dialogue: 0,0:41:51.97,0:41:52.64,Default,,0000,0000,0000,,Say it again.
Dialogue: 0,0:41:52.64,0:41:55.19,Default,,0000,0000,0000,,T is exactly S over 5.
Dialogue: 0,0:41:55.19,0:41:57.64,Default,,0000,0000,0000,,So alpha of S would be S over 5.
Dialogue: 0,0:41:57.64,0:42:01.77,Default,,0000,0000,0000,,In this case, alpha prime of\NS would simply be 1 over 5.
Dialogue: 0,0:42:01.77,0:42:04.41,Default,,0000,0000,0000,,Oh, so that's how I got it.
Dialogue: 0,0:42:04.41,0:42:06.36,Default,,0000,0000,0000,,That's another way to get it.
Dialogue: 0,0:42:06.36,0:42:07.50,Default,,0000,0000,0000,,Much faster.
Dialogue: 0,0:42:07.50,0:42:09.29,Default,,0000,0000,0000,,Much simpler.
Dialogue: 0,0:42:09.29,0:42:13.64,Default,,0000,0000,0000,,So just think of replacing\N5T by the S knowing
Dialogue: 0,0:42:13.64,0:42:19.02,Default,,0000,0000,0000,,that you put S here, the whole\Nthing will have speed of 1.
Dialogue: 0,0:42:19.02,0:42:19.61,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:42:19.61,0:42:21.56,Default,,0000,0000,0000,,So what do I do?
Dialogue: 0,0:42:21.56,0:42:24.64,Default,,0000,0000,0000,,I say OK, alpha prime\Nof S is 1 over 5.
Dialogue: 0,0:42:24.64,0:42:28.26,Default,,0000,0000,0000,,The whole chain rule also\Nspit out alpha prime of S
Dialogue: 0,0:42:28.26,0:42:29.50,Default,,0000,0000,0000,,to B1 over 5.
Dialogue: 0,0:42:29.50,0:42:32.54,Default,,0000,0000,0000,,Now I understand the\Nrelationship between S and T.
Dialogue: 0,0:42:32.54,0:42:33.68,Default,,0000,0000,0000,,It's very simple.
Dialogue: 0,0:42:33.68,0:42:39.80,Default,,0000,0000,0000,,S is 5T in this example,\Nor T equals S over 5.
Dialogue: 0,0:42:39.80,0:42:40.30,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:42:40.30,0:42:46.43,Default,,0000,0000,0000,,So if somebody gives you a curve\Nthat looks like cosine 5T, sine
Dialogue: 0,0:42:46.43,0:42:52.40,Default,,0000,0000,0000,,5T, 1, and that is in speed\N5, as we were able to find,
Dialogue: 0,0:42:52.40,0:42:56.80,Default,,0000,0000,0000,,how do you re-parametrize\Nthat in arc length?
Dialogue: 0,0:42:56.80,0:43:01.49,Default,,0000,0000,0000,,You just change\Nsomething inside so
Dialogue: 0,0:43:01.49,0:43:08.19,Default,,0000,0000,0000,,that you make this curve be\Nrepresentative-- representable
Dialogue: 0,0:43:08.19,0:43:12.33,Default,,0000,0000,0000,,as little r of S.\NThis is in arc length.
Dialogue: 0,0:43:12.33,0:43:13.80,Default,,0000,0000,0000,,In arc length.
Dialogue: 0,0:43:13.80,0:43:17.70,Default,,0000,0000,0000,,
Dialogue: 0,0:43:17.70,0:43:18.20,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:43:18.20,0:43:20.33,Default,,0000,0000,0000,,Finally, this is\Njust an example.
Dialogue: 0,0:43:20.33,0:43:23.68,Default,,0000,0000,0000,,Can you tell me how that\Narc length parameter
Dialogue: 0,0:43:23.68,0:43:25.87,Default,,0000,0000,0000,,is introduced in general?
Dialogue: 0,0:43:25.87,0:43:29.71,Default,,0000,0000,0000,,What is S of T by definition?
Dialogue: 0,0:43:29.71,0:43:34.20,Default,,0000,0000,0000,,What if I have\Nsomething really wild?
Dialogue: 0,0:43:34.20,0:43:36.41,Default,,0000,0000,0000,,How do I get to that\NS of T by definition?
Dialogue: 0,0:43:36.41,0:43:38.95,Default,,0000,0000,0000,,
Dialogue: 0,0:43:38.95,0:43:41.36,Default,,0000,0000,0000,,What is S of T in terms\Nof the function R?
Dialogue: 0,0:43:41.36,0:43:45.23,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] velocity\N[? of the ?] [INAUDIBLE]?
Dialogue: 0,0:43:45.23,0:43:47.84,Default,,0000,0000,0000,,PROFESSOR: S prime of T will\Nbe one of the [INAUDIBLE].
Dialogue: 0,0:43:47.84,0:43:48.67,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,0:43:48.67,0:43:49.34,Default,,0000,0000,0000,,PROFESSOR: OK.
Dialogue: 0,0:43:49.34,0:43:58.77,Default,,0000,0000,0000,,So let's see what we\Nhave if we define S of T
Dialogue: 0,0:43:58.77,0:44:12.46,Default,,0000,0000,0000,,as being integral from 0 to\NT of the speed R prime of T.
Dialogue: 0,0:44:12.46,0:44:14.33,Default,,0000,0000,0000,,And instead of T, we put tau.
Dialogue: 0,0:44:14.33,0:44:14.83,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:44:14.83,0:44:15.83,Default,,0000,0000,0000,,P tau.
Dialogue: 0,0:44:15.83,0:44:18.33,Default,,0000,0000,0000,,STUDENT: What is that?
Dialogue: 0,0:44:18.33,0:44:20.45,Default,,0000,0000,0000,,PROFESSOR: We cannot\Nput T, T, and T.
Dialogue: 0,0:44:20.45,0:44:21.26,Default,,0000,0000,0000,,STUDENT: Oh.
Dialogue: 0,0:44:21.26,0:44:22.08,Default,,0000,0000,0000,,PROFESSOR: OK?
Dialogue: 0,0:44:22.08,0:44:25.70,Default,,0000,0000,0000,,So tau is the Greek T\Nthat runs between zero
Dialogue: 0,0:44:25.70,0:44:29.49,Default,,0000,0000,0000,,and T. This is the\Ndefinition of S
Dialogue: 0,0:44:29.49,0:44:44.20,Default,,0000,0000,0000,,of T. General definition\Nof the arc length parameter
Dialogue: 0,0:44:44.20,0:44:49.65,Default,,0000,0000,0000,,that is according to the chain\Nrule, given by the chain rule.
Dialogue: 0,0:44:49.65,0:44:57.11,Default,,0000,0000,0000,,
Dialogue: 0,0:44:57.11,0:45:00.04,Default,,0000,0000,0000,,Can we verify really\Nquickly in our case,
Dialogue: 0,0:45:00.04,0:45:02.50,Default,,0000,0000,0000,,is it easy to see that\Nin our case it's correct?
Dialogue: 0,0:45:02.50,0:45:03.26,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:45:03.26,0:45:05.53,Default,,0000,0000,0000,,PROFESSOR: Oh yeah,\NS of T will be,
Dialogue: 0,0:45:05.53,0:45:08.44,Default,,0000,0000,0000,,in our case,\Nintegral from 0 to T.
Dialogue: 0,0:45:08.44,0:45:14.34,Default,,0000,0000,0000,,We are lucky our prime of tau\Nis a constant, which is 5.
Dialogue: 0,0:45:14.34,0:45:16.36,Default,,0000,0000,0000,,So I'm going to\Nhave integral from 0
Dialogue: 0,0:45:16.36,0:45:20.72,Default,,0000,0000,0000,,to T absolute value of\N5 [INAUDIBLE] d tau.
Dialogue: 0,0:45:20.72,0:45:23.10,Default,,0000,0000,0000,,And what in the world\Nis absolute value of 5?
Dialogue: 0,0:45:23.10,0:45:27.81,Default,,0000,0000,0000,,It's 5 integral from 0\Nto T [? of the ?] tau.
Dialogue: 0,0:45:27.81,0:45:30.99,Default,,0000,0000,0000,,What is integral from\N0 to T of the tau?
Dialogue: 0,0:45:30.99,0:45:33.66,Default,,0000,0000,0000,,T. 5T.
Dialogue: 0,0:45:33.66,0:45:36.94,Default,,0000,0000,0000,,So S is 5T.
Dialogue: 0,0:45:36.94,0:45:39.53,Default,,0000,0000,0000,,And that's what I\Nsaid before, right?
Dialogue: 0,0:45:39.53,0:45:41.84,Default,,0000,0000,0000,,S is 5T.
Dialogue: 0,0:45:41.84,0:45:46.72,Default,,0000,0000,0000,,S equals 5T, and\NT equals S over 5.
Dialogue: 0,0:45:46.72,0:45:51.30,Default,,0000,0000,0000,,So this thing, in general,\Nis told to us by who?
Dialogue: 0,0:45:51.30,0:45:53.16,Default,,0000,0000,0000,,It has to match the chain rule.
Dialogue: 0,0:45:53.16,0:45:55.15,Default,,0000,0000,0000,,It matches the chain rule.
Dialogue: 0,0:45:55.15,0:46:19.58,Default,,0000,0000,0000,,
Dialogue: 0,0:46:19.58,0:46:20.10,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:46:20.10,0:46:24.72,Default,,0000,0000,0000,,So again, why does that\Nmatch the chain rule?
Dialogue: 0,0:46:24.72,0:46:31.29,Default,,0000,0000,0000,,We have that-- we\Nhave R-- or how
Dialogue: 0,0:46:31.29,0:46:34.55,Default,,0000,0000,0000,,should I start, the little f,\Nthe little r, little r of S,
Dialogue: 0,0:46:34.55,0:46:35.53,Default,,0000,0000,0000,,right?
Dialogue: 0,0:46:35.53,0:46:41.40,Default,,0000,0000,0000,,Little r of S is\Nlittle r of S of T.
Dialogue: 0,0:46:41.40,0:46:45.37,Default,,0000,0000,0000,,How do I differentiate\Nthat with respect to T?
Dialogue: 0,0:46:45.37,0:46:53.24,Default,,0000,0000,0000,,Well DDT of R will be R\Nprimed with respect to S.
Dialogue: 0,0:46:53.24,0:47:01.72,Default,,0000,0000,0000,,So I'll say DRDS of\NS of T times DSDT.
Dialogue: 0,0:47:01.72,0:47:04.51,Default,,0000,0000,0000,,
Dialogue: 0,0:47:04.51,0:47:06.19,Default,,0000,0000,0000,,Now what is DSDT?
Dialogue: 0,0:47:06.19,0:47:09.22,Default,,0000,0000,0000,,DSDT was the derivative of that.
Dialogue: 0,0:47:09.22,0:47:15.87,Default,,0000,0000,0000,,It's exactly the speed\Nabsolute value of R prime of T.
Dialogue: 0,0:47:15.87,0:47:18.19,Default,,0000,0000,0000,,So when you prime\Nhere, S prime of T
Dialogue: 0,0:47:18.19,0:47:22.95,Default,,0000,0000,0000,,will be exactly that,\Nwith T replacing tau.
Dialogue: 0,0:47:22.95,0:47:24.45,Default,,0000,0000,0000,,We learned that in Calc 1.
Dialogue: 0,0:47:24.45,0:47:26.53,Default,,0000,0000,0000,,I know it's been a long time.
Dialogue: 0,0:47:26.53,0:47:28.70,Default,,0000,0000,0000,,I can feel you're\Na little bit rusty.
Dialogue: 0,0:47:28.70,0:47:29.62,Default,,0000,0000,0000,,But it doesn't matter.
Dialogue: 0,0:47:29.62,0:47:32.82,Default,,0000,0000,0000,,So S prime of T,\NDSDT will simply
Dialogue: 0,0:47:32.82,0:47:36.22,Default,,0000,0000,0000,,be absolute value\Nof R prime of T.
Dialogue: 0,0:47:36.22,0:47:40.67,Default,,0000,0000,0000,,That's the speed of\Nthe original curve.
Dialogue: 0,0:47:40.67,0:47:43.58,Default,,0000,0000,0000,,This one.
Dialogue: 0,0:47:43.58,0:47:46.18,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:47:46.18,0:47:46.91,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:47:46.91,0:47:58.96,Default,,0000,0000,0000,,So here, when I look at\NDRDS, this is going to be 1.
Dialogue: 0,0:47:58.96,0:48:02.25,Default,,0000,0000,0000,,
Dialogue: 0,0:48:02.25,0:48:06.34,Default,,0000,0000,0000,,And if you think of\Nthis as a function of T,
Dialogue: 0,0:48:06.34,0:48:11.74,Default,,0000,0000,0000,,you have DR of S of\NT. Who is R of S of T?
Dialogue: 0,0:48:11.74,0:48:15.23,Default,,0000,0000,0000,,This is R-- big\NR-- of T. So this
Dialogue: 0,0:48:15.23,0:48:21.72,Default,,0000,0000,0000,,is the DRDT Which is exactly\Nthe same as R prime of T
Dialogue: 0,0:48:21.72,0:48:24.72,Default,,0000,0000,0000,,when you put the absolute\Nvalues [INAUDIBLE].
Dialogue: 0,0:48:24.72,0:48:26.47,Default,,0000,0000,0000,,It has to fit.
Dialogue: 0,0:48:26.47,0:48:33.08,Default,,0000,0000,0000,,So indeed, you have R prime\Nof T, R prime of T, and 1.
Dialogue: 0,0:48:33.08,0:48:35.13,Default,,0000,0000,0000,,It's an identity.
Dialogue: 0,0:48:35.13,0:48:38.91,Default,,0000,0000,0000,,If I didn't put DSDT to\N[? P, ?] our prime of T
Dialogue: 0,0:48:38.91,0:48:42.22,Default,,0000,0000,0000,,in absolute value,\Nit wouldn't work out.
Dialogue: 0,0:48:42.22,0:48:48.35,Default,,0000,0000,0000,,DSDT has to be R prime\Nof T in absolute value.
Dialogue: 0,0:48:48.35,0:48:51.48,Default,,0000,0000,0000,,And this is how we\Ngot, again-- are
Dialogue: 0,0:48:51.48,0:48:54.42,Default,,0000,0000,0000,,you going to remember\Nthis without having
Dialogue: 0,0:48:54.42,0:48:56.22,Default,,0000,0000,0000,,to re-do the whole thing?
Dialogue: 0,0:48:56.22,0:49:11.31,Default,,0000,0000,0000,,Integral from 0 to T of R\Nprime of T or tau d tau.
Dialogue: 0,0:49:11.31,0:49:13.74,Default,,0000,0000,0000,,When you prime this\Nguy with respect to T
Dialogue: 0,0:49:13.74,0:49:17.57,Default,,0000,0000,0000,,as soon as it's positive--\Nwhen it is positive-- assume--
Dialogue: 0,0:49:17.57,0:49:20.14,Default,,0000,0000,0000,,why is this positive, S of T?
Dialogue: 0,0:49:20.14,0:49:23.69,Default,,0000,0000,0000,,Because you integrate from\Ntime 0 to another time
Dialogue: 0,0:49:23.69,0:49:24.67,Default,,0000,0000,0000,,a positive number.
Dialogue: 0,0:49:24.67,0:49:29.11,Default,,0000,0000,0000,,So it has to be\Npositive derivative.
Dialogue: 0,0:49:29.11,0:49:30.49,Default,,0000,0000,0000,,It's an increasing function.
Dialogue: 0,0:49:30.49,0:49:34.21,Default,,0000,0000,0000,,This function is increasing.
Dialogue: 0,0:49:34.21,0:49:37.36,Default,,0000,0000,0000,,So DSDT again will be the speed.
Dialogue: 0,0:49:37.36,0:49:38.57,Default,,0000,0000,0000,,Say it again, Magdalena?
Dialogue: 0,0:49:38.57,0:49:44.19,Default,,0000,0000,0000,,DSDT will be the speed\Nof the original line.
Dialogue: 0,0:49:44.19,0:49:47.11,Default,,0000,0000,0000,,DSDT in our case was 5.
Dialogue: 0,0:49:47.11,0:49:48.05,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:49:48.05,0:49:50.20,Default,,0000,0000,0000,,DSDT was 5.
Dialogue: 0,0:49:50.20,0:49:54.59,Default,,0000,0000,0000,,S was 5 times T.\NS was 5 times T.
Dialogue: 0,0:49:54.59,0:49:55.09,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:49:55.09,0:49:58.03,Default,,0000,0000,0000,,That was a simple\Nexample, sort of, kind of.
Dialogue: 0,0:49:58.03,0:49:59.99,Default,,0000,0000,0000,,What do we want to remember?
Dialogue: 0,0:49:59.99,0:50:03.62,Default,,0000,0000,0000,,We remember the formula\Nof the arc length.
Dialogue: 0,0:50:03.62,0:50:05.53,Default,,0000,0000,0000,,Formula of arc length.
Dialogue: 0,0:50:05.53,0:50:08.72,Default,,0000,0000,0000,,
Dialogue: 0,0:50:08.72,0:50:11.31,Default,,0000,0000,0000,,So the formula of\Narc length exists
Dialogue: 0,0:50:11.31,0:50:15.44,Default,,0000,0000,0000,,in this form because of\Nthe chain rule [INAUDIBLE]
Dialogue: 0,0:50:15.44,0:50:18.60,Default,,0000,0000,0000,,from this diagram.
Dialogue: 0,0:50:18.60,0:50:24.84,Default,,0000,0000,0000,,So always remember, we have\Na composition of functions.
Dialogue: 0,0:50:24.84,0:50:27.51,Default,,0000,0000,0000,,We use that composition of\Nfunction for the chain rule
Dialogue: 0,0:50:27.51,0:50:28.96,Default,,0000,0000,0000,,to re-parametrize it.
Dialogue: 0,0:50:28.96,0:50:30.90,Default,,0000,0000,0000,,And finally, the drunken bug.
Dialogue: 0,0:50:30.90,0:50:34.06,Default,,0000,0000,0000,,
Dialogue: 0,0:50:34.06,0:50:35.35,Default,,0000,0000,0000,,what did I take [INAUDIBLE] 14?
Dialogue: 0,0:50:35.35,0:50:37.11,Default,,0000,0000,0000,,R of t.
Dialogue: 0,0:50:37.11,0:50:44.48,Default,,0000,0000,0000,,Let's say this is 2\Ncosine t, 2 sine t.
Dialogue: 0,0:50:44.48,0:50:46.46,Default,,0000,0000,0000,,Let me make it more beautiful.
Dialogue: 0,0:50:46.46,0:50:53.50,Default,,0000,0000,0000,,Let me put 4-- 4, 4, and 3t.
Dialogue: 0,0:50:53.50,0:50:56.74,Default,,0000,0000,0000,,Can anybody tell\Nme why I did that?
Dialogue: 0,0:50:56.74,0:50:59.99,Default,,0000,0000,0000,,Maybe you can guess my mind.
Dialogue: 0,0:50:59.99,0:51:04.00,Default,,0000,0000,0000,,Find the following things.
Dialogue: 0,0:51:04.00,0:51:11.31,Default,,0000,0000,0000,,The unit vector T, by\Ndefinition R prime over R prime
Dialogue: 0,0:51:11.31,0:51:16.45,Default,,0000,0000,0000,,of t in absolute value.
Dialogue: 0,0:51:16.45,0:51:22.21,Default,,0000,0000,0000,,Find the speed of\Nthis motion R of t.
Dialogue: 0,0:51:22.21,0:51:24.71,Default,,0000,0000,0000,,This is a law of motion.
Dialogue: 0,0:51:24.71,0:51:32.43,Default,,0000,0000,0000,,And reparametrize in arclength--\Nthis curve in arclength.
Dialogue: 0,0:51:32.43,0:51:36.65,Default,,0000,0000,0000,,
Dialogue: 0,0:51:36.65,0:51:39.91,Default,,0000,0000,0000,,And you go, oh my God, I\Nhave a problem with a, b,c.
Dialogue: 0,0:51:39.91,0:51:43.26,Default,,0000,0000,0000,,The is a typical problem for\Nthe final exam, by the way.
Dialogue: 0,0:51:43.26,0:51:46.29,Default,,0000,0000,0000,,This problem popped up on\Nmany, many final exams.
Dialogue: 0,0:51:46.29,0:51:47.26,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:51:47.26,0:51:49.21,Default,,0000,0000,0000,,Is it easy?
Dialogue: 0,0:51:49.21,0:51:53.38,Default,,0000,0000,0000,,First of all, how did I\Nknow what it looked like?
Dialogue: 0,0:51:53.38,0:51:57.09,Default,,0000,0000,0000,,I should give at\Nleast an explanation.
Dialogue: 0,0:51:57.09,0:52:00.89,Default,,0000,0000,0000,,If instead of 3t I\Nwould have 3, then I
Dialogue: 0,0:52:00.89,0:52:04.72,Default,,0000,0000,0000,,would have the plane\Nz equals 3 constant.
Dialogue: 0,0:52:04.72,0:52:07.61,Default,,0000,0000,0000,,And then I'll say, I'm moving\Nin circles, in circles,
Dialogue: 0,0:52:07.61,0:52:11.16,Default,,0000,0000,0000,,in circles, in circles,\Nwith t as a real parameter,
Dialogue: 0,0:52:11.16,0:52:13.56,Default,,0000,0000,0000,,and I'm not evolving.
Dialogue: 0,0:52:13.56,0:52:16.96,Default,,0000,0000,0000,,But this is like, what, this\Nlike in in the avatar OK?
Dialogue: 0,0:52:16.96,0:52:22.06,Default,,0000,0000,0000,,So I'm performing the circular\Nmotion, but at the same time
Dialogue: 0,0:52:22.06,0:52:25.07,Default,,0000,0000,0000,,going on a different level.
Dialogue: 0,0:52:25.07,0:52:26.72,Default,,0000,0000,0000,,Assume another life.
Dialogue: 0,0:52:26.72,0:52:31.12,Default,,0000,0000,0000,,I'm starting another life\Non the next spiritual level.
Dialogue: 0,0:52:31.12,0:52:34.02,Default,,0000,0000,0000,,OK, I have no religious\Nbeliefs in that area,
Dialogue: 0,0:52:34.02,0:52:36.30,Default,,0000,0000,0000,,but it's a good physical\Nexample to give.
Dialogue: 0,0:52:36.30,0:52:38.29,Default,,0000,0000,0000,,So I go circular.
Dialogue: 0,0:52:38.29,0:52:41.53,Default,,0000,0000,0000,,Instead of going again\Ncircular and again circular,
Dialogue: 0,0:52:41.53,0:52:45.47,Default,,0000,0000,0000,,I go, oh, I go up and\Nup and up, and this 3t
Dialogue: 0,0:52:45.47,0:52:49.21,Default,,0000,0000,0000,,tells me I should also\Nevolve on the vertical.
Dialogue: 0,0:52:49.21,0:52:50.33,Default,,0000,0000,0000,,Ah-hah.
Dialogue: 0,0:52:50.33,0:52:55.37,Default,,0000,0000,0000,,So instead of circular motion\NI get a helicoidal motion.
Dialogue: 0,0:52:55.37,0:52:56.14,Default,,0000,0000,0000,,This is a helix.
Dialogue: 0,0:52:56.14,0:52:58.65,Default,,0000,0000,0000,,
Dialogue: 0,0:52:58.65,0:53:01.92,Default,,0000,0000,0000,,Could somebody tell me how I'm\Ngoing to draw such a helix?
Dialogue: 0,0:53:01.92,0:53:02.56,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:53:02.56,0:53:04.28,Default,,0000,0000,0000,,Is it easy?
Dialogue: 0,0:53:04.28,0:53:05.39,Default,,0000,0000,0000,,This helix-- yes, sir.
Dialogue: 0,0:53:05.39,0:53:08.38,Default,,0000,0000,0000,,
Dialogue: 0,0:53:08.38,0:53:09.19,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:53:09.19,0:53:10.91,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:53:10.91,0:53:12.35,Default,,0000,0000,0000,,PROFESSOR: It's like a tornado.
Dialogue: 0,0:53:12.35,0:53:14.41,Default,,0000,0000,0000,,It's like a tornado,\Nhurricane, but how
Dialogue: 0,0:53:14.41,0:53:18.44,Default,,0000,0000,0000,,do I draw the cylinder on\Nwhich this helix exists?
Dialogue: 0,0:53:18.44,0:53:22.50,Default,,0000,0000,0000,,I have to be a smart girl and\Nremember what I learned before.
Dialogue: 0,0:53:22.50,0:53:25.41,Default,,0000,0000,0000,,What is x squared\Nplus y squared?
Dialogue: 0,0:53:25.41,0:53:28.75,Default,,0000,0000,0000,,Suppose that z is not\Nplaying in the picture.
Dialogue: 0,0:53:28.75,0:53:32.56,Default,,0000,0000,0000,,If I take Mr. x and Mr. y\Nand I square them and I add
Dialogue: 0,0:53:32.56,0:53:34.70,Default,,0000,0000,0000,,them together, what do I get?
Dialogue: 0,0:53:34.70,0:53:35.75,Default,,0000,0000,0000,,STUDENT: It's the radius.
Dialogue: 0,0:53:35.75,0:53:38.26,Default,,0000,0000,0000,,PROFESSOR: What is\Nthe radius squared?
Dialogue: 0,0:53:38.26,0:53:38.91,Default,,0000,0000,0000,,4 squared.
Dialogue: 0,0:53:38.91,0:53:41.30,Default,,0000,0000,0000,,I'm gonna write 4\Nsquared because it's
Dialogue: 0,0:53:41.30,0:53:43.13,Default,,0000,0000,0000,,easier than writing 16.
Dialogue: 0,0:53:43.13,0:53:44.35,Default,,0000,0000,0000,,Thank you for your help.
Dialogue: 0,0:53:44.35,0:53:51.37,Default,,0000,0000,0000,,So I simply have to go ahead and\Ndraw the frame first, x, y, z,
Dialogue: 0,0:53:51.37,0:53:54.90,Default,,0000,0000,0000,,and then I'll say, OK, smart.
Dialogue: 0,0:53:54.90,0:53:57.79,Default,,0000,0000,0000,,R is 4.
Dialogue: 0,0:53:57.79,0:53:59.61,Default,,0000,0000,0000,,The radius should be 4.
Dialogue: 0,0:53:59.61,0:54:02.24,Default,,0000,0000,0000,,This is the cylinder\Nwhere I'm at.
Dialogue: 0,0:54:02.24,0:54:06.57,Default,,0000,0000,0000,,Where do I start\Nmy physical motion?
Dialogue: 0,0:54:06.57,0:54:10.13,Default,,0000,0000,0000,,This bug is drunk,\Nbut sort of not.
Dialogue: 0,0:54:10.13,0:54:11.52,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:54:11.52,0:54:16.02,Default,,0000,0000,0000,,It's a bug that can keep\Nthe same radius, which
Dialogue: 0,0:54:16.02,0:54:16.99,Default,,0000,0000,0000,,is quite something.
Dialogue: 0,0:54:16.99,0:54:17.82,Default,,0000,0000,0000,,STUDENT: It's tipsy.
Dialogue: 0,0:54:17.82,0:54:19.60,Default,,0000,0000,0000,,PROFESSOR: Yeah,\Nexactly, tipsy one.
Dialogue: 0,0:54:19.60,0:54:22.54,Default,,0000,0000,0000,,So how about t equals 0.
Dialogue: 0,0:54:22.54,0:54:24.91,Default,,0000,0000,0000,,Where do I start my motion?
Dialogue: 0,0:54:24.91,0:54:26.90,Default,,0000,0000,0000,,At 4, 0, 0.
Dialogue: 0,0:54:26.90,0:54:28.50,Default,,0000,0000,0000,,Where is 4, 0, 0?
Dialogue: 0,0:54:28.50,0:54:29.29,Default,,0000,0000,0000,,Over here.
Dialogue: 0,0:54:29.29,0:54:31.59,Default,,0000,0000,0000,,So that's my first\Npoint where the bug
Dialogue: 0,0:54:31.59,0:54:33.08,Default,,0000,0000,0000,,will start at t equals 0.
Dialogue: 0,0:54:33.08,0:54:34.37,Default,,0000,0000,0000,,STUDENT: How'd you get 4, 0, 0?
Dialogue: 0,0:54:34.37,0:54:36.30,Default,,0000,0000,0000,,PROFESSOR: Because I'm--\Nvery good question.
Dialogue: 0,0:54:36.30,0:54:38.64,Default,,0000,0000,0000,,I'm on x, y, z axes.
Dialogue: 0,0:54:38.64,0:54:42.05,Default,,0000,0000,0000,,4, y is 0, z is 0.
Dialogue: 0,0:54:42.05,0:54:46.65,Default,,0000,0000,0000,,I plug in t, would be 0,\Nand I get 4 times 1, 4 times
Dialogue: 0,0:54:46.65,0:54:50.62,Default,,0000,0000,0000,,0, 3 times 0, so I\Nknow I'm starting here.
Dialogue: 0,0:54:50.62,0:54:55.88,Default,,0000,0000,0000,,And when I move, I move\Nalong the cylinder like that.
Dialogue: 0,0:54:55.88,0:55:00.24,Default,,0000,0000,0000,,Can somebody tell me at\Nwhat time I'm gonna be here?
Dialogue: 0,0:55:00.24,0:55:03.91,Default,,0000,0000,0000,,Not at 1:50, but what time am\NI going to be at this point?
Dialogue: 0,0:55:03.91,0:55:08.09,Default,,0000,0000,0000,,And then I continue, and I go\Nup, and I continue and I go up.
Dialogue: 0,0:55:08.09,0:55:09.71,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:55:09.71,0:55:11.03,Default,,0000,0000,0000,,PROFESSOR: Pi over 2.
Dialogue: 0,0:55:11.03,0:55:12.67,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:55:12.67,0:55:14.48,Default,,0000,0000,0000,,And can you-- can\Nyou tell me what
Dialogue: 0,0:55:14.48,0:55:16.97,Default,,0000,0000,0000,,point it is in space in R 3?
Dialogue: 0,0:55:16.97,0:55:18.17,Default,,0000,0000,0000,,Plug in pi over 2.
Dialogue: 0,0:55:18.17,0:55:19.62,Default,,0000,0000,0000,,You can do it faster than me.
Dialogue: 0,0:55:19.62,0:55:20.33,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:55:20.33,0:55:23.94,Default,,0000,0000,0000,,PROFESSOR: 0, 4 and 3 pi over 2.
Dialogue: 0,0:55:23.94,0:55:25.58,Default,,0000,0000,0000,,And I keep going.
Dialogue: 0,0:55:25.58,0:55:28.85,Default,,0000,0000,0000,,So this is the helicoidal\Nmotion I'm talking about.
Dialogue: 0,0:55:28.85,0:55:31.69,Default,,0000,0000,0000,,The unit vector-- is it easy\Nto write it on the final?
Dialogue: 0,0:55:31.69,0:55:33.15,Default,,0000,0000,0000,,Can do that in no time.
Dialogue: 0,0:55:33.15,0:55:39.14,Default,,0000,0000,0000,,So we get like, let's say, 30%,\N30%, 30%, and 10% for drawing.
Dialogue: 0,0:55:39.14,0:55:40.51,Default,,0000,0000,0000,,How about that?
Dialogue: 0,0:55:40.51,0:55:44.21,Default,,0000,0000,0000,,That would be a typical\Ngrid for the problem.
Dialogue: 0,0:55:44.21,0:55:49.90,Default,,0000,0000,0000,,So t will be minus 4 sine t.
Dialogue: 0,0:55:49.90,0:55:53.87,Default,,0000,0000,0000,,If I make a mistake, are\Nyou gonna shout, please?
Dialogue: 0,0:55:53.87,0:55:58.66,Default,,0000,0000,0000,,4 cosine t and 3\Ndivided by what?
Dialogue: 0,0:55:58.66,0:56:00.95,Default,,0000,0000,0000,,What is the tangent unit vector?
Dialogue: 0,0:56:00.95,0:56:03.84,Default,,0000,0000,0000,,At every point in\Nspace, I'm gonna
Dialogue: 0,0:56:03.84,0:56:05.58,Default,,0000,0000,0000,,have this tangent unit vector.
Dialogue: 0,0:56:05.58,0:56:08.14,Default,,0000,0000,0000,,It has to have\Nlength 1, and it has
Dialogue: 0,0:56:08.14,0:56:11.10,Default,,0000,0000,0000,,to be tangent to my trajectory.
Dialogue: 0,0:56:11.10,0:56:12.20,Default,,0000,0000,0000,,I'll draw him.
Dialogue: 0,0:56:12.20,0:56:15.67,Default,,0000,0000,0000,,So he gives me a\Nfield, a vector field--
Dialogue: 0,0:56:15.67,0:56:19.08,Default,,0000,0000,0000,,this is beautiful-- T\Nof t is a vector field.
Dialogue: 0,0:56:19.08,0:56:20.85,Default,,0000,0000,0000,,At every point of\Nthe trajectory,
Dialogue: 0,0:56:20.85,0:56:23.46,Default,,0000,0000,0000,,I have only one such vector.
Dialogue: 0,0:56:23.46,0:56:27.07,Default,,0000,0000,0000,,That's what we mean\Nby vector field.
Dialogue: 0,0:56:27.07,0:56:29.99,Default,,0000,0000,0000,,What's the magnitude?
Dialogue: 0,0:56:29.99,0:56:31.45,Default,,0000,0000,0000,,It's buzzing.
Dialogue: 0,0:56:31.45,0:56:33.40,Default,,0000,0000,0000,,It's buzzing.
Dialogue: 0,0:56:33.40,0:56:35.22,Default,,0000,0000,0000,,How did you do it?
Dialogue: 0,0:56:35.22,0:56:40.14,Default,,0000,0000,0000,,4, 16 times sine squared\Nplus cosine squared.
Dialogue: 0,0:56:40.14,0:56:42.40,Default,,0000,0000,0000,,16 plus 9 is 25.
Dialogue: 0,0:56:42.40,0:56:46.12,Default,,0000,0000,0000,,Square root of 25 is 5.
Dialogue: 0,0:56:46.12,0:56:47.87,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:56:47.87,0:56:49.86,Default,,0000,0000,0000,,Do I have to write this down?
Dialogue: 0,0:56:49.86,0:56:51.78,Default,,0000,0000,0000,,Are you guys sure?
Dialogue: 0,0:56:51.78,0:56:53.11,Default,,0000,0000,0000,,STUDENT: You plugged in 0 for t?
Dialogue: 0,0:56:53.11,0:56:56.39,Default,,0000,0000,0000,,Is that what you did\Nwhen you [INAUDIBLE]
Dialogue: 0,0:56:56.39,0:56:58.98,Default,,0000,0000,0000,,PROFESSOR: No, I plugged\N0 for t when I started.
Dialogue: 0,0:56:58.98,0:57:01.68,Default,,0000,0000,0000,,But when I'm computing,\NI don't plug anything,
Dialogue: 0,0:57:01.68,0:57:03.94,Default,,0000,0000,0000,,I just do it in general.
Dialogue: 0,0:57:03.94,0:57:07.71,Default,,0000,0000,0000,,I said 16 sine squared\Nplus 16 cosine squared
Dialogue: 0,0:57:07.71,0:57:10.40,Default,,0000,0000,0000,,is 16 times 1 plus 9.
Dialogue: 0,0:57:10.40,0:57:13.41,Default,,0000,0000,0000,,My son would know this\None and he's 10, right?
Dialogue: 0,0:57:13.41,0:57:16.03,Default,,0000,0000,0000,,16 plus 9 square root of 25.
Dialogue: 0,0:57:16.03,0:57:17.81,Default,,0000,0000,0000,,And I taught him\Nabout square roots.
Dialogue: 0,0:57:17.81,0:57:20.59,Default,,0000,0000,0000,,So square root of 25,\Nhe knows that's 5.
Dialogue: 0,0:57:20.59,0:57:22.25,Default,,0000,0000,0000,,And if he knows\Nthat's 5, then you
Dialogue: 0,0:57:22.25,0:57:24.47,Default,,0000,0000,0000,,should do that in a\Nminute-- in a second.
Dialogue: 0,0:57:24.47,0:57:25.33,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:57:25.33,0:57:32.43,Default,,0000,0000,0000,,So t will simply be-- if you\Ndon't simplify 1/5 minus 4 sine
Dialogue: 0,0:57:32.43,0:57:37.46,Default,,0000,0000,0000,,t 4 cosine t 3 in the final,\Nit wouldn't be a big deal,
Dialogue: 0,0:57:37.46,0:57:39.32,Default,,0000,0000,0000,,I would give you\Nstill partial credit,
Dialogue: 0,0:57:39.32,0:57:42.26,Default,,0000,0000,0000,,but what if we raise this\Nas a multiple choice?
Dialogue: 0,0:57:42.26,0:57:46.52,Default,,0000,0000,0000,,Then you have to be able\Nto find where the 5 is.
Dialogue: 0,0:57:46.52,0:57:47.27,Default,,0000,0000,0000,,What is the speed?
Dialogue: 0,0:57:47.27,0:57:49.41,Default,,0000,0000,0000,,Was that hard for you to find?
Dialogue: 0,0:57:49.41,0:57:50.99,Default,,0000,0000,0000,,Where is the speed hiding?
Dialogue: 0,0:57:50.99,0:57:53.80,Default,,0000,0000,0000,,It's exactly the\Ndenominator of R.
Dialogue: 0,0:57:53.80,0:57:57.07,Default,,0000,0000,0000,,This is the speed\Nof the curve in t.
Dialogue: 0,0:57:57.07,0:57:58.57,Default,,0000,0000,0000,,And that was 5.
Dialogue: 0,0:57:58.57,0:58:01.19,Default,,0000,0000,0000,,You told me the speed was\N5, and I'm very happy.
Dialogue: 0,0:58:01.19,0:58:07.80,Default,,0000,0000,0000,,So you got 30%, 30%, 10% from\Nthe picture-- no, this picture.
Dialogue: 0,0:58:07.80,0:58:09.30,Default,,0000,0000,0000,,This picture's no good.
Dialogue: 0,0:58:09.30,0:58:13.19,Default,,0000,0000,0000,,STUDENT: What does the\Nfirst word of c say?
Dialogue: 0,0:58:13.19,0:58:15.29,Default,,0000,0000,0000,,Question c, what does\Nthe first word say?
Dialogue: 0,0:58:15.29,0:58:16.37,Default,,0000,0000,0000,,PROFESSOR: The first what?
Dialogue: 0,0:58:16.37,0:58:17.73,Default,,0000,0000,0000,,STUDENT: The word.
Dialogue: 0,0:58:17.73,0:58:19.38,Default,,0000,0000,0000,,PROFESSOR: Reparametrize.
Dialogue: 0,0:58:19.38,0:58:23.07,Default,,0000,0000,0000,,Reparametrize this\Ncurve in arclength.
Dialogue: 0,0:58:23.07,0:58:26.43,Default,,0000,0000,0000,,Oh my God, so according\Nto that chain rule,
Dialogue: 0,0:58:26.43,0:58:31.43,Default,,0000,0000,0000,,could you guys remember-- if you\Nremember, what is the s of t?
Dialogue: 0,0:58:31.43,0:58:39.08,Default,,0000,0000,0000,,If I want to reparametrize\Nin arclength integral from 0
Dialogue: 0,0:58:39.08,0:58:45.58,Default,,0000,0000,0000,,to t of the speed, how\Nis the speed defined?
Dialogue: 0,0:58:45.58,0:58:49.04,Default,,0000,0000,0000,,Absolute value of r prime of t.
Dialogue: 0,0:58:49.04,0:58:54.37,Default,,0000,0000,0000,,dt, but I don't like t,\NI write-- I write tau.
Dialogue: 0,0:58:54.37,0:58:56.61,Default,,0000,0000,0000,,Like Dr. [? Solinger, ?]\Nyou know him,
Dialogue: 0,0:58:56.61,0:58:59.37,Default,,0000,0000,0000,,he's one of my colleagues,\Ncalls that-- that's
Dialogue: 0,0:58:59.37,0:59:00.88,Default,,0000,0000,0000,,the dummy dummy variable.
Dialogue: 0,0:59:00.88,0:59:03.77,Default,,0000,0000,0000,,In many books, tau is\Nthe dummy variable.
Dialogue: 0,0:59:03.77,0:59:08.48,Default,,0000,0000,0000,,Or you can-- some people even\Nput t by inclusive notation.
Dialogue: 0,0:59:08.48,0:59:09.90,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:59:09.90,0:59:12.58,Default,,0000,0000,0000,,So in my case, what is s of t?
Dialogue: 0,0:59:12.58,0:59:14.07,Default,,0000,0000,0000,,It should be easy.
Dialogue: 0,0:59:14.07,0:59:18.67,Default,,0000,0000,0000,,Because although this\Nnot a circular motion,
Dialogue: 0,0:59:18.67,0:59:20.61,Default,,0000,0000,0000,,I still have constant speed.
Dialogue: 0,0:59:20.61,0:59:23.59,Default,,0000,0000,0000,,So who is that special speed?
Dialogue: 0,0:59:23.59,0:59:24.35,Default,,0000,0000,0000,,5.
Dialogue: 0,0:59:24.35,0:59:31.40,Default,,0000,0000,0000,,Integral from 0 to t5 d tau,\Nand that is 5t, am I right?
Dialogue: 0,0:59:31.40,0:59:32.05,Default,,0000,0000,0000,,5t.
Dialogue: 0,0:59:32.05,0:59:36.98,Default,,0000,0000,0000,,So-- so if I want to\Nreparametrize this helix,
Dialogue: 0,0:59:36.98,0:59:41.60,Default,,0000,0000,0000,,keeping in mind\Nthat s is simply 5t,
Dialogue: 0,0:59:41.60,0:59:47.39,Default,,0000,0000,0000,,what do I have to do to\Nget 100% on this problem?
Dialogue: 0,0:59:47.39,0:59:57.59,Default,,0000,0000,0000,,All I have to do is say little r\Nof s, which represents actually
Dialogue: 0,0:59:57.59,1:00:00.69,Default,,0000,0000,0000,,big R of t of s.
Dialogue: 0,1:00:00.69,1:00:02.38,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:00:02.38,1:00:04.45,Default,,0000,0000,0000,,Do you have to write\Nall this story down?
Dialogue: 0,1:00:04.45,1:00:04.95,Default,,0000,0000,0000,,No.
Dialogue: 0,1:00:04.95,1:00:07.65,Default,,0000,0000,0000,,But that will remind\Nyou of the diagram.
Dialogue: 0,1:00:07.65,1:00:12.00,Default,,0000,0000,0000,,So I have R of t of s.
Dialogue: 0,1:00:12.00,1:00:13.27,Default,,0000,0000,0000,,Or alpha of s.
Dialogue: 0,1:00:13.27,1:00:15.36,Default,,0000,0000,0000,,And this is t of s.
Dialogue: 0,1:00:15.36,1:00:16.39,Default,,0000,0000,0000,,t of s.
Dialogue: 0,1:00:16.39,1:00:19.64,Default,,0000,0000,0000,,R of t of s is R of s, right?
Dialogue: 0,1:00:19.64,1:00:21.10,Default,,0000,0000,0000,,Do you have to remind me?
Dialogue: 0,1:00:21.10,1:00:21.60,Default,,0000,0000,0000,,No.
Dialogue: 0,1:00:21.60,1:00:23.24,Default,,0000,0000,0000,,The heck with the diagram.
Dialogue: 0,1:00:23.24,1:00:26.90,Default,,0000,0000,0000,,As long as you understood\Nit was about a composition
Dialogue: 0,1:00:26.90,1:00:28.25,Default,,0000,0000,0000,,of functions.
Dialogue: 0,1:00:28.25,1:00:30.64,Default,,0000,0000,0000,,And then R of s\Nwill simply be what?
Dialogue: 0,1:00:30.64,1:00:33.06,Default,,0000,0000,0000,,How do we do that fast?
Dialogue: 0,1:00:33.06,1:00:37.43,Default,,0000,0000,0000,,We replaced t by s over 5.
Dialogue: 0,1:00:37.43,1:00:38.79,Default,,0000,0000,0000,,Where from?
Dialogue: 0,1:00:38.79,1:00:42.28,Default,,0000,0000,0000,,Little s equals 5t,\Nwe just computed it.
Dialogue: 0,1:00:42.28,1:00:43.68,Default,,0000,0000,0000,,Little s equals 5t.
Dialogue: 0,1:00:43.68,1:00:44.78,Default,,0000,0000,0000,,That's all you need to do.
Dialogue: 0,1:00:44.78,1:00:49.11,Default,,0000,0000,0000,,To pull out t, replace\Nthe third sub s.
Dialogue: 0,1:00:49.11,1:00:52.87,Default,,0000,0000,0000,,So what is the function\Nt in terms of s?
Dialogue: 0,1:00:52.87,1:00:55.10,Default,,0000,0000,0000,,It's s over 5.
Dialogue: 0,1:00:55.10,1:00:59.60,Default,,0000,0000,0000,,What is the function t, what's\Nthe parameter t, in terms of s?
Dialogue: 0,1:00:59.60,1:01:01.48,Default,,0000,0000,0000,,s over 5.
Dialogue: 0,1:01:01.48,1:01:07.14,Default,,0000,0000,0000,,And finally, at the end, 3\Ntimes what is the stinking t?
Dialogue: 0,1:01:07.14,1:01:09.14,Default,,0000,0000,0000,,s over 5.
Dialogue: 0,1:01:09.14,1:01:11.00,Default,,0000,0000,0000,,I'm done.
Dialogue: 0,1:01:11.00,1:01:16.24,Default,,0000,0000,0000,,I got 100% I don't want\Nto say how much time it's
Dialogue: 0,1:01:16.24,1:01:18.18,Default,,0000,0000,0000,,gonna take me to\Ndo it, but I think
Dialogue: 0,1:01:18.18,1:01:20.48,Default,,0000,0000,0000,,I can do it in like, 2\Nor 3 minutes, 5 minutes.
Dialogue: 0,1:01:20.48,1:01:24.29,Default,,0000,0000,0000,,If I know the problem I'll\Ndo it in a few minutes.
Dialogue: 0,1:01:24.29,1:01:26.64,Default,,0000,0000,0000,,If I waste too\Nmuch time thinking,
Dialogue: 0,1:01:26.64,1:01:28.71,Default,,0000,0000,0000,,I'm not gonna do it at all.
Dialogue: 0,1:01:28.71,1:01:30.47,Default,,0000,0000,0000,,So what do you have to remember?
Dialogue: 0,1:01:30.47,1:01:35.01,Default,,0000,0000,0000,,You have to remember the\Nformula that says s of t,
Dialogue: 0,1:01:35.01,1:01:40.78,Default,,0000,0000,0000,,the arclength parameter--\Nthe arclength parameter
Dialogue: 0,1:01:40.78,1:01:46.74,Default,,0000,0000,0000,,equals integral from 0 to\Nt is 0 to t of the speed.
Dialogue: 0,1:01:46.74,1:01:53.14,Default,,0000,0000,0000,,Does this element of information\Nremind you of something?
Dialogue: 0,1:01:53.14,1:01:56.46,Default,,0000,0000,0000,,Of course, s will be the\Narclength, practically.
Dialogue: 0,1:01:56.46,1:01:58.45,Default,,0000,0000,0000,,What kind of parameter is that?
Dialogue: 0,1:01:58.45,1:02:03.51,Default,,0000,0000,0000,,Is you're measuring how\Nbig-- how much you travel.
Dialogue: 0,1:02:03.51,1:02:06.96,Default,,0000,0000,0000,,s of t is the time you\Ntravel-- the distance
Dialogue: 0,1:02:06.96,1:02:10.68,Default,,0000,0000,0000,,you travel in time t.
Dialogue: 0,1:02:10.68,1:02:15.57,Default,,0000,0000,0000,,
Dialogue: 0,1:02:15.57,1:02:20.01,Default,,0000,0000,0000,,So it's a space-time continuum.
Dialogue: 0,1:02:20.01,1:02:23.60,Default,,0000,0000,0000,,It's a space-time relationship.
Dialogue: 0,1:02:23.60,1:02:27.40,Default,,0000,0000,0000,,So it's the space you\Ntravel in times t.
Dialogue: 0,1:02:27.40,1:02:30.44,Default,,0000,0000,0000,,Now, if I drive to Amarillo\Nat 60 miles an hour,
Dialogue: 0,1:02:30.44,1:02:35.00,Default,,0000,0000,0000,,I'm happy and sassy, and I\Nsay OK, it's gonna be s of t.
Dialogue: 0,1:02:35.00,1:02:37.67,Default,,0000,0000,0000,,My displacement to\NAmarillo is given
Dialogue: 0,1:02:37.67,1:02:41.54,Default,,0000,0000,0000,,by this linear law, 60 times t.
Dialogue: 0,1:02:41.54,1:02:42.91,Default,,0000,0000,0000,,Suppose I'm on cruise control.
Dialogue: 0,1:02:42.91,1:02:44.37,Default,,0000,0000,0000,,But I've never on\Ncruise control.
Dialogue: 0,1:02:44.37,1:02:47.26,Default,,0000,0000,0000,,
Dialogue: 0,1:02:47.26,1:02:50.75,Default,,0000,0000,0000,,So this is going to\Nbe very variable.
Dialogue: 0,1:02:50.75,1:02:54.67,Default,,0000,0000,0000,,And the only way you can compute\Nthis displacement or distance
Dialogue: 0,1:02:54.67,1:02:57.14,Default,,0000,0000,0000,,traveled, it'll\Nbe as an integral.
Dialogue: 0,1:02:57.14,1:03:01.33,Default,,0000,0000,0000,,From time 0, when I start\Ndriving, to time t of my speed,
Dialogue: 0,1:03:01.33,1:03:02.22,Default,,0000,0000,0000,,and that's it.
Dialogue: 0,1:03:02.22,1:03:04.36,Default,,0000,0000,0000,,That's all you have to remember.
Dialogue: 0,1:03:04.36,1:03:08.20,Default,,0000,0000,0000,,It's actually-- mathematics\Nshould not be memorized.
Dialogue: 0,1:03:08.20,1:03:11.52,Default,,0000,0000,0000,,It should be sort of\Nunderstood, just like physics.
Dialogue: 0,1:03:11.52,1:03:15.17,Default,,0000,0000,0000,,What if you take your\Nfirst test, quiz,
Dialogue: 0,1:03:15.17,1:03:18.81,Default,,0000,0000,0000,,whatever, on WeBWorK or in\Nperson, and you freak out.
Dialogue: 0,1:03:18.81,1:03:22.92,Default,,0000,0000,0000,,You get such a\Nproblem, and you blank.
Dialogue: 0,1:03:22.92,1:03:24.95,Default,,0000,0000,0000,,You just blank.
Dialogue: 0,1:03:24.95,1:03:27.67,Default,,0000,0000,0000,,What do you do?
Dialogue: 0,1:03:27.67,1:03:31.62,Default,,0000,0000,0000,,You sort of know this,\Nbut you have a blank.
Dialogue: 0,1:03:31.62,1:03:34.10,Default,,0000,0000,0000,,Always tell me, right?
Dialogue: 0,1:03:34.10,1:03:36.03,Default,,0000,0000,0000,,Always email, say I'm\Nfreaking out here.
Dialogue: 0,1:03:36.03,1:03:38.69,Default,,0000,0000,0000,,I don't know what's\Nthe matter with me.
Dialogue: 0,1:03:38.69,1:03:46.40,Default,,0000,0000,0000,,Don't cut our correspondence,\Neither by speaking or by email.
Dialogue: 0,1:03:46.40,1:03:48.81,Default,,0000,0000,0000,,Very few of you email me.
Dialogue: 0,1:03:48.81,1:03:51.57,Default,,0000,0000,0000,,I'd like you to be\Nmore like my friends,
Dialogue: 0,1:03:51.57,1:03:53.25,Default,,0000,0000,0000,,and I would be more\Nlike your tutor,
Dialogue: 0,1:03:53.25,1:03:55.40,Default,,0000,0000,0000,,and when you\Nencounter an obstacle,
Dialogue: 0,1:03:55.40,1:03:58.34,Default,,0000,0000,0000,,you email me and\NI email you back.
Dialogue: 0,1:03:58.34,1:04:00.66,Default,,0000,0000,0000,,This is what I want.
Dialogue: 0,1:04:00.66,1:04:03.64,Default,,0000,0000,0000,,The WeBWorK, this is what I\Nwant our model of interaction
Dialogue: 0,1:04:03.64,1:04:05.75,Default,,0000,0000,0000,,to become.
Dialogue: 0,1:04:05.75,1:04:06.88,Default,,0000,0000,0000,,Don't be shy.
Dialogue: 0,1:04:06.88,1:04:10.60,Default,,0000,0000,0000,,Many of you are shy even to\Nask questions in the classroom.
Dialogue: 0,1:04:10.60,1:04:12.50,Default,,0000,0000,0000,,And I'm not going\Nto let you be shy.
Dialogue: 0,1:04:12.50,1:04:16.64,Default,,0000,0000,0000,,At 2 o'clock I'm going to let\Nyou ask all the questions you
Dialogue: 0,1:04:16.64,1:04:19.69,Default,,0000,0000,0000,,have about homework,\Nand we will do
Dialogue: 0,1:04:19.69,1:04:21.25,Default,,0000,0000,0000,,more homework-like questions.
Dialogue: 0,1:04:21.25,1:04:24.06,Default,,0000,0000,0000,,I want to imitate some\NWeBWorK questions.
Dialogue: 0,1:04:24.06,1:04:27.81,Default,,0000,0000,0000,,And we will work them out.
Dialogue: 0,1:04:27.81,1:04:32.31,Default,,0000,0000,0000,,So any questions right now?
Dialogue: 0,1:04:32.31,1:04:32.84,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:04:32.84,1:04:35.68,Default,,0000,0000,0000,,STUDENT: You emailed-- did\Nyou email us this weekend
Dialogue: 0,1:04:35.68,1:04:37.58,Default,,0000,0000,0000,,the numbers for WeBWorK?
Dialogue: 0,1:04:37.58,1:04:41.04,Default,,0000,0000,0000,,PROFESSOR: I emailed you the\NWeBWorK assignment completely.
Dialogue: 0,1:04:41.04,1:04:44.91,Default,,0000,0000,0000,,I mean, the link-- you\Nget in and you of see it.
Dialogue: 0,1:04:44.91,1:04:48.34,Default,,0000,0000,0000,,STUDENT: Which email\Ndid you send that to?
Dialogue: 0,1:04:48.34,1:04:49.66,Default,,0000,0000,0000,,PROFESSOR: To your TTU.
Dialogue: 0,1:04:49.66,1:04:51.86,Default,,0000,0000,0000,,All the emails go to your TTU.
Dialogue: 0,1:04:51.86,1:04:56.40,Default,,0000,0000,0000,,You have one week\Nstarting yesterday until,
Dialogue: 0,1:04:56.40,1:04:58.14,Default,,0000,0000,0000,,was it the 2nd?
Dialogue: 0,1:04:58.14,1:05:00.01,Default,,0000,0000,0000,,I gave you a little\Nbit more time.
Dialogue: 0,1:05:00.01,1:05:03.01,Default,,0000,0000,0000,,So it's due on the\N2nd of February at,
Dialogue: 0,1:05:03.01,1:05:03.98,Default,,0000,0000,0000,,I forgot what time.
Dialogue: 0,1:05:03.98,1:05:05.44,Default,,0000,0000,0000,,1 o'clock or something.
Dialogue: 0,1:05:05.44,1:05:06.42,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:05:06.42,1:05:07.88,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\NI was confused
Dialogue: 0,1:05:07.88,1:05:10.31,Default,,0000,0000,0000,,at the beginning where you got\Nx squared plus y squared equals
Dialogue: 0,1:05:10.31,1:05:10.81,Default,,0000,0000,0000,,4 squared.
Dialogue: 0,1:05:10.81,1:05:13.60,Default,,0000,0000,0000,,Where did you get that?
Dialogue: 0,1:05:13.60,1:05:14.18,Default,,0000,0000,0000,,PROFESSOR: Oh.
Dialogue: 0,1:05:14.18,1:05:15.03,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:05:15.03,1:05:19.16,Default,,0000,0000,0000,,I eliminated the t between\Nthe first two guys.
Dialogue: 0,1:05:19.16,1:05:24.94,Default,,0000,0000,0000,,This is called eliminating a\Nparameter, which was the time
Dialogue: 0,1:05:24.94,1:05:27.97,Default,,0000,0000,0000,,parameter between x and y.
Dialogue: 0,1:05:27.97,1:05:32.09,Default,,0000,0000,0000,,When I do that, I get a\Nbeautiful equation which
Dialogue: 0,1:05:32.09,1:05:36.67,Default,,0000,0000,0000,,is x squared plus y squared\Nequals 16, which tells me, hey,
Dialogue: 0,1:05:36.67,1:05:39.83,Default,,0000,0000,0000,,your curve sits on\Nthe surface x squared
Dialogue: 0,1:05:39.83,1:05:42.23,Default,,0000,0000,0000,,plus y squared equals 16.
Dialogue: 0,1:05:42.23,1:05:44.32,Default,,0000,0000,0000,,It's not the same\Nwith the surface,
Dialogue: 0,1:05:44.32,1:05:47.52,Default,,0000,0000,0000,,because you have additional\Nconstraints on the z.
Dialogue: 0,1:05:47.52,1:05:52.37,Default,,0000,0000,0000,,So the z is constrained\Nto follow this thing.
Dialogue: 0,1:05:52.37,1:05:59.91,Default,,0000,0000,0000,,Now, could anybody tell me how\NI'm gonna write eventually--
Dialogue: 0,1:05:59.91,1:06:02.40,Default,,0000,0000,0000,,this is a harder\Ntask, OK, but I'm
Dialogue: 0,1:06:02.40,1:06:09.13,Default,,0000,0000,0000,,glad you asked because I\Nwanted to discuss that.
Dialogue: 0,1:06:09.13,1:06:13.02,Default,,0000,0000,0000,,How do I express t\Nin terms of x and y?
Dialogue: 0,1:06:13.02,1:06:16.79,Default,,0000,0000,0000,,I mean, I'm going to have an\Nintersection of two surfaces.
Dialogue: 0,1:06:16.79,1:06:18.46,Default,,0000,0000,0000,,How?
Dialogue: 0,1:06:18.46,1:06:21.01,Default,,0000,0000,0000,,This is just practically\Ndifferential geometry
Dialogue: 0,1:06:21.01,1:06:24.25,Default,,0000,0000,0000,,or advanced calculus\Nat the same time.
Dialogue: 0,1:06:24.25,1:06:27.97,Default,,0000,0000,0000,,x squared plus y squared\Nequals our first surface
Dialogue: 0,1:06:27.97,1:06:31.95,Default,,0000,0000,0000,,that I'm thinking about, which\NI'm sitting with my curve.
Dialogue: 0,1:06:31.95,1:06:35.37,Default,,0000,0000,0000,,But I also have my curve\Nto be at the intersection
Dialogue: 0,1:06:35.37,1:06:39.49,Default,,0000,0000,0000,,between the cylinder\Nand something else.
Dialogue: 0,1:06:39.49,1:06:45.21,Default,,0000,0000,0000,,And it's hard to figure out how\NI'm going to do the other one.
Dialogue: 0,1:06:45.21,1:06:49.27,Default,,0000,0000,0000,,Can anybody figure\Nout how another
Dialogue: 0,1:06:49.27,1:06:51.38,Default,,0000,0000,0000,,surface-- what is the surface?
Dialogue: 0,1:06:51.38,1:06:56.38,Default,,0000,0000,0000,,A surface will have an implicit\Nequation of the type f of x, y,
Dialogue: 0,1:06:56.38,1:06:58.00,Default,,0000,0000,0000,,z equals a constant.
Dialogue: 0,1:06:58.00,1:07:01.14,Default,,0000,0000,0000,,So you have to sort of\Neliminate your parameter t.
Dialogue: 0,1:07:01.14,1:07:02.47,Default,,0000,0000,0000,,The heck with the time.
Dialogue: 0,1:07:02.47,1:07:05.31,Default,,0000,0000,0000,,We don't care about time,\Nwe only care about space.
Dialogue: 0,1:07:05.31,1:07:07.37,Default,,0000,0000,0000,,So is there any other\Nway to eliminate
Dialogue: 0,1:07:07.37,1:07:09.96,Default,,0000,0000,0000,,t between the equations?
Dialogue: 0,1:07:09.96,1:07:13.86,Default,,0000,0000,0000,,I have to use the information\Nthat I haven't used yet.
Dialogue: 0,1:07:13.86,1:07:15.34,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:07:15.34,1:07:19.58,Default,,0000,0000,0000,,Now my question is\Nthat, how can I do that?
Dialogue: 0,1:07:19.58,1:07:22.62,Default,,0000,0000,0000,,z is beautiful.
Dialogue: 0,1:07:22.62,1:07:23.77,Default,,0000,0000,0000,,3 is beautiful.
Dialogue: 0,1:07:23.77,1:07:25.70,Default,,0000,0000,0000,,t drives me nuts.
Dialogue: 0,1:07:25.70,1:07:30.28,Default,,0000,0000,0000,,How do I get the t out of\Nthe first two equations?
Dialogue: 0,1:07:30.28,1:07:32.78,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:07:32.78,1:07:35.82,Default,,0000,0000,0000,,Yeah, I divide them\None to the other one.
Dialogue: 0,1:07:35.82,1:07:39.73,Default,,0000,0000,0000,,So if I-- for example,\NI go y over x.
Dialogue: 0,1:07:39.73,1:07:42.75,Default,,0000,0000,0000,,What is y over x?
Dialogue: 0,1:07:42.75,1:07:45.42,Default,,0000,0000,0000,,It's tangent of t.
Dialogue: 0,1:07:45.42,1:07:48.80,Default,,0000,0000,0000,,How do I pull Mr. t out?
Dialogue: 0,1:07:48.80,1:07:51.65,Default,,0000,0000,0000,,Say t, get out.
Dialogue: 0,1:07:51.65,1:07:54.72,Default,,0000,0000,0000,,Well, I have to think about\Nif I'm not losing anything.
Dialogue: 0,1:07:54.72,1:07:58.45,Default,,0000,0000,0000,,But in principle, t would\Nbe arctangent of y over x.
Dialogue: 0,1:07:58.45,1:08:01.54,Default,,0000,0000,0000,,
Dialogue: 0,1:08:01.54,1:08:02.04,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:08:02.04,1:08:06.08,Default,,0000,0000,0000,,So, I'm having two\Nequations of this type.
Dialogue: 0,1:08:06.08,1:08:08.50,Default,,0000,0000,0000,,I'm eliminating t\Nbetween the two.
Dialogue: 0,1:08:08.50,1:08:10.46,Default,,0000,0000,0000,,I don't care about\Nthe other one.
Dialogue: 0,1:08:10.46,1:08:13.77,Default,,0000,0000,0000,,I only cared for you\Nto draw the cylinder.
Dialogue: 0,1:08:13.77,1:08:17.25,Default,,0000,0000,0000,,So we can draw point\Nby point the helix.
Dialogue: 0,1:08:17.25,1:08:18.86,Default,,0000,0000,0000,,I don't draw many points.
Dialogue: 0,1:08:18.86,1:08:22.58,Default,,0000,0000,0000,,I draw only t equals 0,\Nwhere I'm starting over here,
Dialogue: 0,1:08:22.58,1:08:25.16,Default,,0000,0000,0000,,t equals pi over 2, which\N[INAUDIBLE] gave me,
Dialogue: 0,1:08:25.16,1:08:27.05,Default,,0000,0000,0000,,then what was it?
Dialogue: 0,1:08:27.05,1:08:29.85,Default,,0000,0000,0000,,At pi I'm here, and so on.
Dialogue: 0,1:08:29.85,1:08:33.65,Default,,0000,0000,0000,,So I move-- when\NI move one time,
Dialogue: 0,1:08:33.65,1:08:39.41,Default,,0000,0000,0000,,so let's say from 0 to\N2 pi, I should be smart.
Dialogue: 0,1:08:39.41,1:08:48.98,Default,,0000,0000,0000,,Pi over 2, pi, 3 pi over 2,\N2 pi just on top of that.
Dialogue: 0,1:08:48.98,1:08:52.44,Default,,0000,0000,0000,,It has to be on the same line.
Dialogue: 0,1:08:52.44,1:08:54.62,Default,,0000,0000,0000,,On top of that--\Non the cylinder.
Dialogue: 0,1:08:54.62,1:08:55.83,Default,,0000,0000,0000,,They are all on the cylinder.
Dialogue: 0,1:08:55.83,1:08:59.44,Default,,0000,0000,0000,,I'm not good enough to draw\Nthem as being on the cylinder.
Dialogue: 0,1:08:59.44,1:09:03.19,Default,,0000,0000,0000,,So I'm coming where I started\Nfrom, but on the higher
Dialogue: 0,1:09:03.19,1:09:08.05,Default,,0000,0000,0000,,level of intelligence-- no, on\Na higher level of experience.
Dialogue: 0,1:09:08.05,1:09:08.84,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:09:08.84,1:09:13.100,Default,,0000,0000,0000,,That's kind of the idea\Nof evolving on the helix?
Dialogue: 0,1:09:13.100,1:09:16.65,Default,,0000,0000,0000,,Any other questions?
Dialogue: 0,1:09:16.65,1:09:17.58,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:09:17.58,1:09:19.76,Default,,0000,0000,0000,,STUDENT: So that\Ncapital R of t is
Dialogue: 0,1:09:19.76,1:09:23.89,Default,,0000,0000,0000,,you position vector, but what's\Nlittle r of t? [INAUDIBLE]
Dialogue: 0,1:09:23.89,1:09:25.52,Default,,0000,0000,0000,,PROFESSOR: It's also\Na position vector.
Dialogue: 0,1:09:25.52,1:09:32.00,Default,,0000,0000,0000,,So practically it depends on\Nthe type of parametrization
Dialogue: 0,1:09:32.00,1:09:33.48,Default,,0000,0000,0000,,you are using.
Dialogue: 0,1:09:33.48,1:09:36.25,Default,,0000,0000,0000,,
Dialogue: 0,1:09:36.25,1:09:39.68,Default,,0000,0000,0000,,The dependence of\Ntime is crucial.
Dialogue: 0,1:09:39.68,1:09:43.34,Default,,0000,0000,0000,,The dependence of the\Ntime parameter is crucial.
Dialogue: 0,1:09:43.34,1:09:50.58,Default,,0000,0000,0000,,So when you draw\Nthis diagram, r of s
Dialogue: 0,1:09:50.58,1:09:59.12,Default,,0000,0000,0000,,will practically be the same\Nas R of s of t-- R of t of s,
Dialogue: 0,1:09:59.12,1:10:00.00,Default,,0000,0000,0000,,I'm sorry.
Dialogue: 0,1:10:00.00,1:10:02.34,Default,,0000,0000,0000,,R of t of s.
Dialogue: 0,1:10:02.34,1:10:05.62,Default,,0000,0000,0000,,So practically it's telling\Nme it's a combination.
Dialogue: 0,1:10:05.62,1:10:11.80,Default,,0000,0000,0000,,Physically, it's the same\Nthing, but at a different time.
Dialogue: 0,1:10:11.80,1:10:20.12,Default,,0000,0000,0000,,So you look at one vector\Nat time-- time is t here,
Dialogue: 0,1:10:20.12,1:10:22.52,Default,,0000,0000,0000,,but s was 5t.
Dialogue: 0,1:10:22.52,1:10:26.05,Default,,0000,0000,0000,,So I'm gonna be-- let\Nme give you an example.
Dialogue: 0,1:10:26.05,1:10:29.89,Default,,0000,0000,0000,,So we had s was 5t, right?
Dialogue: 0,1:10:29.89,1:10:32.56,Default,,0000,0000,0000,,I don't remember how it went.
Dialogue: 0,1:10:32.56,1:10:36.26,Default,,0000,0000,0000,,So when I have\Nlittle r of s, that
Dialogue: 0,1:10:36.26,1:10:42.77,Default,,0000,0000,0000,,means the same as\Nlittle r of 5t,
Dialogue: 0,1:10:42.77,1:10:47.47,Default,,0000,0000,0000,,which means this kind of guy.
Dialogue: 0,1:10:47.47,1:10:57.55,Default,,0000,0000,0000,,Now assume that I have something\Nlike cosine 5t, sine 5t, and 0.
Dialogue: 0,1:10:57.55,1:11:00.81,Default,,0000,0000,0000,,And what does this mean?
Dialogue: 0,1:11:00.81,1:11:10.33,Default,,0000,0000,0000,,It means that R of 2 pi over\N5 is the same as little r of 2
Dialogue: 0,1:11:10.33,1:11:16.26,Default,,0000,0000,0000,,pi where R of t is cosine\Nof 5t, and little r of s
Dialogue: 0,1:11:16.26,1:11:20.79,Default,,0000,0000,0000,,is cosine of s, sine s, 0.
Dialogue: 0,1:11:20.79,1:11:23.86,Default,,0000,0000,0000,,So Mr. t says, I'm\Nrunning, I'm time.
Dialogue: 0,1:11:23.86,1:11:29.29,Default,,0000,0000,0000,,I'm running from 0 to 2 pi\Nover 5, and that's when I stop.
Dialogue: 0,1:11:29.29,1:11:31.46,Default,,0000,0000,0000,,And little s says,\NI'm running too.
Dialogue: 0,1:11:31.46,1:11:34.76,Default,,0000,0000,0000,,I'm also time, but I'm\Na special kind of time,
Dialogue: 0,1:11:34.76,1:11:38.40,Default,,0000,0000,0000,,and I'm running from 0 to\N2 pi, and I stop at 2 pi
Dialogue: 0,1:11:38.40,1:11:40.92,Default,,0000,0000,0000,,where the circle will stop.
Dialogue: 0,1:11:40.92,1:11:44.39,Default,,0000,0000,0000,,Then physically,\Nthe two vectors,
Dialogue: 0,1:11:44.39,1:11:48.75,Default,,0000,0000,0000,,at two different moments\Nin time, are the same.
Dialogue: 0,1:11:48.75,1:11:51.31,Default,,0000,0000,0000,,Where-- why-- why is that?
Dialogue: 0,1:11:51.31,1:11:53.30,Default,,0000,0000,0000,,So I start here.
Dialogue: 0,1:11:53.30,1:11:55.77,Default,,0000,0000,0000,,And I end here.
Dialogue: 0,1:11:55.77,1:12:01.30,Default,,0000,0000,0000,,So physically, these two guys\Nhave the same, the red vector,
Dialogue: 0,1:12:01.30,1:12:05.45,Default,,0000,0000,0000,,but they are there at\Ndifferent moments in time.
Dialogue: 0,1:12:05.45,1:12:06.75,Default,,0000,0000,0000,,All right?
Dialogue: 0,1:12:06.75,1:12:12.84,Default,,0000,0000,0000,,So imagine that you have sister.
Dialogue: 0,1:12:12.84,1:12:17.76,Default,,0000,0000,0000,,And she is five times faster\Nthan you in a competition.
Dialogue: 0,1:12:17.76,1:12:20.98,Default,,0000,0000,0000,,It's a math competition,\Nathletic, it doesn't matter.
Dialogue: 0,1:12:20.98,1:12:25.57,Default,,0000,0000,0000,,You both get there, but you\Nget there in different times,
Dialogue: 0,1:12:25.57,1:12:27.48,Default,,0000,0000,0000,,in different amounts of time.
Dialogue: 0,1:12:27.48,1:12:31.03,Default,,0000,0000,0000,,And unfortunately, this is--\NI will do philosophy still
Dialogue: 0,1:12:31.03,1:12:35.79,Default,,0000,0000,0000,,in mathematics-- this is the\Nsituation with many of us
Dialogue: 0,1:12:35.79,1:12:39.25,Default,,0000,0000,0000,,when it comes to\Nunderstanding a material,
Dialogue: 0,1:12:39.25,1:12:42.47,Default,,0000,0000,0000,,like calculus or advanced\Ncalculus or geometry.
Dialogue: 0,1:12:42.47,1:12:47.42,Default,,0000,0000,0000,,We get to the understanding\Nin different times.
Dialogue: 0,1:12:47.42,1:12:51.35,Default,,0000,0000,0000,,In my class-- I was\Ntalking to my old--
Dialogue: 0,1:12:51.35,1:12:55.59,Default,,0000,0000,0000,,they are all old now,\Nall in their 40s--
Dialogue: 0,1:12:55.59,1:12:59.01,Default,,0000,0000,0000,,when did you\Nunderstand this helix
Dialogue: 0,1:12:59.01,1:13:01.71,Default,,0000,0000,0000,,thing being on a cylinder?
Dialogue: 0,1:13:01.71,1:13:04.00,Default,,0000,0000,0000,,Because I think I\Nunderstood it when
Dialogue: 0,1:13:04.00,1:13:07.56,Default,,0000,0000,0000,,I was in third-- like a\Njunior level, sophomore level,
Dialogue: 0,1:13:07.56,1:13:09.82,Default,,0000,0000,0000,,and I understood nothing\Nof this kind of stuff
Dialogue: 0,1:13:09.82,1:13:14.83,Default,,0000,0000,0000,,in my freshman [INAUDIBLE]\NAnd one of my colleagues
Dialogue: 0,1:13:14.83,1:13:17.100,Default,,0000,0000,0000,,who was really smart,\Nhad a big background,
Dialogue: 0,1:13:17.100,1:13:20.87,Default,,0000,0000,0000,,was in a Math\NOlympiad, said, I think
Dialogue: 0,1:13:20.87,1:13:22.98,Default,,0000,0000,0000,,I understood it as a freshman.
Dialogue: 0,1:13:22.98,1:13:25.11,Default,,0000,0000,0000,,So then the other two that\NI was talking-- actually
Dialogue: 0,1:13:25.11,1:13:27.62,Default,,0000,0000,0000,,I never understood it.
Dialogue: 0,1:13:27.62,1:13:32.00,Default,,0000,0000,0000,,So we all eventually get to\Nthat point, that position,
Dialogue: 0,1:13:32.00,1:13:34.88,Default,,0000,0000,0000,,but at a different\Nmoment in time.
Dialogue: 0,1:13:34.88,1:13:39.08,Default,,0000,0000,0000,,And it's also unfortunate it\Nhappens about relationships.
Dialogue: 0,1:13:39.08,1:13:42.29,Default,,0000,0000,0000,,You are in a relationship\Nwith somebody,
Dialogue: 0,1:13:42.29,1:13:44.70,Default,,0000,0000,0000,,and one is faster\Nthan the other one.
Dialogue: 0,1:13:44.70,1:13:46.76,Default,,0000,0000,0000,,One grows faster\Nthan the other one.
Dialogue: 0,1:13:46.76,1:13:50.43,Default,,0000,0000,0000,,Eventually both get to the\Nsame level of understanding,
Dialogue: 0,1:13:50.43,1:13:53.48,Default,,0000,0000,0000,,but since it's at\Ndifferent moments in time,
Dialogue: 0,1:13:53.48,1:13:55.86,Default,,0000,0000,0000,,the relationship could\Nbreak by the time
Dialogue: 0,1:13:55.86,1:13:58.84,Default,,0000,0000,0000,,both reach that level\Nof understanding.
Dialogue: 0,1:13:58.84,1:14:02.62,Default,,0000,0000,0000,,So physical phenomena,\Nreally tricky.
Dialogue: 0,1:14:02.62,1:14:05.49,Default,,0000,0000,0000,,It's-- physically you\Nsee where everything is,
Dialogue: 0,1:14:05.49,1:14:08.49,Default,,0000,0000,0000,,but you have to think\Ndynamically, in time.
Dialogue: 0,1:14:08.49,1:14:11.30,Default,,0000,0000,0000,,Everything evolves in time.
Dialogue: 0,1:14:11.30,1:14:15.17,Default,,0000,0000,0000,,Any other questions?
Dialogue: 0,1:14:15.17,1:14:17.88,Default,,0000,0000,0000,,I'm gonna do problems\Nwith you next time,
Dialogue: 0,1:14:17.88,1:14:22.69,Default,,0000,0000,0000,,but you need a break because\Nyour brain is overheated.
Dialogue: 0,1:14:22.69,1:14:27.84,Default,,0000,0000,0000,,And so, we will take a\Nbreak of 10-12 minutes.
Dialogue: 0,1:14:27.84,1:14:30.65,Default,,0000,0000,0000,,