[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:03.19,Default,,0000,0000,0000,,PROFESSOR TODA: And Calc II.
Dialogue: 0,0:00:03.19,0:00:08.19,Default,,0000,0000,0000,,And I will go ahead and\Nsolve some problems today out
Dialogue: 0,0:00:08.19,0:00:10.78,Default,,0000,0000,0000,,of chapter 10 as a review.
Dialogue: 0,0:00:10.78,0:00:14.49,Default,,0000,0000,0000,,
Dialogue: 0,0:00:14.49,0:00:15.48,Default,,0000,0000,0000,,Meaning what?
Dialogue: 0,0:00:15.48,0:00:22.54,Default,,0000,0000,0000,,Meaning, that you have\Nsection 10.1 followed by 10.2
Dialogue: 0,0:00:22.54,0:00:24.74,Default,,0000,0000,0000,,followed by 10.4.
Dialogue: 0,0:00:24.74,0:00:26.92,Default,,0000,0000,0000,,These ones are\Nrequired sections,
Dialogue: 0,0:00:26.92,0:00:34.52,Default,,0000,0000,0000,,but I'm putting the material\Nall together as a compact set.
Dialogue: 0,0:00:34.52,0:00:38.82,Default,,0000,0000,0000,,So, if we cannot officially\Ncut between, as I told you,
Dialogue: 0,0:00:38.82,0:00:41.72,Default,,0000,0000,0000,,cut between the sections.
Dialogue: 0,0:00:41.72,0:00:47.27,Default,,0000,0000,0000,,One thing that I did\Nnot work examples on,
Dialogue: 0,0:00:47.27,0:00:50.40,Default,,0000,0000,0000,,trusting that you'd\Nremember it was integration.
Dialogue: 0,0:00:50.40,0:00:53.27,Default,,0000,0000,0000,,In particular, I didn't\Ncover integration
Dialogue: 0,0:00:53.27,0:00:56.22,Default,,0000,0000,0000,,of vector valued functions\Nand examples that
Dialogue: 0,0:00:56.22,0:00:58.07,Default,,0000,0000,0000,,are very very important.
Dialogue: 0,0:00:58.07,0:01:02.82,Default,,0000,0000,0000,,Now, do you need to learn\Nsomething special for that?
Dialogue: 0,0:01:02.82,0:01:03.32,Default,,0000,0000,0000,,No.
Dialogue: 0,0:01:03.32,0:01:07.82,Default,,0000,0000,0000,,But just like you cannot learn\Norganic chemistry without
Dialogue: 0,0:01:07.82,0:01:11.54,Default,,0000,0000,0000,,knowing inorganic chemistry,\Nthen you could not know how
Dialogue: 0,0:01:11.54,0:01:17.03,Default,,0000,0000,0000,,to integrate a vector value\Nfunction r prime of d to get r
Dialogue: 0,0:01:17.03,0:01:21.22,Default,,0000,0000,0000,,of d unless you know calculus\None and caluculus two, right?
Dialogue: 0,0:01:21.22,0:01:25.68,Default,,0000,0000,0000,,So let's say first\Na bunch of formulas
Dialogue: 0,0:01:25.68,0:01:30.57,Default,,0000,0000,0000,,that you use going back\Nto last week's knowledge
Dialogue: 0,0:01:30.57,0:01:32.10,Default,,0000,0000,0000,,what have we learned?
Dialogue: 0,0:01:32.10,0:01:38.90,Default,,0000,0000,0000,,We work with regular\Ncurves in r3.
Dialogue: 0,0:01:38.90,0:01:42.20,Default,,0000,0000,0000,,And in particular if\Nthey are part of R2,
Dialogue: 0,0:01:42.20,0:01:45.17,Default,,0000,0000,0000,,they are plain curves.
Dialogue: 0,0:01:45.17,0:01:47.69,Default,,0000,0000,0000,,I want to encourage\Nyou to ask questions
Dialogue: 0,0:01:47.69,0:01:50.09,Default,,0000,0000,0000,,about the example\N[INAUDIBLE] now.
Dialogue: 0,0:01:50.09,0:01:57.11,Default,,0000,0000,0000,,In the review session we\Nhave applications [INAUDIBLE]
Dialogue: 0,0:01:57.11,0:01:58.87,Default,,0000,0000,0000,,from 2 2 3.
Dialogue: 0,0:01:58.87,0:02:00.87,Default,,0000,0000,0000,,What was a regular curve?
Dialogue: 0,0:02:00.87,0:02:04.18,Default,,0000,0000,0000,,Is anybody willing to tell\Nme what a regular curve was?
Dialogue: 0,0:02:04.18,0:02:08.09,Default,,0000,0000,0000,,Was it vector value function?
Dialogue: 0,0:02:08.09,0:02:09.70,Default,,0000,0000,0000,,Do you like big r or little r?
Dialogue: 0,0:02:09.70,0:02:10.70,Default,,0000,0000,0000,,STUDENT: Doesn't matter.
Dialogue: 0,0:02:10.70,0:02:11.98,Default,,0000,0000,0000,,PROFESSOR TODA: Big r of t.
Dialogue: 0,0:02:11.98,0:02:14.12,Default,,0000,0000,0000,,Vector value function.
Dialogue: 0,0:02:14.12,0:02:17.94,Default,,0000,0000,0000,,x of t [INAUDIBLE] You know,\NI told you that sometimes we
Dialogue: 0,0:02:17.94,0:02:19.29,Default,,0000,0000,0000,,use brackets here.
Dialogue: 0,0:02:19.29,0:02:25.04,Default,,0000,0000,0000,,Sometimes we use round\Nparentheses depending
Dialogue: 0,0:02:25.04,0:02:31.13,Default,,0000,0000,0000,,how you represent a vector in r3\Nin our book they use brackets,
Dialogue: 0,0:02:31.13,0:02:37.07,Default,,0000,0000,0000,,but in other calculus books,\Nthey use round parentheses
Dialogue: 0,0:02:37.07,0:02:38.51,Default,,0000,0000,0000,,around it.
Dialogue: 0,0:02:38.51,0:02:44.27,Default,,0000,0000,0000,,So these are the coordinates\Nof the moving particle in time.
Dialogue: 0,0:02:44.27,0:02:47.60,Default,,0000,0000,0000,,Doesn't have to be a specific\Nobject, could be a fly,
Dialogue: 0,0:02:47.60,0:02:50.78,Default,,0000,0000,0000,,could be just a\Nparticle, anything
Dialogue: 0,0:02:50.78,0:02:58.00,Default,,0000,0000,0000,,in physical motion between this\Npoint a of b equals a and b
Dialogue: 0,0:02:58.00,0:03:00.14,Default,,0000,0000,0000,,of t equals b.
Dialogue: 0,0:03:00.14,0:03:02.34,Default,,0000,0000,0000,,So at time a and\Ntime b you are there.
Dialogue: 0,0:03:02.34,0:03:03.21,Default,,0000,0000,0000,,What have we learned?
Dialogue: 0,0:03:03.21,0:03:11.28,Default,,0000,0000,0000,,We've learned that a regular\Ncurve means its differentiable
Dialogue: 0,0:03:11.28,0:03:15.12,Default,,0000,0000,0000,,and the derivative is\Ncontinuous, it's a c1 function.
Dialogue: 0,0:03:15.12,0:03:16.29,Default,,0000,0000,0000,,And what else?
Dialogue: 0,0:03:16.29,0:03:19.93,Default,,0000,0000,0000,,The derivative of\Nthe position vector
Dialogue: 0,0:03:19.93,0:03:22.85,Default,,0000,0000,0000,,called velocity never vanishes.
Dialogue: 0,0:03:22.85,0:03:26.67,Default,,0000,0000,0000,,So it's different from 0\Nfor every t in the interval
Dialogue: 0,0:03:26.67,0:03:30.05,Default,,0000,0000,0000,,that you take, like ab.
Dialogue: 0,0:03:30.05,0:03:31.84,Default,,0000,0000,0000,,That's a regular curve.
Dialogue: 0,0:03:31.84,0:03:38.72,Default,,0000,0000,0000,,Regular curve was something we\Ntalked about at least 5 times.
Dialogue: 0,0:03:38.72,0:03:44.17,Default,,0000,0000,0000,,The point is how do we\Nsee the backwards process?
Dialogue: 0,0:03:44.17,0:03:52.16,Default,,0000,0000,0000,,That means if somebody gives you\Nthe velocity of a vector curve,
Dialogue: 0,0:03:52.16,0:03:55.19,Default,,0000,0000,0000,,they ask you for\Nthe position vector.
Dialogue: 0,0:03:55.19,0:03:57.32,Default,,0000,0000,0000,,So let's see an example.
Dialogue: 0,0:03:57.32,0:04:02.99,Default,,0000,0000,0000,,Integration example\N1 says I gave you
Dialogue: 0,0:04:02.99,0:04:07.82,Default,,0000,0000,0000,,the veclocity vector or\Na certain law of motion
Dialogue: 0,0:04:07.82,0:04:09.03,Default,,0000,0000,0000,,that I don't know.
Dialogue: 0,0:04:09.03,0:04:13.18,Default,,0000,0000,0000,,I just know the velocity\Nvector is being 1 over 1
Dialogue: 0,0:04:13.18,0:04:15.60,Default,,0000,0000,0000,,plus t squared.
Dialogue: 0,0:04:15.60,0:04:17.06,Default,,0000,0000,0000,,Should I put the brace here?
Dialogue: 0,0:04:17.06,0:04:19.00,Default,,0000,0000,0000,,An angular bracket?
Dialogue: 0,0:04:19.00,0:04:20.72,Default,,0000,0000,0000,,One over one plus t squared.
Dialogue: 0,0:04:20.72,0:04:32.86,Default,,0000,0000,0000,,And I'm gonna put a cosign\Non 2t, and t squared
Dialogue: 0,0:04:32.86,0:04:37.50,Default,,0000,0000,0000,,plus equal to minus t.
Dialogue: 0,0:04:37.50,0:04:43.29,Default,,0000,0000,0000,,And somebody says,\Nthat's all I know for P
Dialogue: 0,0:04:43.29,0:04:47.21,Default,,0000,0000,0000,,on an arbitrary real integral.
Dialogue: 0,0:04:47.21,0:04:54.98,Default,,0000,0000,0000,,And we know via the\N0 as being even.
Dialogue: 0,0:04:54.98,0:05:03.09,Default,,0000,0000,0000,,Let's say it's even\Nas 0 0 and that
Dialogue: 0,0:05:03.09,0:05:06.61,Default,,0000,0000,0000,,takes a little bit of thinking.
Dialogue: 0,0:05:06.61,0:05:09.00,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:05:09.00,0:05:20.26,Default,,0000,0000,0000,,How about a 1, which\Nwould be just k.
Dialogue: 0,0:05:20.26,0:05:24.45,Default,,0000,0000,0000,,Using this velocity vector\Nfind me being normal,
Dialogue: 0,0:05:24.45,0:05:26.75,Default,,0000,0000,0000,,which means find\Nthe position vector
Dialogue: 0,0:05:26.75,0:05:29.69,Default,,0000,0000,0000,,corresponding to this velocity.
Dialogue: 0,0:05:29.69,0:05:31.37,Default,,0000,0000,0000,,What is this?
Dialogue: 0,0:05:31.37,0:05:34.34,Default,,0000,0000,0000,,It's actually initial value
Dialogue: 0,0:05:34.34,0:05:40.62,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\N1, 1, and 1?
Dialogue: 0,0:05:40.62,0:05:42.40,Default,,0000,0000,0000,,PROFESSOR TODA: 0, what is it?
Dialogue: 0,0:05:42.40,0:05:44.06,Default,,0000,0000,0000,,When place 0 in?
Dialogue: 0,0:05:44.06,0:05:45.43,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:05:45.43,0:05:47.67,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:05:47.67,0:05:49.53,Default,,0000,0000,0000,,STUDENT: Are these\Nthe initial conditions
Dialogue: 0,0:05:49.53,0:05:50.80,Default,,0000,0000,0000,,for the location, or--
Dialogue: 0,0:05:50.80,0:05:51.88,Default,,0000,0000,0000,,PROFESSOR TODA: I'm sorry.
Dialogue: 0,0:05:51.88,0:05:58.78,Default,,0000,0000,0000,,I wrote r the intial\Ncondition for the location.
Dialogue: 0,0:05:58.78,0:06:01.04,Default,,0000,0000,0000,,Thank you so much, OK?
Dialogue: 0,0:06:01.04,0:06:04.52,Default,,0000,0000,0000,,I probably would've realized\Nit as soon as possible.
Dialogue: 0,0:06:04.52,0:06:07.03,Default,,0000,0000,0000,,Not the initial velocity\NI wanted to give you,
Dialogue: 0,0:06:07.03,0:06:11.12,Default,,0000,0000,0000,,but the initial position.
Dialogue: 0,0:06:11.12,0:06:17.73,Default,,0000,0000,0000,,All right, so how do\NI get to the r of d?
Dialogue: 0,0:06:17.73,0:06:20.00,Default,,0000,0000,0000,,I would say integrate,\Nand when I integrate,
Dialogue: 0,0:06:20.00,0:06:25.68,Default,,0000,0000,0000,,I have to keep in mind that\NI have to add the constants.
Dialogue: 0,0:06:25.68,0:06:26.32,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:06:26.32,0:06:27.00,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:06:27.00,0:06:29.44,Default,,0000,0000,0000,,So from v, v is our priority.
Dialogue: 0,0:06:29.44,0:06:32.40,Default,,0000,0000,0000,,
Dialogue: 0,0:06:32.40,0:06:37.88,Default,,0000,0000,0000,,It follows that r will\Nbe-- who tells me?
Dialogue: 0,0:06:37.88,0:06:42.51,Default,,0000,0000,0000,,Do you guys remember the\Nintegral of 1 plus t squared?
Dialogue: 0,0:06:42.51,0:06:43.46,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:06:43.46,0:06:45.36,Default,,0000,0000,0000,,PROFESSOR TODA: So\Nthat's the inverse.
Dialogue: 0,0:06:45.36,0:06:49.01,Default,,0000,0000,0000,,Or, I'll write it [? arc tan, ?]\Nand I'm very happy that you
Dialogue: 0,0:06:49.01,0:06:51.22,Default,,0000,0000,0000,,remember that, but there\Nare many students who don't.
Dialogue: 0,0:06:51.22,0:06:54.53,Default,,0000,0000,0000,,If you feel you don't, that\Nmeans that you have to open
Dialogue: 0,0:06:54.53,0:06:59.74,Default,,0000,0000,0000,,the -- where? -- Between\Nchapters 5 and chapter 7.
Dialogue: 0,0:06:59.74,0:07:03.68,Default,,0000,0000,0000,,You have all these\Nintegration chapters--
Dialogue: 0,0:07:03.68,0:07:05.97,Default,,0000,0000,0000,,the main ones over there.
Dialogue: 0,0:07:05.97,0:07:08.14,Default,,0000,0000,0000,,It's a function definted\Non the whole real interval,
Dialogue: 0,0:07:08.14,0:07:11.86,Default,,0000,0000,0000,,so I don't care\Nto worry about it.
Dialogue: 0,0:07:11.86,0:07:14.70,Default,,0000,0000,0000,,This what we call an IVP,\Ninitial value problem.
Dialogue: 0,0:07:14.70,0:07:18.65,Default,,0000,0000,0000,,
Dialogue: 0,0:07:18.65,0:07:20.94,Default,,0000,0000,0000,,So what kind of problem is that?
Dialogue: 0,0:07:20.94,0:07:23.03,Default,,0000,0000,0000,,It's a problem\Nlike somebody would
Dialogue: 0,0:07:23.03,0:07:29.57,Default,,0000,0000,0000,,give you knowing that f\Nprime of t is the little f,
Dialogue: 0,0:07:29.57,0:07:32.82,Default,,0000,0000,0000,,and knowing that big f\Nof 0 is the initial value
Dialogue: 0,0:07:32.82,0:07:37.34,Default,,0000,0000,0000,,for your function of find f.
Dialogue: 0,0:07:37.34,0:07:42.86,Default,,0000,0000,0000,,So you have actually an initial\Nvalue problem of the calc
Dialogue: 0,0:07:42.86,0:07:47.16,Default,,0000,0000,0000,,that you've seen\Nin previous class.
Dialogue: 0,0:07:47.16,0:07:54.68,Default,,0000,0000,0000,,arctangent of t plus c1 and then\Nif you miss the c1 in general,
Dialogue: 0,0:07:54.68,0:07:59.69,Default,,0000,0000,0000,,this can mess up the whole thing\Nbecause-- see, in your case,
Dialogue: 0,0:07:59.69,0:08:02.28,Default,,0000,0000,0000,,you're really lucky.
Dialogue: 0,0:08:02.28,0:08:06.65,Default,,0000,0000,0000,,If you plug in the 0 here,\Nwhat are you gonna have?
Dialogue: 0,0:08:06.65,0:08:10.50,Default,,0000,0000,0000,,You're gonna have arctangent\Nof 0, and that is 0.
Dialogue: 0,0:08:10.50,0:08:12.73,Default,,0000,0000,0000,,So in that case c1 is just 0.
Dialogue: 0,0:08:12.73,0:08:15.19,Default,,0000,0000,0000,,And [? three ?] [? not ?] and\Nif you forgot it would not be
Dialogue: 0,0:08:15.19,0:08:19.52,Default,,0000,0000,0000,,the end of the world, but\Nif you forgot it in general,
Dialogue: 0,0:08:19.52,0:08:20.87,Default,,0000,0000,0000,,it would be a big problem.
Dialogue: 0,0:08:20.87,0:08:22.99,Default,,0000,0000,0000,,So don't forget\Nabout the constant.
Dialogue: 0,0:08:22.99,0:08:25.26,Default,,0000,0000,0000,,When you integrate-- the\Nfamiliar of antiderivatives
Dialogue: 0,0:08:25.26,0:08:26.45,Default,,0000,0000,0000,,is cosine 2t.
Dialogue: 0,0:08:26.45,0:08:29.98,Default,,0000,0000,0000,,
Dialogue: 0,0:08:29.98,0:08:32.51,Default,,0000,0000,0000,,I know you know it.
Dialogue: 0,0:08:32.51,0:08:35.87,Default,,0000,0000,0000,,1/2 sine of t.
Dialogue: 0,0:08:35.87,0:08:37.24,Default,,0000,0000,0000,,Am I done?
Dialogue: 0,0:08:37.24,0:08:40.39,Default,,0000,0000,0000,,No, I should say plus C2.
Dialogue: 0,0:08:40.39,0:08:43.04,Default,,0000,0000,0000,,And finally the familiar\Nof antiderivatives of t
Dialogue: 0,0:08:43.04,0:08:45.70,Default,,0000,0000,0000,,squared plus e to minus t.
Dialogue: 0,0:08:45.70,0:08:48.23,Default,,0000,0000,0000,,STUDENT: 2t minus e\Nto the negative t.
Dialogue: 0,0:08:48.23,0:08:50.12,Default,,0000,0000,0000,,PROFESSOR TODA: No, integral of.
Dialogue: 0,0:08:50.12,0:08:52.57,Default,,0000,0000,0000,,So what's the integral of--
Dialogue: 0,0:08:52.57,0:08:53.70,Default,,0000,0000,0000,,STUDENT: t 2 squared.
Dialogue: 0,0:08:53.70,0:08:57.86,Default,,0000,0000,0000,,PROFESSOR TODA: t cubed\Nover 3-- minus, excellent.
Dialogue: 0,0:08:57.86,0:09:01.31,Default,,0000,0000,0000,,Now, do you want one\Nof you guys almost
Dialogue: 0,0:09:01.31,0:09:03.43,Default,,0000,0000,0000,,kill me during the weekend.
Dialogue: 0,0:09:03.43,0:09:04.01,Default,,0000,0000,0000,,But that's OK.
Dialogue: 0,0:09:04.01,0:09:06.34,Default,,0000,0000,0000,,I mean, this problem\Nhad something
Dialogue: 0,0:09:06.34,0:09:08.10,Default,,0000,0000,0000,,to do with integral minus.
Dialogue: 0,0:09:08.10,0:09:12.13,Default,,0000,0000,0000,,He put that integral of e to the\Nminus t was equal to minus t.
Dialogue: 0,0:09:12.13,0:09:14.45,Default,,0000,0000,0000,,So pay attention to the sign.
Dialogue: 0,0:09:14.45,0:09:16.92,Default,,0000,0000,0000,,Remember that integral\Nof e to the at,
Dialogue: 0,0:09:16.92,0:09:22.22,Default,,0000,0000,0000,,the t is to the at over a plus.
Dialogue: 0,0:09:22.22,0:09:23.02,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:09:23.02,0:09:26.81,Default,,0000,0000,0000,,OK, so this is what you\Nhave, a minus plus C3.
Dialogue: 0,0:09:26.81,0:09:28.77,Default,,0000,0000,0000,,Pay attention also to the exam.
Dialogue: 0,0:09:28.77,0:09:30.71,Default,,0000,0000,0000,,Because in the\Nexams, when you rush,
Dialogue: 0,0:09:30.71,0:09:33.03,Default,,0000,0000,0000,,you make lots of\Nmistakes like that.
Dialogue: 0,0:09:33.03,0:09:36.81,Default,,0000,0000,0000,,R of 0 is even.
Dialogue: 0,0:09:36.81,0:09:43.09,Default,,0000,0000,0000,,So the initial position\Nis given as C1.
Dialogue: 0,0:09:43.09,0:09:44.82,Default,,0000,0000,0000,,I'm replacing in my formula.
Dialogue: 0,0:09:44.82,0:09:49.43,Default,,0000,0000,0000,,It's going to be\NC1, C2, and what?
Dialogue: 0,0:09:49.43,0:09:52.26,Default,,0000,0000,0000,,When I replace the 0 here,\Nwhat am I going to get?
Dialogue: 0,0:09:52.26,0:09:53.93,Default,,0000,0000,0000,,STUDENT: You're going\Nto get negative 1.
Dialogue: 0,0:09:53.93,0:09:59.06,Default,,0000,0000,0000,,PROFESSOR TODA: Minus 1 plus C3.
Dialogue: 0,0:09:59.06,0:10:03.04,Default,,0000,0000,0000,,Note that I fabricated this\Nexample, so that C3 is not
Dialogue: 0,0:10:03.04,0:10:04.44,Default,,0000,0000,0000,,going to be 0.
Dialogue: 0,0:10:04.44,0:10:06.79,Default,,0000,0000,0000,,I wanted some customs to\Nbe zero and some customs
Dialogue: 0,0:10:06.79,0:10:10.24,Default,,0000,0000,0000,,to not be 0, just for\Nyou to realize it's
Dialogue: 0,0:10:10.24,0:10:12.56,Default,,0000,0000,0000,,important to pay attention.
Dialogue: 0,0:10:12.56,0:10:14.72,Default,,0000,0000,0000,,OK, minus 1 plus C3.
Dialogue: 0,0:10:14.72,0:10:22.49,Default,,0000,0000,0000,,And then I have 0, 0, 1 as\Ngiven as initial position.
Dialogue: 0,0:10:22.49,0:10:28.13,Default,,0000,0000,0000,,So what do you get by solving\Nthis linear system that's
Dialogue: 0,0:10:28.13,0:10:29.23,Default,,0000,0000,0000,,very simple?
Dialogue: 0,0:10:29.23,0:10:32.30,Default,,0000,0000,0000,,In general, you can get\Nmore complicated stuff.
Dialogue: 0,0:10:32.30,0:10:35.74,Default,,0000,0000,0000,,C1 is 0, C2 is 0, C3 is a--
Dialogue: 0,0:10:35.74,0:10:36.36,Default,,0000,0000,0000,,STUDENT: 2.
Dialogue: 0,0:10:36.36,0:10:37.11,Default,,0000,0000,0000,,PROFESSOR TODA: 2.
Dialogue: 0,0:10:37.11,0:10:38.86,Default,,0000,0000,0000,,And so it was a piece of cake.
Dialogue: 0,0:10:38.86,0:10:40.88,Default,,0000,0000,0000,,What is my formula?
Dialogue: 0,0:10:40.88,0:10:43.59,Default,,0000,0000,0000,,If you leave it like\Nthat, generally you're
Dialogue: 0,0:10:43.59,0:10:44.67,Default,,0000,0000,0000,,going to get full credit.
Dialogue: 0,0:10:44.67,0:10:47.40,Default,,0000,0000,0000,,What would you need to\Ndo to get full credit?
Dialogue: 0,0:10:47.40,0:10:53.42,Default,,0000,0000,0000,,STUDENT: Rt is equal to R10\Nplus 1/2 sine of 2t plus tq--
Dialogue: 0,0:10:53.42,0:10:55.42,Default,,0000,0000,0000,,PROFESSOR TODA: Precisely,\Nand thank you so much
Dialogue: 0,0:10:55.42,0:10:56.81,Default,,0000,0000,0000,,for your help.
Dialogue: 0,0:10:56.81,0:11:01.55,Default,,0000,0000,0000,,So you have R10 of\Nt, 1/2 sine of 2t
Dialogue: 0,0:11:01.55,0:11:09.90,Default,,0000,0000,0000,,and t cubed over 3 minus\Ne to the minus e plus 2.
Dialogue: 0,0:11:09.90,0:11:11.65,Default,,0000,0000,0000,,And close, and that's it.
Dialogue: 0,0:11:11.65,0:11:14.24,Default,,0000,0000,0000,,And box your answer.
Dialogue: 0,0:11:14.24,0:11:16.26,Default,,0000,0000,0000,,So I got the long motion back.
Dialogue: 0,0:11:16.26,0:11:22.00,Default,,0000,0000,0000,,Similarly, you could find,\Nif somebody gives you
Dialogue: 0,0:11:22.00,0:11:27.84,Default,,0000,0000,0000,,the acceleration of a\Nlong motion and asks you
Dialogue: 0,0:11:27.84,0:11:29.83,Default,,0000,0000,0000,,this is the acceleration.
Dialogue: 0,0:11:29.83,0:11:31.91,Default,,0000,0000,0000,,And I give you some\Ninitial values.
Dialogue: 0,0:11:31.91,0:11:34.52,Default,,0000,0000,0000,,And you have to find\Nfirst the velocity,
Dialogue: 0,0:11:34.52,0:11:36.27,Default,,0000,0000,0000,,going backwards one step.
Dialogue: 0,0:11:36.27,0:11:39.68,Default,,0000,0000,0000,,And from the velocity,\Nbackwards a second step,
Dialogue: 0,0:11:39.68,0:11:42.05,Default,,0000,0000,0000,,get the position vector.
Dialogue: 0,0:11:42.05,0:11:44.05,Default,,0000,0000,0000,,And that sounds a little\Nbit more elaborate.
Dialogue: 0,0:11:44.05,0:11:47.11,Default,,0000,0000,0000,,But it doesn't have to\Nbe a long computation.
Dialogue: 0,0:11:47.11,0:11:50.96,Default,,0000,0000,0000,,In general, we do not\Nfocus on giving you
Dialogue: 0,0:11:50.96,0:11:54.10,Default,,0000,0000,0000,,an awfully long computation.
Dialogue: 0,0:11:54.10,0:11:58.87,Default,,0000,0000,0000,,We just want to test your\Nunderstanding of the concepts.
Dialogue: 0,0:11:58.87,0:12:04.49,Default,,0000,0000,0000,,And having this in mind,\NI picked another example.
Dialogue: 0,0:12:04.49,0:12:08.57,Default,,0000,0000,0000,,I would like to\Nsee what that is.
Dialogue: 0,0:12:08.57,0:12:14.38,Default,,0000,0000,0000,,And the initial velocity\Nwill be given in this case.
Dialogue: 0,0:12:14.38,0:12:16.81,Default,,0000,0000,0000,,This is what I was thinking\Na little bit ahead of that.
Dialogue: 0,0:12:16.81,0:12:22.89,Default,,0000,0000,0000,,So somebody gives you the\Nacceleration in the velocity
Dialogue: 0,0:12:22.89,0:12:31.13,Default,,0000,0000,0000,,vector at 0 and is asking you\Nto find the velocity vector So
Dialogue: 0,0:12:31.13,0:12:36.30,Default,,0000,0000,0000,,let me give it to you\Nfor t between 0 and 2 pi.
Dialogue: 0,0:12:36.30,0:12:38.05,Default,,0000,0000,0000,,I give you the\Nacceleration vector,
Dialogue: 0,0:12:38.05,0:12:40.37,Default,,0000,0000,0000,,it will be nice and sassy.
Dialogue: 0,0:12:40.37,0:12:45.69,Default,,0000,0000,0000,,Let's see, that's going to be\Ncosine of t, sine of t and 0.
Dialogue: 0,0:12:45.69,0:12:48.34,Default,,0000,0000,0000,,And you'll say, oh, I\Nknow how to do those.
Dialogue: 0,0:12:48.34,0:12:49.77,Default,,0000,0000,0000,,Of course you know.
Dialogue: 0,0:12:49.77,0:12:52.37,Default,,0000,0000,0000,,But I want you to pay\Nattention to the constraints
Dialogue: 0,0:12:52.37,0:12:53.14,Default,,0000,0000,0000,,of integration.
Dialogue: 0,0:12:53.14,0:12:58.25,Default,,0000,0000,0000,,This is why I do this\Nkind of exercise again.
Dialogue: 0,0:12:58.25,0:13:02.81,Default,,0000,0000,0000,,So what do we have for V of t.
Dialogue: 0,0:13:02.81,0:13:09.98,Default,,0000,0000,0000,,V of 0 is-- somebody will say,\Nlet's give something nice,
Dialogue: 0,0:13:09.98,0:13:15.53,Default,,0000,0000,0000,,and let's say this would be--\NI have no idea what I want.
Dialogue: 0,0:13:15.53,0:13:21.77,Default,,0000,0000,0000,,Let's say i, j, and that's it.
Dialogue: 0,0:13:21.77,0:13:24.59,Default,,0000,0000,0000,,
Dialogue: 0,0:13:24.59,0:13:26.95,Default,,0000,0000,0000,,How do you do that?
Dialogue: 0,0:13:26.95,0:13:27.64,Default,,0000,0000,0000,,V of t.
Dialogue: 0,0:13:27.64,0:13:30.01,Default,,0000,0000,0000,,Let's integrate together.
Dialogue: 0,0:13:30.01,0:13:31.53,Default,,0000,0000,0000,,You don't like this?
Dialogue: 0,0:13:31.53,0:13:35.27,Default,,0000,0000,0000,,I hope that by now,\Nyou've got used to it.
Dialogue: 0,0:13:35.27,0:13:38.56,Default,,0000,0000,0000,,A bracket, I'm doing a\Nbracket, like in the book.
Dialogue: 0,0:13:38.56,0:13:42.70,Default,,0000,0000,0000,,So sine t plus a constant.
Dialogue: 0,0:13:42.70,0:13:45.09,Default,,0000,0000,0000,,What's the integral\Nof sine, class?
Dialogue: 0,0:13:45.09,0:13:48.20,Default,,0000,0000,0000,,V equals sine t plus a constant.
Dialogue: 0,0:13:48.20,0:13:51.38,Default,,0000,0000,0000,,And C3 is a constant.
Dialogue: 0,0:13:51.38,0:13:52.79,Default,,0000,0000,0000,,And there I go.
Dialogue: 0,0:13:52.79,0:13:54.67,Default,,0000,0000,0000,,You say, oh my god,\Nwhat am I having?
Dialogue: 0,0:13:54.67,0:13:58.36,Default,,0000,0000,0000,,V of 0-- is as a\Nvector, I presented it
Dialogue: 0,0:13:58.36,0:14:03.81,Default,,0000,0000,0000,,in the canonical standard\Nbasis as 1, 1, and 0.
Dialogue: 0,0:14:03.81,0:14:07.35,Default,,0000,0000,0000,,So from that one, you\Ncan jump to this one
Dialogue: 0,0:14:07.35,0:14:10.74,Default,,0000,0000,0000,,and say, yes, I'm going to\Nplug in 0, see what I get.
Dialogue: 0,0:14:10.74,0:14:13.38,Default,,0000,0000,0000,,In the general formula,\Nwhen you plug in 0,
Dialogue: 0,0:14:13.38,0:14:19.07,Default,,0000,0000,0000,,you get C1-- what\Nis cosine of 0?
Dialogue: 0,0:14:19.07,0:14:22.47,Default,,0000,0000,0000,,Minus 1, I have here, plus C2.
Dialogue: 0,0:14:22.47,0:14:27.90,Default,,0000,0000,0000,,And C3, that is always there.
Dialogue: 0,0:14:27.90,0:14:34.53,Default,,0000,0000,0000,,And then V of 0 is\Nwhat I got here.
Dialogue: 0,0:14:34.53,0:14:39.73,Default,,0000,0000,0000,,V of 0 has to be compared to\Nwhat your initial data was.
Dialogue: 0,0:14:39.73,0:14:48.38,Default,,0000,0000,0000,,So C1 is 1, C2 is 2, and C3 is--
Dialogue: 0,0:14:48.38,0:14:51.46,Default,,0000,0000,0000,,So let me replace it.
Dialogue: 0,0:14:51.46,0:15:05.43,Default,,0000,0000,0000,,I say the answer will be--\Ncosine t plus 1, sine t plus 2,
Dialogue: 0,0:15:05.43,0:15:13.32,Default,,0000,0000,0000,,and the constants.
Dialogue: 0,0:15:13.32,0:15:19.24,Default,,0000,0000,0000,,
Dialogue: 0,0:15:19.24,0:15:24.74,Default,,0000,0000,0000,,But then somebody, who is\Nreally an experimental guy,
Dialogue: 0,0:15:24.74,0:15:25.54,Default,,0000,0000,0000,,says well--
Dialogue: 0,0:15:25.54,0:15:26.83,Default,,0000,0000,0000,,STUDENT: You have it backwards.
Dialogue: 0,0:15:26.83,0:15:28.97,Default,,0000,0000,0000,,It's sine of t plus\N1, and then you
Dialogue: 0,0:15:28.97,0:15:31.05,Default,,0000,0000,0000,,have the cosine of t plus 2.
Dialogue: 0,0:15:31.05,0:15:32.10,Default,,0000,0000,0000,,PROFESSOR TODA: Oh, yeah.
Dialogue: 0,0:15:32.10,0:15:35.47,Default,,0000,0000,0000,,
Dialogue: 0,0:15:35.47,0:15:36.49,Default,,0000,0000,0000,,Wait a minute.
Dialogue: 0,0:15:36.49,0:15:41.55,Default,,0000,0000,0000,,This is-- I\Nmiscopied looking up.
Dialogue: 0,0:15:41.55,0:15:52.21,Default,,0000,0000,0000,,So I have sine t, I was\Nsupposed to-- minus cosine t
Dialogue: 0,0:15:52.21,0:15:56.55,Default,,0000,0000,0000,,and I'm done.
Dialogue: 0,0:15:56.55,0:15:58.38,Default,,0000,0000,0000,,So thank you for telling me.
Dialogue: 0,0:15:58.38,0:16:04.03,Default,,0000,0000,0000,,So sum t plus 1 minus\Ncosine t plus 2 and 0
Dialogue: 0,0:16:04.03,0:16:13.07,Default,,0000,0000,0000,,are the functions that I put\Nhere by replacing C1, C2, C3.
Dialogue: 0,0:16:13.07,0:16:15.26,Default,,0000,0000,0000,,And then, somebody\Nsays, wait a minute,
Dialogue: 0,0:16:15.26,0:16:18.50,Default,,0000,0000,0000,,now let me give you V of 0.
Dialogue: 0,0:16:18.50,0:16:20.86,Default,,0000,0000,0000,,Let me give you R of 0.
Dialogue: 0,0:16:20.86,0:16:22.50,Default,,0000,0000,0000,,We have zeroes already there.
Dialogue: 0,0:16:22.50,0:16:25.29,Default,,0000,0000,0000,,
Dialogue: 0,0:16:25.29,0:16:28.80,Default,,0000,0000,0000,,And you were supposed\Nto get R from here.
Dialogue: 0,0:16:28.80,0:16:36.32,Default,,0000,0000,0000,,So what is R of t, the\Nposition vector, find it.
Dialogue: 0,0:16:36.32,0:16:38.00,Default,,0000,0000,0000,,V of t is given.
Dialogue: 0,0:16:38.00,0:16:40.23,Default,,0000,0000,0000,,Actually, it's given by\Nyou, because you found it
Dialogue: 0,0:16:40.23,0:16:41.94,Default,,0000,0000,0000,,at the previous step.
Dialogue: 0,0:16:41.94,0:16:46.30,Default,,0000,0000,0000,,And R of 0 is given as well.
Dialogue: 0,0:16:46.30,0:16:59.22,Default,,0000,0000,0000,,And let's say that would\Nbe-- let's say 1, 1, and 1.
Dialogue: 0,0:16:59.22,0:17:02.63,Default,,0000,0000,0000,,
Dialogue: 0,0:17:02.63,0:17:05.22,Default,,0000,0000,0000,,So what do you need to do next?
Dialogue: 0,0:17:05.22,0:17:14.39,Default,,0000,0000,0000,,
Dialogue: 0,0:17:14.39,0:17:18.07,Default,,0000,0000,0000,,You have R prime given.
Dialogue: 0,0:17:18.07,0:17:22.14,Default,,0000,0000,0000,,That leaves you to\Nintegrate to get R t.
Dialogue: 0,0:17:22.14,0:17:24.52,Default,,0000,0000,0000,,And R of t is going to be what?
Dialogue: 0,0:17:24.52,0:17:29.03,Default,,0000,0000,0000,,Who is going to tell me\Nwhat I have to write down?
Dialogue: 0,0:17:29.03,0:17:39.48,Default,,0000,0000,0000,,Minus cosine t plus t plus--\Nlet's use the constant K1
Dialogue: 0,0:17:39.48,0:17:40.95,Default,,0000,0000,0000,,integration.
Dialogue: 0,0:17:40.95,0:17:42.42,Default,,0000,0000,0000,,And then what?
Dialogue: 0,0:17:42.42,0:17:43.41,Default,,0000,0000,0000,,STUDENT: Sine of t.
Dialogue: 0,0:17:43.41,0:17:45.38,Default,,0000,0000,0000,,PROFESSOR TODA: I think\Nit's minus sine, right?
Dialogue: 0,0:17:45.38,0:17:56.12,Default,,0000,0000,0000,,Minus sine of t plus 2t\Nplus K2 and K3, right?
Dialogue: 0,0:17:56.12,0:18:04.20,Default,,0000,0000,0000,,So R of 0 is going to be what?
Dialogue: 0,0:18:04.20,0:18:07.93,Default,,0000,0000,0000,,First of all, we use this\Npiece of information.
Dialogue: 0,0:18:07.93,0:18:12.48,Default,,0000,0000,0000,,Second of all, we identify\Nfrom the formula we got.
Dialogue: 0,0:18:12.48,0:18:16.30,Default,,0000,0000,0000,,So from the formula I\Ngot, just plugging in 0,
Dialogue: 0,0:18:16.30,0:18:22.69,Default,,0000,0000,0000,,it should come out straight\Nas minus 1 plus K1.
Dialogue: 0,0:18:22.69,0:18:28.07,Default,,0000,0000,0000,,0 for this guy, 0 for the\Nsecond term, K2 and K3.
Dialogue: 0,0:18:28.07,0:18:31.83,Default,,0000,0000,0000,,
Dialogue: 0,0:18:31.83,0:18:36.51,Default,,0000,0000,0000,,So who is helping me solve\Nthe system really quickly?
Dialogue: 0,0:18:36.51,0:18:39.75,Default,,0000,0000,0000,,K1 is 2.
Dialogue: 0,0:18:39.75,0:18:41.03,Default,,0000,0000,0000,,K2 is--
Dialogue: 0,0:18:41.03,0:18:41.72,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,0:18:41.72,0:18:44.01,Default,,0000,0000,0000,,PROFESSOR TODA: K3 is 1.
Dialogue: 0,0:18:44.01,0:18:50.61,Default,,0000,0000,0000,,And I'm going back\Nto R and replace it.
Dialogue: 0,0:18:50.61,0:18:55.47,Default,,0000,0000,0000,,And that's my final answer\Nfor this two-step problem.
Dialogue: 0,0:18:55.47,0:18:57.87,Default,,0000,0000,0000,,So I have a two-step integration\Nfrom the acceleration
Dialogue: 0,0:18:57.87,0:19:00.27,Default,,0000,0000,0000,,to the velocity,\Nfrom the velocity
Dialogue: 0,0:19:00.27,0:19:05.08,Default,,0000,0000,0000,,to the position vector.
Dialogue: 0,0:19:05.08,0:19:08.25,Default,,0000,0000,0000,,Minus cosine t plus t plus 2.
Dialogue: 0,0:19:08.25,0:19:11.63,Default,,0000,0000,0000,,Remind me, because I have\Na tendency to miscopy,
Dialogue: 0,0:19:11.63,0:19:13.28,Default,,0000,0000,0000,,an I looking in the right place?
Dialogue: 0,0:19:13.28,0:19:14.34,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:19:14.34,0:19:24.93,Default,,0000,0000,0000,,So I have minus sine t plus\N2t plus 1 and K3 is one.
Dialogue: 0,0:19:24.93,0:19:29.75,Default,,0000,0000,0000,,So this is the process you\Nare supposed to remember
Dialogue: 0,0:19:29.75,0:19:32.16,Default,,0000,0000,0000,,for the rest of the semester.
Dialogue: 0,0:19:32.16,0:19:33.42,Default,,0000,0000,0000,,It's not a hard one.
Dialogue: 0,0:19:33.42,0:19:36.96,Default,,0000,0000,0000,,It's something that\Neverybody should master.
Dialogue: 0,0:19:36.96,0:19:38.06,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:19:38.06,0:19:39.81,Default,,0000,0000,0000,,How many of you understood this?
Dialogue: 0,0:19:39.81,0:19:41.82,Default,,0000,0000,0000,,Please raise hands.
Dialogue: 0,0:19:41.82,0:19:45.11,Default,,0000,0000,0000,,Oh, no problem, good.
Dialogue: 0,0:19:45.11,0:19:52.16,Default,,0000,0000,0000,,Now would you tell me--\NI'm not going to ask you
Dialogue: 0,0:19:52.16,0:19:53.80,Default,,0000,0000,0000,,what kind of motion this is.
Dialogue: 0,0:19:53.80,0:19:57.03,Default,,0000,0000,0000,,It's a little bit close to\Na circular motion but not
Dialogue: 0,0:19:57.03,0:19:58.38,Default,,0000,0000,0000,,a circular motion.
Dialogue: 0,0:19:58.38,0:20:00.67,Default,,0000,0000,0000,,However, can you tell\Nme anything interesting
Dialogue: 0,0:20:00.67,0:20:04.80,Default,,0000,0000,0000,,about the type of trajectory\Nthat I have, in terms
Dialogue: 0,0:20:04.80,0:20:06.38,Default,,0000,0000,0000,,of the acceleration vector?
Dialogue: 0,0:20:06.38,0:20:10.82,Default,,0000,0000,0000,,The acceleration\Nvector is beautiful,
Dialogue: 0,0:20:10.82,0:20:13.84,Default,,0000,0000,0000,,just like in the\Ncase of the washer.
Dialogue: 0,0:20:13.84,0:20:18.87,Default,,0000,0000,0000,,That was a vector\Nthat-- like this
Dialogue: 0,0:20:18.87,0:20:20.86,Default,,0000,0000,0000,,would be the circular motion.
Dialogue: 0,0:20:20.86,0:20:23.04,Default,,0000,0000,0000,,The acceleration would\Nbe this unique vector
Dialogue: 0,0:20:23.04,0:20:25.05,Default,,0000,0000,0000,,that comes inside.
Dialogue: 0,0:20:25.05,0:20:26.92,Default,,0000,0000,0000,,Is this going outside\Nor coming inside?
Dialogue: 0,0:20:26.92,0:20:29.60,Default,,0000,0000,0000,,Is it a unit vector?
Dialogue: 0,0:20:29.60,0:20:32.72,Default,,0000,0000,0000,,Yes, it is a unit vector.
Dialogue: 0,0:20:32.72,0:20:37.43,Default,,0000,0000,0000,,So suppose that I'm\Nlooking at the trajectory,
Dialogue: 0,0:20:37.43,0:20:40.29,Default,,0000,0000,0000,,if it were more or\Nless a motion that has
Dialogue: 0,0:20:40.29,0:20:44.76,Default,,0000,0000,0000,,to do with mixing into a bowl.
Dialogue: 0,0:20:44.76,0:20:48.83,Default,,0000,0000,0000,,Would this go inside or outside?
Dialogue: 0,0:20:48.83,0:20:51.96,Default,,0000,0000,0000,,Towards the outside\Nor towards the inside?
Dialogue: 0,0:20:51.96,0:20:57.65,Default,,0000,0000,0000,,I plugged j-- depends on\Nwhat I'm looking at, in terms
Dialogue: 0,0:20:57.65,0:21:00.15,Default,,0000,0000,0000,,of surface that I'm on, right?
Dialogue: 0,0:21:00.15,0:21:01.56,Default,,0000,0000,0000,,Do you remember\Nfrom last time we
Dialogue: 0,0:21:01.56,0:21:04.06,Default,,0000,0000,0000,,had that helix that\Nwas on a cylinder.
Dialogue: 0,0:21:04.06,0:21:07.92,Default,,0000,0000,0000,,And we asked ourselves, how\Nis that [INAUDIBLE] pointing?
Dialogue: 0,0:21:07.92,0:21:11.78,Default,,0000,0000,0000,,And it was pointing\Noutside of the cylinder,
Dialogue: 0,0:21:11.78,0:21:16.05,Default,,0000,0000,0000,,in the direction\Ntowards the outside.
Dialogue: 0,0:21:16.05,0:21:26.93,Default,,0000,0000,0000,,Coming back to the\Nreview, there are
Dialogue: 0,0:21:26.93,0:21:31.44,Default,,0000,0000,0000,,several things I'd like to\Nreview but not all of them.
Dialogue: 0,0:21:31.44,0:21:34.44,Default,,0000,0000,0000,,Because some of the\Nexamples we have there,
Dialogue: 0,0:21:34.44,0:21:38.15,Default,,0000,0000,0000,,you understood them really well.
Dialogue: 0,0:21:38.15,0:21:40.30,Default,,0000,0000,0000,,I was very proud\Nof you, and I saw
Dialogue: 0,0:21:40.30,0:21:43.76,Default,,0000,0000,0000,,that you finished--\Nalmost all of you
Dialogue: 0,0:21:43.76,0:21:45.75,Default,,0000,0000,0000,,finished the\Nhomework number one.
Dialogue: 0,0:21:45.75,0:21:49.10,Default,,0000,0000,0000,,So I was looking outside\Nat homework number
Dialogue: 0,0:21:49.10,0:21:53.18,Default,,0000,0000,0000,,two that is over\Nthese three sections.
Dialogue: 0,0:21:53.18,0:21:58.47,Default,,0000,0000,0000,,So I was hoping you would ask\Nme today, between two and three,
Dialogue: 0,0:21:58.47,0:22:00.86,Default,,0000,0000,0000,,if you have any difficulties\Nwith homework two.
Dialogue: 0,0:22:00.86,0:22:03.73,Default,,0000,0000,0000,,That's due February 11.
Dialogue: 0,0:22:03.73,0:22:12.73,Default,,0000,0000,0000,,And then the latest homework\Nthat I posted yesterday, I
Dialogue: 0,0:22:12.73,0:22:14.98,Default,,0000,0000,0000,,don't know how many\Nof you logged in.
Dialogue: 0,0:22:14.98,0:22:18.62,Default,,0000,0000,0000,,But last night I\Nposted a homework
Dialogue: 0,0:22:18.62,0:22:21.80,Default,,0000,0000,0000,,that is getting a huge\Nextended deadline, which
Dialogue: 0,0:22:21.80,0:22:23.37,Default,,0000,0000,0000,,is the 28th of February.
Dialogue: 0,0:22:23.37,0:22:29.01,Default,,0000,0000,0000,,Because somebody's\Nbirthday is February 29.
Dialogue: 0,0:22:29.01,0:22:34.74,Default,,0000,0000,0000,,I was just thinking why would\Nsomebody need be a whole month?
Dialogue: 0,0:22:34.74,0:22:37.30,Default,,0000,0000,0000,,You would need the whole\Nmonth to have a good view
Dialogue: 0,0:22:37.30,0:22:39.02,Default,,0000,0000,0000,,of the whole chapter 11.
Dialogue: 0,0:22:39.02,0:22:40.97,Default,,0000,0000,0000,,I sent you the videos\Nfor chapter 11.
Dialogue: 0,0:22:40.97,0:22:43.54,Default,,0000,0000,0000,,And for chapter 11, you\Nhave this huge homework
Dialogue: 0,0:22:43.54,0:22:46.77,Default,,0000,0000,0000,,which is 49 problems.
Dialogue: 0,0:22:46.77,0:22:50.43,Default,,0000,0000,0000,,So please do not,\Ndo not leave it
Dialogue: 0,0:22:50.43,0:22:52.26,Default,,0000,0000,0000,,to the last five\Ndays or six days,
Dialogue: 0,0:22:52.26,0:22:55.71,Default,,0000,0000,0000,,because it's going to kill you.
Dialogue: 0,0:22:55.71,0:22:57.50,Default,,0000,0000,0000,,There are people who\Nsay, I can finish
Dialogue: 0,0:22:57.50,0:22:58.62,Default,,0000,0000,0000,,this in the next five days.
Dialogue: 0,0:22:58.62,0:23:00.32,Default,,0000,0000,0000,,I know you can.
Dialogue: 0,0:23:00.32,0:23:01.95,Default,,0000,0000,0000,,I know you can,\NI don't doubt it.
Dialogue: 0,0:23:01.95,0:23:04.42,Default,,0000,0000,0000,,That's why I left\Nyou so much freedom.
Dialogue: 0,0:23:04.42,0:23:07.61,Default,,0000,0000,0000,,But you have-- today is\Nthe second or the third?
Dialogue: 0,0:23:07.61,0:23:10.86,Default,,0000,0000,0000,,So practically you have\N25 days to work on this.
Dialogue: 0,0:23:10.86,0:23:15.20,Default,,0000,0000,0000,,On the 28th at 11 PM\Nit's going to close.
Dialogue: 0,0:23:15.20,0:23:18.82,Default,,0000,0000,0000,,I would work a few\Nproblems every other day.
Dialogue: 0,0:23:18.82,0:23:22.05,Default,,0000,0000,0000,,Because I need a break,\Nso I would alternate.
Dialogue: 0,0:23:22.05,0:23:25.03,Default,,0000,0000,0000,,But don't leave it--\Neven if you have help,
Dialogue: 0,0:23:25.03,0:23:27.69,Default,,0000,0000,0000,,especially if you have help,\Nlike a tutor or tutoring
Dialogue: 0,0:23:27.69,0:23:30.28,Default,,0000,0000,0000,,services here that are\Nfree in the department.
Dialogue: 0,0:23:30.28,0:23:32.22,Default,,0000,0000,0000,,Do not leave it\Nto the last days.
Dialogue: 0,0:23:32.22,0:23:35.16,Default,,0000,0000,0000,,Because you're putting pressure\Non yourself, on your brain,
Dialogue: 0,0:23:35.16,0:23:37.22,Default,,0000,0000,0000,,on your tutor, on everybody.
Dialogue: 0,0:23:37.22,0:23:37.72,Default,,0000,0000,0000,,Yes sir.
Dialogue: 0,0:23:37.72,0:23:38.64,Default,,0000,0000,0000,,STUDENT: So that's\Nhomework three?
Dialogue: 0,0:23:38.64,0:23:40.22,Default,,0000,0000,0000,,PROFESSOR TODA:\NThat's homework three,
Dialogue: 0,0:23:40.22,0:23:43.30,Default,,0000,0000,0000,,and it's a huge homework\Nover chapter 11.
Dialogue: 0,0:23:43.30,0:23:45.61,Default,,0000,0000,0000,,STUDENT: You said\Nthere are 49 problems?
Dialogue: 0,0:23:45.61,0:23:49.16,Default,,0000,0000,0000,,PROFESSOR TODA: I don't\Nremember exactly but 47, 49.
Dialogue: 0,0:23:49.16,0:23:50.24,Default,,0000,0000,0000,,I don't remember how many.
Dialogue: 0,0:23:50.24,0:23:52.85,Default,,0000,0000,0000,,STUDENT: Between 45 and 50.
Dialogue: 0,0:23:52.85,0:23:55.91,Default,,0000,0000,0000,,PROFESSOR TODA:\NBetween 45 and 50, yes.
Dialogue: 0,0:23:55.91,0:23:59.21,Default,,0000,0000,0000,,If you encounter any bug--\Nalthough there shouldn't
Dialogue: 0,0:23:59.21,0:24:02.02,Default,,0000,0000,0000,,be bugs, maybe 1 in 1,000.
Dialogue: 0,0:24:02.02,0:24:04.53,Default,,0000,0000,0000,,If you encounter any\Nbug that the programmer
Dialogue: 0,0:24:04.53,0:24:09.71,Default,,0000,0000,0000,,of those problems may\Nhave accidentally put in,
Dialogue: 0,0:24:09.71,0:24:11.37,Default,,0000,0000,0000,,you let me know.
Dialogue: 0,0:24:11.37,0:24:13.58,Default,,0000,0000,0000,,So I can contact them.
Dialogue: 0,0:24:13.58,0:24:17.39,Default,,0000,0000,0000,,If there is a problem that I\Nconsider shouldn't be there,
Dialogue: 0,0:24:17.39,0:24:19.79,Default,,0000,0000,0000,,I will eliminate that later on.
Dialogue: 0,0:24:19.79,0:24:23.45,Default,,0000,0000,0000,,But hopefully, everything\Nwill be doable,
Dialogue: 0,0:24:23.45,0:24:28.10,Default,,0000,0000,0000,,everything will be fair and\Nyou will be able to solve it.
Dialogue: 0,0:24:28.10,0:24:32.14,Default,,0000,0000,0000,,
Dialogue: 0,0:24:32.14,0:24:34.59,Default,,0000,0000,0000,,Any questions?
Dialogue: 0,0:24:34.59,0:24:37.02,Default,,0000,0000,0000,,Particular questions\Nfrom the homework?
Dialogue: 0,0:24:37.02,0:24:39.92,Default,,0000,0000,0000,,
Dialogue: 0,0:24:39.92,0:24:43.80,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] is it to\Nparametrize a circle of a set,
Dialogue: 0,0:24:43.80,0:24:47.86,Default,,0000,0000,0000,,like of a certain\Nradius on the xy-plane?
Dialogue: 0,0:24:47.86,0:24:49.26,Default,,0000,0000,0000,,PROFESSOR TODA:\NShall we do that?
Dialogue: 0,0:24:49.26,0:24:53.23,Default,,0000,0000,0000,,Do you want me to do that\Nin general, in xy-plane, OK.
Dialogue: 0,0:24:53.23,0:24:55.22,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nin the xy-plane.
Dialogue: 0,0:24:55.22,0:24:59.22,Default,,0000,0000,0000,,
Dialogue: 0,0:24:59.22,0:25:04.66,Default,,0000,0000,0000,,PROFESSOR TODA: xy-plane and\Nthen what was the equation?
Dialogue: 0,0:25:04.66,0:25:10.16,Default,,0000,0000,0000,,Was it like a equals sine\Nof t or a equals sine of bt?
Dialogue: 0,0:25:10.16,0:25:12.43,Default,,0000,0000,0000,,Because it's a\Nlittle bit different,
Dialogue: 0,0:25:12.43,0:25:15.79,Default,,0000,0000,0000,,depending on how the\Nparametrization was given.
Dialogue: 0,0:25:15.79,0:25:17.16,Default,,0000,0000,0000,,What's your name\Nagain, I forgot.
Dialogue: 0,0:25:17.16,0:25:18.92,Default,,0000,0000,0000,,I don't know what to refer you.
Dialogue: 0,0:25:18.92,0:25:19.54,Default,,0000,0000,0000,,STUDENT: Ryder.
Dialogue: 0,0:25:19.54,0:25:22.40,Default,,0000,0000,0000,,
Dialogue: 0,0:25:22.40,0:25:24.73,Default,,0000,0000,0000,,PROFESSOR TODA: Was that part\Nof what's due on the 11th?
Dialogue: 0,0:25:24.73,0:25:27.89,Default,,0000,0000,0000,,STUDENT: It doesn't-- yes, it\Ndoesn't give a revision set.
Dialogue: 0,0:25:27.89,0:25:29.05,Default,,0000,0000,0000,,It says--
Dialogue: 0,0:25:29.05,0:25:33.00,Default,,0000,0000,0000,,PROFESSOR TODA: Let me quickly\Nread-- find parametrization
Dialogue: 0,0:25:33.00,0:25:38.44,Default,,0000,0000,0000,,of the circle of radius 7 in\Nthe xy-plane, centered at 3, 1,
Dialogue: 0,0:25:38.44,0:25:40.62,Default,,0000,0000,0000,,oriented counterclockwise.
Dialogue: 0,0:25:40.62,0:25:43.24,Default,,0000,0000,0000,,The point 10, 1\Nshould be connected--
Dialogue: 0,0:25:43.24,0:25:44.68,Default,,0000,0000,0000,,STUDENT: Just one more second.
Dialogue: 0,0:25:44.68,0:25:45.63,Default,,0000,0000,0000,,PROFESSOR TODA: Do\Nyou mind if I put it.
Dialogue: 0,0:25:45.63,0:25:47.07,Default,,0000,0000,0000,,I'll take good care of it.
Dialogue: 0,0:25:47.07,0:25:48.03,Default,,0000,0000,0000,,I won't drop it.
Dialogue: 0,0:25:48.03,0:25:51.86,Default,,0000,0000,0000,,
Dialogue: 0,0:25:51.86,0:25:57.88,Default,,0000,0000,0000,,So the point-- parametrization\Nof the circle of radius
Dialogue: 0,0:25:57.88,0:26:01.98,Default,,0000,0000,0000,,7 in the xy-plane,\Ncentered at 3, 1.
Dialogue: 0,0:26:01.98,0:26:12.09,Default,,0000,0000,0000,,So circle centered at-- and\NI'll say it x0, 1 0, being 3, 1.
Dialogue: 0,0:26:12.09,0:26:16.34,Default,,0000,0000,0000,,
Dialogue: 0,0:26:16.34,0:26:19.04,Default,,0000,0000,0000,,No, because then I'm\Nsolving your problem.
Dialogue: 0,0:26:19.04,0:26:20.71,Default,,0000,0000,0000,,But I'm solving\Nyour problem anyway,
Dialogue: 0,0:26:20.71,0:26:23.48,Default,,0000,0000,0000,,even if I change\Nchange the numbers.
Dialogue: 0,0:26:23.48,0:26:26.06,Default,,0000,0000,0000,,
Dialogue: 0,0:26:26.06,0:26:27.96,Default,,0000,0000,0000,,Why don't I change\Nthe numbers, and then
Dialogue: 0,0:26:27.96,0:26:30.85,Default,,0000,0000,0000,,you do it for the given numbers.
Dialogue: 0,0:26:30.85,0:26:33.88,Default,,0000,0000,0000,,Let's say 1, 0.
Dialogue: 0,0:26:33.88,0:26:39.61,Default,,0000,0000,0000,,And it's the same type\Nof problem, right?
Dialogue: 0,0:26:39.61,0:26:42.66,Default,,0000,0000,0000,,Oriented counterclockwise.
Dialogue: 0,0:26:42.66,0:26:43.37,Default,,0000,0000,0000,,That's important.
Dialogue: 0,0:26:43.37,0:26:51.72,Default,,0000,0000,0000,,
Dialogue: 0,0:26:51.72,0:26:54.29,Default,,0000,0000,0000,,So you have circle radius 7.
Dialogue: 0,0:26:54.29,0:26:56.85,Default,,0000,0000,0000,,I think people could\Nhave any other,
Dialogue: 0,0:26:56.85,0:27:01.37,Default,,0000,0000,0000,,because problems are-- sometimes\Nyou get a random assignment.
Dialogue: 0,0:27:01.37,0:27:05.47,Default,,0000,0000,0000,,So you have R\Nequals 2, let's say.
Dialogue: 0,0:27:05.47,0:27:08.33,Default,,0000,0000,0000,,
Dialogue: 0,0:27:08.33,0:27:14.23,Default,,0000,0000,0000,,And you have the point,\Nhow to make up something.
Dialogue: 0,0:27:14.23,0:27:21.29,Default,,0000,0000,0000,,The point corresponding\Nto t equals
Dialogue: 0,0:27:21.29,0:27:29.79,Default,,0000,0000,0000,,0 will be given as you have\N[INAUDIBLE], 1, 0, whatever.
Dialogue: 0,0:27:29.79,0:27:32.00,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:27:32.00,0:27:36.85,Default,,0000,0000,0000,,Use the t as the parameter\Nfor all your answers.
Dialogue: 0,0:27:36.85,0:27:39.18,Default,,0000,0000,0000,,So use t as a parameter\Nfor all your answers,
Dialogue: 0,0:27:39.18,0:27:42.92,Default,,0000,0000,0000,,and the answers are written in\Nthe interactive field as x of t
Dialogue: 0,0:27:42.92,0:27:45.31,Default,,0000,0000,0000,,equals what and y\Nof t equals what,
Dialogue: 0,0:27:45.31,0:27:47.41,Default,,0000,0000,0000,,and it's waiting for\Nyou to fill them in.
Dialogue: 0,0:27:47.41,0:27:49.23,Default,,0000,0000,0000,,You know.
Dialogue: 0,0:27:49.23,0:27:54.47,Default,,0000,0000,0000,,OK, now I was talking\Nto [INAUDIBLE].
Dialogue: 0,0:27:54.47,0:27:56.89,Default,,0000,0000,0000,,I'm going to give\Nthis back to you.
Dialogue: 0,0:27:56.89,0:27:57.68,Default,,0000,0000,0000,,Thank you, Ryan.
Dialogue: 0,0:27:57.68,0:28:02.76,Default,,0000,0000,0000,,So when you said it's a\Nlittle bit frustrating,
Dialogue: 0,0:28:02.76,0:28:07.80,Default,,0000,0000,0000,,and I agree wit you, that\Nin this variant of webwork
Dialogue: 0,0:28:07.80,0:28:10.76,Default,,0000,0000,0000,,problems you have to enter\Nboth of them correctly
Dialogue: 0,0:28:10.76,0:28:14.64,Default,,0000,0000,0000,,in order to say yes, correct.
Dialogue: 0,0:28:14.64,0:28:18.07,Default,,0000,0000,0000,,I was used to another library--\Nthe library was outdated
Dialogue: 0,0:28:18.07,0:28:22.48,Default,,0000,0000,0000,,[INAUDIBLE]-- where if I\Nenter this correctly I get 50%
Dialogue: 0,0:28:22.48,0:28:25.66,Default,,0000,0000,0000,,credit, and if I enter this\Nincorrectly it's not going
Dialogue: 0,0:28:25.66,0:28:26.81,Default,,0000,0000,0000,,to penalize me.
Dialogue: 0,0:28:26.81,0:28:29.92,Default,,0000,0000,0000,,So I a little bit\Ncomplained about it,
Dialogue: 0,0:28:29.92,0:28:32.31,Default,,0000,0000,0000,,and I was shown the\Nold library where
Dialogue: 0,0:28:32.31,0:28:35.55,Default,,0000,0000,0000,,I can go ahead and go\Nback and assign problems
Dialogue: 0,0:28:35.55,0:28:38.87,Default,,0000,0000,0000,,where you get the answer\Ncorrect for this one
Dialogue: 0,0:28:38.87,0:28:42.13,Default,,0000,0000,0000,,and incorrect for this one,\Nand you get partial credit.
Dialogue: 0,0:28:42.13,0:28:46.59,Default,,0000,0000,0000,,So I'm probably going\Nto switch to that.
Dialogue: 0,0:28:46.59,0:28:47.31,Default,,0000,0000,0000,,Let's do that.
Dialogue: 0,0:28:47.31,0:28:48.53,Default,,0000,0000,0000,,This is a very good problem.
Dialogue: 0,0:28:48.53,0:28:51.76,Default,,0000,0000,0000,,I'm glad you brought it up.
Dialogue: 0,0:28:51.76,0:28:56.85,Default,,0000,0000,0000,,What have you learned about\Nconics in high school?
Dialogue: 0,0:28:56.85,0:28:59.77,Default,,0000,0000,0000,,You've learned about--\Nwell, it depends.
Dialogue: 0,0:28:59.77,0:29:01.38,Default,,0000,0000,0000,,You've learned about ellipse.
Dialogue: 0,0:29:01.38,0:29:03.00,Default,,0000,0000,0000,,You've learned about hyperbola.
Dialogue: 0,0:29:03.00,0:29:04.50,Default,,0000,0000,0000,,You've learned about parabola.
Dialogue: 0,0:29:04.50,0:29:07.19,Default,,0000,0000,0000,,Some of you put them down\Nfor me for extra credit.
Dialogue: 0,0:29:07.19,0:29:08.98,Default,,0000,0000,0000,,I was very happy you did that.
Dialogue: 0,0:29:08.98,0:29:10.49,Default,,0000,0000,0000,,It's a good exercise.
Dialogue: 0,0:29:10.49,0:29:12.17,Default,,0000,0000,0000,,If you have-- Alex, yes?
Dialogue: 0,0:29:12.17,0:29:14.21,Default,,0000,0000,0000,,STUDENT: I was just\Nthinking, does that say 1, 0?
Dialogue: 0,0:29:14.21,0:29:17.60,Default,,0000,0000,0000,,
Dialogue: 0,0:29:17.60,0:29:19.41,Default,,0000,0000,0000,,The point corresponding\Nto t0 [INAUDIBLE]?
Dialogue: 0,0:29:19.41,0:29:20.45,Default,,0000,0000,0000,,PROFESSOR TODA: I think\Nthat's what I meant.
Dialogue: 0,0:29:20.45,0:29:22.06,Default,,0000,0000,0000,,I don't know, I just\Ncame up with it.
Dialogue: 0,0:29:22.06,0:29:22.62,Default,,0000,0000,0000,,I made it.
Dialogue: 0,0:29:22.62,0:29:23.23,Default,,0000,0000,0000,,1, 0.
Dialogue: 0,0:29:23.23,0:29:24.31,Default,,0000,0000,0000,,I make up all my problems.
Dialogue: 0,0:29:24.31,0:29:26.12,Default,,0000,0000,0000,,STUDENT: But the center\Nof the circle isn't 1, 0.
Dialogue: 0,0:29:26.12,0:29:27.19,Default,,0000,0000,0000,,PROFESSOR TODA: Oh, oops.
Dialogue: 0,0:29:27.19,0:29:29.74,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:29:29.74,0:29:31.59,Default,,0000,0000,0000,,Sorry.
Dialogue: 0,0:29:31.59,0:29:33.33,Default,,0000,0000,0000,,So 2, 0.
Dialogue: 0,0:29:33.33,0:29:34.10,Default,,0000,0000,0000,,No--
Dialogue: 0,0:29:34.10,0:29:35.29,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:29:35.29,0:29:38.30,Default,,0000,0000,0000,,PROFESSOR TODA:\N--because the radius.
Dialogue: 0,0:29:38.30,0:29:40.72,Default,,0000,0000,0000,,This is the problem when you\Ndon't think very [INAUDIBLE].
Dialogue: 0,0:29:40.72,0:29:44.24,Default,,0000,0000,0000,,I always like to make\Nup my own problems.
Dialogue: 0,0:29:44.24,0:29:48.40,Default,,0000,0000,0000,,When an author, when we came up\Nwith the problems in the book,
Dialogue: 0,0:29:48.40,0:29:51.70,Default,,0000,0000,0000,,of course we had to think, draw,\Nand make sure they made sense.
Dialogue: 0,0:29:51.70,0:29:55.18,Default,,0000,0000,0000,,But when you just come up with\Na problem out of the middle
Dialogue: 0,0:29:55.18,0:29:57.48,Default,,0000,0000,0000,,of nowhere-- thank you so much.
Dialogue: 0,0:29:57.48,0:29:58.85,Default,,0000,0000,0000,,Of course, we\Nwould have realized
Dialogue: 0,0:29:58.85,0:30:01.09,Default,,0000,0000,0000,,that was nonsense\Nin just a minute.
Dialogue: 0,0:30:01.09,0:30:04.47,Default,,0000,0000,0000,,But it's good that you told me.
Dialogue: 0,0:30:04.47,0:30:06.53,Default,,0000,0000,0000,,So x of t, y of t.
Dialogue: 0,0:30:06.53,0:30:10.81,Default,,0000,0000,0000,,
Dialogue: 0,0:30:10.81,0:30:11.90,Default,,0000,0000,0000,,Let's find it.
Dialogue: 0,0:30:11.90,0:30:13.35,Default,,0000,0000,0000,,Based on what?
Dialogue: 0,0:30:13.35,0:30:15.56,Default,,0000,0000,0000,,What is the general\Nequation of a circle?
Dialogue: 0,0:30:15.56,0:30:22.34,Default,,0000,0000,0000,,x minus x0 squared plus y minus\Ny0 squared equals R squared.
Dialogue: 0,0:30:22.34,0:30:24.53,Default,,0000,0000,0000,,And you have learned\Nthat in high school.
Dialogue: 0,0:30:24.53,0:30:26.59,Default,,0000,0000,0000,,Am I right or not?
Dialogue: 0,0:30:26.59,0:30:27.39,Default,,0000,0000,0000,,You have.
Dialogue: 0,0:30:27.39,0:30:28.24,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:30:28.24,0:30:29.13,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:30:29.13,0:30:36.19,Default,,0000,0000,0000,,Now, in our case what\Nis x0 and what is y0?
Dialogue: 0,0:30:36.19,0:30:40.00,Default,,0000,0000,0000,,x0 is 1 and y0 is 0.
Dialogue: 0,0:30:40.00,0:30:42.86,Default,,0000,0000,0000,,Because that's\Nwhy-- I don't know.
Dialogue: 0,0:30:42.86,0:30:44.06,Default,,0000,0000,0000,,I just made it up.
Dialogue: 0,0:30:44.06,0:30:47.07,Default,,0000,0000,0000,,And I said that's the center.
Dialogue: 0,0:30:47.07,0:30:49.14,Default,,0000,0000,0000,,I'll draw.
Dialogue: 0,0:30:49.14,0:30:50.81,Default,,0000,0000,0000,,I should have drawn\Nit in the beginning,
Dialogue: 0,0:30:50.81,0:30:54.40,Default,,0000,0000,0000,,and that would have\Nhelped me not come up
Dialogue: 0,0:30:54.40,0:31:00.43,Default,,0000,0000,0000,,with some nonsensical data.
Dialogue: 0,0:31:00.43,0:31:01.61,Default,,0000,0000,0000,,c is 1, 0.
Dialogue: 0,0:31:01.61,0:31:03.01,Default,,0000,0000,0000,,Radius is 2.
Dialogue: 0,0:31:03.01,0:31:04.64,Default,,0000,0000,0000,,So I'm going this way.
Dialogue: 0,0:31:04.64,0:31:06.79,Default,,0000,0000,0000,,What point is this way, guys?
Dialogue: 0,0:31:06.79,0:31:08.71,Default,,0000,0000,0000,,Just by the way.
Dialogue: 0,0:31:08.71,0:31:10.24,Default,,0000,0000,0000,,Because [INAUDIBLE]\Nis 1, 0, right?
Dialogue: 0,0:31:10.24,0:31:16.24,Default,,0000,0000,0000,,And this way the other\Nextreme, the antipode is 3, 0.
Dialogue: 0,0:31:16.24,0:31:20.09,Default,,0000,0000,0000,,So that's exactly what\NAlexander was saying.
Dialogue: 0,0:31:20.09,0:31:22.02,Default,,0000,0000,0000,,And now it makes sense.
Dialogue: 0,0:31:22.02,0:31:25.40,Default,,0000,0000,0000,,
Dialogue: 0,0:31:25.40,0:31:26.50,Default,,0000,0000,0000,,Well, I cannot draw today.
Dialogue: 0,0:31:26.50,0:31:27.33,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:31:27.33,0:31:30.35,Default,,0000,0000,0000,,
Dialogue: 0,0:31:30.35,0:31:31.77,Default,,0000,0000,0000,,PROFESSOR TODA:\NIt looks horrible.
Dialogue: 0,0:31:31.77,0:31:37.06,Default,,0000,0000,0000,,It looks like an egg that\Nis shaped-- disabled egg.
Dialogue: 0,0:31:37.06,0:31:41.47,Default,,0000,0000,0000,,
Dialogue: 0,0:31:41.47,0:31:42.48,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:31:42.48,0:31:43.06,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:31:43.06,0:31:49.64,Default,,0000,0000,0000,,So the motion of-- the\Nmotion will come like that.
Dialogue: 0,0:31:49.64,0:31:53.74,Default,,0000,0000,0000,,From t equals 0, when I'm\Nhere, counterclockwise,
Dialogue: 0,0:31:53.74,0:31:57.31,Default,,0000,0000,0000,,I have to draw-- any kind of\Ncircle you have in the homework
Dialogue: 0,0:31:57.31,0:32:00.61,Default,,0000,0000,0000,,should be drawn on the board.
Dialogue: 0,0:32:00.61,0:32:06.21,Default,,0000,0000,0000,,If you have a general, you\Ndon't know what the data is.
Dialogue: 0,0:32:06.21,0:32:08.80,Default,,0000,0000,0000,,I want to help you solve\Nthe general problem.
Dialogue: 0,0:32:08.80,0:32:10.78,Default,,0000,0000,0000,,For the original problem,\Nwhich is a circle
Dialogue: 0,0:32:10.78,0:32:15.47,Default,,0000,0000,0000,,of center x, 0, y, 0 and\Nradius R, generic one,
Dialogue: 0,0:32:15.47,0:32:19.57,Default,,0000,0000,0000,,what is the parametrization\Nwithout data?
Dialogue: 0,0:32:19.57,0:32:20.49,Default,,0000,0000,0000,,Without specific data.
Dialogue: 0,0:32:20.49,0:32:23.24,Default,,0000,0000,0000,,What is the parametrization?
Dialogue: 0,0:32:23.24,0:32:25.64,Default,,0000,0000,0000,,And I want you to pay\Nattention very well.
Dialogue: 0,0:32:25.64,0:32:26.93,Default,,0000,0000,0000,,You are paying attention.
Dialogue: 0,0:32:26.93,0:32:29.73,Default,,0000,0000,0000,,You are very careful today.
Dialogue: 0,0:32:29.73,0:32:31.19,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:32:31.19,0:32:33.94,Default,,0000,0000,0000,,So what do you have?
Dialogue: 0,0:32:33.94,0:32:35.74,Default,,0000,0000,0000,,STUDENT: Cosine.
Dialogue: 0,0:32:35.74,0:32:37.52,Default,,0000,0000,0000,,PROFESSOR TODA:\NBefore that cosine
Dialogue: 0,0:32:37.52,0:32:39.58,Default,,0000,0000,0000,,there is an R, excellent.
Dialogue: 0,0:32:39.58,0:32:43.98,Default,,0000,0000,0000,,So [INAUDIBLE]\Nthere R cosine of t.
Dialogue: 0,0:32:43.98,0:32:46.55,Default,,0000,0000,0000,,I'm not done.
Dialogue: 0,0:32:46.55,0:32:47.48,Default,,0000,0000,0000,,What do I put here?
Dialogue: 0,0:32:47.48,0:32:48.34,Default,,0000,0000,0000,,STUDENT: Over d.
Dialogue: 0,0:32:48.34,0:32:49.30,Default,,0000,0000,0000,,PROFESSOR TODA: No, no.
Dialogue: 0,0:32:49.30,0:32:51.28,Default,,0000,0000,0000,,I'm continuing.
Dialogue: 0,0:32:51.28,0:32:52.24,Default,,0000,0000,0000,,STUDENT: Plus x0.
Dialogue: 0,0:32:52.24,0:32:53.69,Default,,0000,0000,0000,,PROFESSOR TODA: Plus x0.
Dialogue: 0,0:32:53.69,0:32:57.46,Default,,0000,0000,0000,,And R sine t plus y0.
Dialogue: 0,0:32:57.46,0:32:59.56,Default,,0000,0000,0000,,Who taught me that?
Dialogue: 0,0:32:59.56,0:33:02.64,Default,,0000,0000,0000,,First of all, this\Nis not unique.
Dialogue: 0,0:33:02.64,0:33:03.65,Default,,0000,0000,0000,,It's not unique.
Dialogue: 0,0:33:03.65,0:33:05.92,Default,,0000,0000,0000,,I could put sine t\Nhere and cosine t here
Dialogue: 0,0:33:05.92,0:33:08.65,Default,,0000,0000,0000,,and it would be the same\Ntype of parametrization.
Dialogue: 0,0:33:08.65,0:33:11.02,Default,,0000,0000,0000,,But we usually put\Nthe cosine first
Dialogue: 0,0:33:11.02,0:33:13.50,Default,,0000,0000,0000,,because we look at the\Nx-axis corresponding
Dialogue: 0,0:33:13.50,0:33:17.77,Default,,0000,0000,0000,,to the cosine and the y-axis\Ncorresponding to the sine.
Dialogue: 0,0:33:17.77,0:33:20.84,Default,,0000,0000,0000,,If I don't know that,\Nbecause I happen to know that
Dialogue: 0,0:33:20.84,0:33:24.17,Default,,0000,0000,0000,,from when I was 16 in high\Nschool, if I don't know that,
Dialogue: 0,0:33:24.17,0:33:25.38,Default,,0000,0000,0000,,what do I know?
Dialogue: 0,0:33:25.38,0:33:27.55,Default,,0000,0000,0000,,I cook up my own\Nparametrization.
Dialogue: 0,0:33:27.55,0:33:29.13,Default,,0000,0000,0000,,And that's a very good thing.
Dialogue: 0,0:33:29.13,0:33:31.16,Default,,0000,0000,0000,,And I'm glad Ryan\Nasked about that.
Dialogue: 0,0:33:31.16,0:33:33.36,Default,,0000,0000,0000,,How does one come up with this?
Dialogue: 0,0:33:33.36,0:33:34.36,Default,,0000,0000,0000,,Do we have to memorize?
Dialogue: 0,0:33:34.36,0:33:38.19,Default,,0000,0000,0000,,In mathematics, thank god,\Nwe don't memorize much.
Dialogue: 0,0:33:38.19,0:33:41.73,Default,,0000,0000,0000,,The way we cook up things\Nis just from, in this case,
Dialogue: 0,0:33:41.73,0:33:44.93,Default,,0000,0000,0000,,from the Pythagorean\Ntheorem of-- no.
Dialogue: 0,0:33:44.93,0:33:47.14,Default,,0000,0000,0000,,Pythagorean theorem\Nof trigonometry?
Dialogue: 0,0:33:47.14,0:33:49.17,Default,,0000,0000,0000,,The fundamental identity\Nof trigonometry,
Dialogue: 0,0:33:49.17,0:33:52.84,Default,,0000,0000,0000,,which is the same thing as\Nthe Pythagorean theorem.
Dialogue: 0,0:33:52.84,0:33:55.30,Default,,0000,0000,0000,,What's the fundamental\Nidentity of trigonometry?
Dialogue: 0,0:33:55.30,0:33:58.21,Default,,0000,0000,0000,,Cosine squared plus\Nsin squared equals 1.
Dialogue: 0,0:33:58.21,0:34:03.68,Default,,0000,0000,0000,,If I have a problem\Nlike that, I must
Dialogue: 0,0:34:03.68,0:34:08.81,Default,,0000,0000,0000,,have that this is R cosine\Nt and this is R sine t.
Dialogue: 0,0:34:08.81,0:34:11.31,Default,,0000,0000,0000,,Because when I take\Nthe red guys and I
Dialogue: 0,0:34:11.31,0:34:13.82,Default,,0000,0000,0000,,square them and I\Nadd them together,
Dialogue: 0,0:34:13.82,0:34:17.65,Default,,0000,0000,0000,,I'm going to have R squared.
Dialogue: 0,0:34:17.65,0:34:18.94,Default,,0000,0000,0000,,All righty, good.
Dialogue: 0,0:34:18.94,0:34:23.37,Default,,0000,0000,0000,,So no matter what\Nkind of data you have,
Dialogue: 0,0:34:23.37,0:34:27.77,Default,,0000,0000,0000,,you should be able to come\Nup with this on your own.
Dialogue: 0,0:34:27.77,0:34:33.51,Default,,0000,0000,0000,,And what else is\Ngoing to be happening?
Dialogue: 0,0:34:33.51,0:34:38.14,Default,,0000,0000,0000,,When I solve for x of-- the\Npoint corresponding to t
Dialogue: 0,0:34:38.14,0:34:39.39,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,0:34:39.39,0:34:43.92,Default,,0000,0000,0000,,x of 0 and y of 0 will\Ntherefore be what?
Dialogue: 0,0:34:43.92,0:34:49.35,Default,,0000,0000,0000,,It will be R plus x0.
Dialogue: 0,0:34:49.35,0:34:51.43,Default,,0000,0000,0000,,This is going to be what?
Dialogue: 0,0:34:51.43,0:34:53.71,Default,,0000,0000,0000,,Just y0.
Dialogue: 0,0:34:53.71,0:34:56.19,Default,,0000,0000,0000,,Does anybody give them to me?
Dialogue: 0,0:34:56.19,0:34:59.11,Default,,0000,0000,0000,,STUDENT: 3, 0.
Dialogue: 0,0:34:59.11,0:35:01.63,Default,,0000,0000,0000,,PROFESSOR TODA: Alexander\Ngave me the correct ones.
Dialogue: 0,0:35:01.63,0:35:05.67,Default,,0000,0000,0000,,They will be 3 and 0.
Dialogue: 0,0:35:05.67,0:35:06.67,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:35:06.67,0:35:10.92,Default,,0000,0000,0000,,They could be anything,\Nanything that makes sense.
Dialogue: 0,0:35:10.92,0:35:15.10,Default,,0000,0000,0000,,All right, for example somebody\Nwould say, I'm starting here.
Dialogue: 0,0:35:15.10,0:35:16.58,Default,,0000,0000,0000,,I give you other points.
Dialogue: 0,0:35:16.58,0:35:19.91,Default,,0000,0000,0000,,Then you put them in, you\Nplug in that initial point,
Dialogue: 0,0:35:19.91,0:35:22.98,Default,,0000,0000,0000,,meaning that you're\Nstarting your motion here.
Dialogue: 0,0:35:22.98,0:35:26.12,Default,,0000,0000,0000,,And you do go around\Nthe circle one
Dialogue: 0,0:35:26.12,0:35:31.69,Default,,0000,0000,0000,,because, you take [INAUDIBLE]\Nonly between 0 and 2 pi.
Dialogue: 0,0:35:31.69,0:35:32.85,Default,,0000,0000,0000,,Alexander.
Dialogue: 0,0:35:32.85,0:35:34.02,Default,,0000,0000,0000,,STUDENT: I have [INAUDIBLE].
Dialogue: 0,0:35:34.02,0:35:34.95,Default,,0000,0000,0000,,PROFESSOR TODA: OK.
Dialogue: 0,0:35:34.95,0:35:35.88,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:35:35.88,0:35:38.31,Default,,0000,0000,0000,,PROFESSOR TODA: No, I thought\Nthat I misprinted something
Dialogue: 0,0:35:38.31,0:35:38.61,Default,,0000,0000,0000,,again.
Dialogue: 0,0:35:38.61,0:35:40.78,Default,,0000,0000,0000,,STUDENT: No, I was about to\Nsay something really dumb.
Dialogue: 0,0:35:40.78,0:35:41.58,Default,,0000,0000,0000,,PROFESSOR TODA: OK.
Dialogue: 0,0:35:41.58,0:35:43.84,Default,,0000,0000,0000,,
Dialogue: 0,0:35:43.84,0:35:48.69,Default,,0000,0000,0000,,So how do we make sense\Nof what we have here?
Dialogue: 0,0:35:48.69,0:35:52.28,Default,,0000,0000,0000,,Well, y0 corresponds\Nto what I said.
Dialogue: 0,0:35:52.28,0:35:55.68,Default,,0000,0000,0000,,So this is a\Nsuperfluous equation.
Dialogue: 0,0:35:55.68,0:35:57.62,Default,,0000,0000,0000,,I don't need that.
Dialogue: 0,0:35:57.62,0:36:01.20,Default,,0000,0000,0000,,What do I know from that?
Dialogue: 0,0:36:01.20,0:36:05.82,Default,,0000,0000,0000,,R will be 2.
Dialogue: 0,0:36:05.82,0:36:07.69,Default,,0000,0000,0000,,x1 is 1.
Dialogue: 0,0:36:07.69,0:36:10.00,Default,,0000,0000,0000,,I have a superfluous equation.
Dialogue: 0,0:36:10.00,0:36:13.89,Default,,0000,0000,0000,,I have to get identities\Nin that case, right?
Dialogue: 0,0:36:13.89,0:36:14.70,Default,,0000,0000,0000,,OK, now.
Dialogue: 0,0:36:14.70,0:36:19.64,Default,,0000,0000,0000,,
Dialogue: 0,0:36:19.64,0:36:26.64,Default,,0000,0000,0000,,What is going to be my--\Nmy bunch of equations
Dialogue: 0,0:36:26.64,0:36:47.64,Default,,0000,0000,0000,,will be x of t equals 2\Ncosine t plus 1 and y of t
Dialogue: 0,0:36:47.64,0:36:49.23,Default,,0000,0000,0000,,equals-- I don't\Nlike this marker.
Dialogue: 0,0:36:49.23,0:36:49.92,Default,,0000,0000,0000,,I hate it.
Dialogue: 0,0:36:49.92,0:36:50.77,Default,,0000,0000,0000,,Where did I get it?
Dialogue: 0,0:36:50.77,0:36:51.73,Default,,0000,0000,0000,,In the math department.
Dialogue: 0,0:36:51.73,0:36:53.46,Default,,0000,0000,0000,,And it's a new one.
Dialogue: 0,0:36:53.46,0:36:54.82,Default,,0000,0000,0000,,I got it as a new one.
Dialogue: 0,0:36:54.82,0:36:56.74,Default,,0000,0000,0000,,It's not working.
Dialogue: 0,0:36:56.74,0:36:58.18,Default,,0000,0000,0000,,OK, y of t.
Dialogue: 0,0:36:58.18,0:37:01.15,Default,,0000,0000,0000,,
Dialogue: 0,0:37:01.15,0:37:03.83,Default,,0000,0000,0000,,The blue contrast is invisible.
Dialogue: 0,0:37:03.83,0:37:07.90,Default,,0000,0000,0000,,I have 2 sine t.
Dialogue: 0,0:37:07.90,0:37:08.40,Default,,0000,0000,0000,,Okey dokey.
Dialogue: 0,0:37:08.40,0:37:11.59,Default,,0000,0000,0000,,When you finish a\Nproblem, always quickly
Dialogue: 0,0:37:11.59,0:37:15.74,Default,,0000,0000,0000,,verify if what you\Ngot makes sense.
Dialogue: 0,0:37:15.74,0:37:19.50,Default,,0000,0000,0000,,And obviously if I\Nlook at everything,
Dialogue: 0,0:37:19.50,0:37:21.13,Default,,0000,0000,0000,,it's matching the whole point.
Dialogue: 0,0:37:21.13,0:37:21.91,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:37:21.91,0:37:22.68,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:37:22.68,0:37:29.84,Default,,0000,0000,0000,,Now going back to-- this is\Nreminding me of something in 3d
Dialogue: 0,0:37:29.84,0:37:34.67,Default,,0000,0000,0000,,that I wanted to talk\Nto you today about.
Dialogue: 0,0:37:34.67,0:37:36.60,Default,,0000,0000,0000,,This is [INAUDIBLE].
Dialogue: 0,0:37:36.60,0:37:42.90,Default,,0000,0000,0000,,
Dialogue: 0,0:37:42.90,0:37:45.12,Default,,0000,0000,0000,,I'm going to give\Nyou, in a similar way
Dialogue: 0,0:37:45.12,0:37:47.50,Default,,0000,0000,0000,,with this simple\Nproblem, I'm going
Dialogue: 0,0:37:47.50,0:37:49.89,Default,,0000,0000,0000,,to give you something\Nmore complicated
Dialogue: 0,0:37:49.89,0:38:16.70,Default,,0000,0000,0000,,and say find the\Nparametrization of a helix.
Dialogue: 0,0:38:16.70,0:38:19.61,Default,,0000,0000,0000,,And you say, well,\NI'm happy that this
Dialogue: 0,0:38:19.61,0:38:21.82,Default,,0000,0000,0000,,isn't a made-up problem again.
Dialogue: 0,0:38:21.82,0:38:23.92,Default,,0000,0000,0000,,I have to be a little\Nbit more careful
Dialogue: 0,0:38:23.92,0:38:27.43,Default,,0000,0000,0000,,with these made-up problems\Nso that they make sense.
Dialogue: 0,0:38:27.43,0:38:44.38,Default,,0000,0000,0000,,Of a helix R of t such that\Nit is contained or it lies,
Dialogue: 0,0:38:44.38,0:38:59.77,Default,,0000,0000,0000,,it lies on the circular\Ncylinder x squared
Dialogue: 0,0:38:59.77,0:39:03.82,Default,,0000,0000,0000,,plus y squared equals 4.
Dialogue: 0,0:39:03.82,0:39:04.78,Default,,0000,0000,0000,,Why is that a cylinder?
Dialogue: 0,0:39:04.78,0:39:07.91,Default,,0000,0000,0000,,The z's missing, so it's\Ngoing to be a cylinder whose
Dialogue: 0,0:39:07.91,0:39:09.47,Default,,0000,0000,0000,,main axis is the z axis.
Dialogue: 0,0:39:09.47,0:39:09.97,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:39:09.97,0:39:11.45,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:39:11.45,0:39:14.75,Default,,0000,0000,0000,,I think we are on the same page.
Dialogue: 0,0:39:14.75,0:39:19.45,Default,,0000,0000,0000,,And you cannot solve the\Nproblem just with this data.
Dialogue: 0,0:39:19.45,0:39:22.16,Default,,0000,0000,0000,,Do you agree with me?
Dialogue: 0,0:39:22.16,0:39:47.19,Default,,0000,0000,0000,,And knowing that, the\Ncurvature of the helix is k
Dialogue: 0,0:39:47.19,0:40:04.03,Default,,0000,0000,0000,,equals 2/5 at every point.
Dialogue: 0,0:40:04.03,0:40:06.08,Default,,0000,0000,0000,,And of course it's an oxymoron.
Dialogue: 0,0:40:06.08,0:40:08.24,Default,,0000,0000,0000,,Because what I\Nproved last time is
Dialogue: 0,0:40:08.24,0:40:12.53,Default,,0000,0000,0000,,that the curvature of\Na helix is a constant.
Dialogue: 0,0:40:12.53,0:40:27.22,Default,,0000,0000,0000,,So remember, we got the\Ncurvature of a helix
Dialogue: 0,0:40:27.22,0:40:30.06,Default,,0000,0000,0000,,as being a constant.
Dialogue: 0,0:40:30.06,0:40:34.46,Default,,0000,0000,0000,,
Dialogue: 0,0:40:34.46,0:40:36.41,Default,,0000,0000,0000,,STUDENT: What's that last\Nword of the sentence?
Dialogue: 0,0:40:36.41,0:40:38.59,Default,,0000,0000,0000,,It's "the curvature\Nis at every" what?
Dialogue: 0,0:40:38.59,0:40:39.88,Default,,0000,0000,0000,,PROFESSOR TODA: At every point.
Dialogue: 0,0:40:39.88,0:40:45.07,Default,,0000,0000,0000,,I'm sorry I said, it very--\NI abbreviated [INAUDIBLE].
Dialogue: 0,0:40:45.07,0:40:48.10,Default,,0000,0000,0000,,So at every point you\Nhave the same curvature.
Dialogue: 0,0:40:48.10,0:40:50.92,Default,,0000,0000,0000,,When you draw a\Nhelix you say, wait,
Dialogue: 0,0:40:50.92,0:40:53.82,Default,,0000,0000,0000,,the helix is bent uniformly.
Dialogue: 0,0:40:53.82,0:40:58.76,Default,,0000,0000,0000,,If you were to play with a\Nspring taken from am old bed,
Dialogue: 0,0:40:58.76,0:41:01.91,Default,,0000,0000,0000,,you would go with your\Nhands along the spring.
Dialogue: 0,0:41:01.91,0:41:04.70,Default,,0000,0000,0000,,And then you say, oh,\Nit bends about the same.
Dialogue: 0,0:41:04.70,0:41:06.01,Default,,0000,0000,0000,,Yes, it does.
Dialogue: 0,0:41:06.01,0:41:08.80,Default,,0000,0000,0000,,And that means the\Ncurvature is the same.
Dialogue: 0,0:41:08.80,0:41:11.72,Default,,0000,0000,0000,,How would you\Nsolve this problem?
Dialogue: 0,0:41:11.72,0:41:16.38,Default,,0000,0000,0000,,This problem is hard,\Nbecause you cannot integrate
Dialogue: 0,0:41:16.38,0:41:17.45,Default,,0000,0000,0000,,the curvature.
Dialogue: 0,0:41:17.45,0:41:19.11,Default,,0000,0000,0000,,Well, what is the curvature?
Dialogue: 0,0:41:19.11,0:41:21.04,Default,,0000,0000,0000,,The curvature would be--
Dialogue: 0,0:41:21.04,0:41:22.04,Default,,0000,0000,0000,,STUDENT: Absolute value.
Dialogue: 0,0:41:22.04,0:41:23.91,Default,,0000,0000,0000,,PROFESSOR TODA: Just\Nabsolute value of R
Dialogue: 0,0:41:23.91,0:41:28.41,Default,,0000,0000,0000,,double prime if it were in s.
Dialogue: 0,0:41:28.41,0:41:31.03,Default,,0000,0000,0000,,And you cannot integrate.
Dialogue: 0,0:41:31.03,0:41:34.00,Default,,0000,0000,0000,,If somebody gave you\Nthe vector equation
Dialogue: 0,0:41:34.00,0:41:36.57,Default,,0000,0000,0000,,of double prime of\Nthis, them you say,
Dialogue: 0,0:41:36.57,0:41:38.73,Default,,0000,0000,0000,,yes, I can integrate\None step going back.
Dialogue: 0,0:41:38.73,0:41:40.33,Default,,0000,0000,0000,,I get R prime of s.
Dialogue: 0,0:41:40.33,0:41:41.73,Default,,0000,0000,0000,,Then I go back to R of s.
Dialogue: 0,0:41:41.73,0:41:43.27,Default,,0000,0000,0000,,But this is a little\Nbit complicated.
Dialogue: 0,0:41:43.27,0:41:45.49,Default,,0000,0000,0000,,I'm giving you a scalar.
Dialogue: 0,0:41:45.49,0:41:50.76,Default,,0000,0000,0000,,You have to be a little bit\Naware of what you did last time
Dialogue: 0,0:41:50.76,0:41:54.60,Default,,0000,0000,0000,,and try to remember\Nwhat we did last time.
Dialogue: 0,0:41:54.60,0:41:56.33,Default,,0000,0000,0000,,What did we do last time?
Dialogue: 0,0:41:56.33,0:41:58.11,Default,,0000,0000,0000,,I would not give you\Na problem like that
Dialogue: 0,0:41:58.11,0:42:03.29,Default,,0000,0000,0000,,on the final, because it would\Nassume that you have solved
Dialogue: 0,0:42:03.29,0:42:06.28,Default,,0000,0000,0000,,the problem we did last\Ntime in terms of R of t
Dialogue: 0,0:42:06.28,0:42:09.95,Default,,0000,0000,0000,,equals A equals sine t.
Dialogue: 0,0:42:09.95,0:42:11.40,Default,,0000,0000,0000,,A sine t and [? vt. ?]
Dialogue: 0,0:42:11.40,0:42:15.77,Default,,0000,0000,0000,,And we said, this is the\Nstandard parametrized helix
Dialogue: 0,0:42:15.77,0:42:22.33,Default,,0000,0000,0000,,that sits on a cylinder of\Nradius A and has the phb.
Dialogue: 0,0:42:22.33,0:42:27.51,Default,,0000,0000,0000,,So the distance between\Nconsecutive spirals
Dialogue: 0,0:42:27.51,0:42:28.77,Default,,0000,0000,0000,,really matters.
Dialogue: 0,0:42:28.77,0:42:30.16,Default,,0000,0000,0000,,That really makes\Nthe difference.
Dialogue: 0,0:42:30.16,0:42:30.62,Default,,0000,0000,0000,,STUDENT: I have a question.
Dialogue: 0,0:42:30.62,0:42:32.58,Default,,0000,0000,0000,,PROFESSOR TODA: You wanted\Nto ask me something.
Dialogue: 0,0:42:32.58,0:42:34.35,Default,,0000,0000,0000,,STUDENT: Is s always\Nthe reciprocal of t?
Dialogue: 0,0:42:34.35,0:42:35.95,Default,,0000,0000,0000,,Are they always--
Dialogue: 0,0:42:35.95,0:42:37.41,Default,,0000,0000,0000,,PROFESSOR TODA:\NNo, not reciprocal.
Dialogue: 0,0:42:37.41,0:42:45.80,Default,,0000,0000,0000,,You mean s of t is a function\Nis from t0 to t of the speed.
Dialogue: 0,0:42:45.80,0:42:50.43,Default,,0000,0000,0000,,R prime and t-- d tau, right?
Dialogue: 0,0:42:50.43,0:42:51.63,Default,,0000,0000,0000,,Tau not t. [INAUDIBLE].
Dialogue: 0,0:42:51.63,0:42:54.22,Default,,0000,0000,0000,,
Dialogue: 0,0:42:54.22,0:43:00.65,Default,,0000,0000,0000,,t and s are\Ndifferent parameters.
Dialogue: 0,0:43:00.65,0:43:01.99,Default,,0000,0000,0000,,Different times.
Dialogue: 0,0:43:01.99,0:43:04.37,Default,,0000,0000,0000,,Different parameter times.
Dialogue: 0,0:43:04.37,0:43:04.91,Default,,0000,0000,0000,,And you say--
Dialogue: 0,0:43:04.91,0:43:06.70,Default,,0000,0000,0000,,STUDENT: Isn't s\Nthe parameter time
Dialogue: 0,0:43:06.70,0:43:08.81,Default,,0000,0000,0000,,when [INAUDIBLE] parametrized?
Dialogue: 0,0:43:08.81,0:43:09.89,Default,,0000,0000,0000,,PROFESSOR TODA: Very good.
Dialogue: 0,0:43:09.89,0:43:12.36,Default,,0000,0000,0000,,So what is the magic s?
Dialogue: 0,0:43:12.36,0:43:13.85,Default,,0000,0000,0000,,I'm proud of you.
Dialogue: 0,0:43:13.85,0:43:15.94,Default,,0000,0000,0000,,This is the important\Nthing to remember.
Dialogue: 0,0:43:15.94,0:43:17.53,Default,,0000,0000,0000,,t could be any time.
Dialogue: 0,0:43:17.53,0:43:19.96,Default,,0000,0000,0000,,I start measuring\Nwherever I want.
Dialogue: 0,0:43:19.96,0:43:23.69,Default,,0000,0000,0000,,I can set my watch to start now.
Dialogue: 0,0:43:23.69,0:43:24.95,Default,,0000,0000,0000,,It could be crazy.
Dialogue: 0,0:43:24.95,0:43:26.64,Default,,0000,0000,0000,,Doesn't have to be uniform.
Dialogue: 0,0:43:26.64,0:43:27.60,Default,,0000,0000,0000,,Motion, I don't care.
Dialogue: 0,0:43:27.60,0:43:30.76,Default,,0000,0000,0000,,
Dialogue: 0,0:43:30.76,0:43:33.33,Default,,0000,0000,0000,,s is a friend of\Nyours that says,
Dialogue: 0,0:43:33.33,0:43:38.29,Default,,0000,0000,0000,,I am that special time\Nso that according to me
Dialogue: 0,0:43:38.29,0:43:40.59,Default,,0000,0000,0000,,the speed will become one.
Dialogue: 0,0:43:40.59,0:43:45.65,Default,,0000,0000,0000,,So for a physicist to measure\Nthe speed with respect to this,
Dialogue: 0,0:43:45.65,0:43:49.41,Default,,0000,0000,0000,,parameter s time, the speed\Nwill always become one.
Dialogue: 0,0:43:49.41,0:43:51.66,Default,,0000,0000,0000,,That is the arclength\Ntime and position.
Dialogue: 0,0:43:51.66,0:43:54.48,Default,,0000,0000,0000,,How you get from one\Nanother, I told you last time
Dialogue: 0,0:43:54.48,0:43:57.38,Default,,0000,0000,0000,,that for both of them\Nyou have-- this is R of t
Dialogue: 0,0:43:57.38,0:43:59.23,Default,,0000,0000,0000,,and this is little r of s.
Dialogue: 0,0:43:59.23,0:44:01.24,Default,,0000,0000,0000,,And there is a composition.
Dialogue: 0,0:44:01.24,0:44:03.46,Default,,0000,0000,0000,,s can be viewed as\Na function of t,
Dialogue: 0,0:44:03.46,0:44:06.34,Default,,0000,0000,0000,,and t can be viewed\Nas a function of s.
Dialogue: 0,0:44:06.34,0:44:09.58,Default,,0000,0000,0000,,As functions they are\Ninverse to one another.
Dialogue: 0,0:44:09.58,0:44:12.65,Default,,0000,0000,0000,,So going back to who they\Nare, a very good question,
Dialogue: 0,0:44:12.65,0:44:15.58,Default,,0000,0000,0000,,because this is a review\Nanyway, [? who wants ?]
Dialogue: 0,0:44:15.58,0:44:19.18,Default,,0000,0000,0000,,s as a function of t for\Nthis particular problem?
Dialogue: 0,0:44:19.18,0:44:23.95,Default,,0000,0000,0000,,I hope you remember, we were\Nlike-- have you seen this movie
Dialogue: 0,0:44:23.95,0:44:28.40,Default,,0000,0000,0000,,with Mickey Mouse going\Non a mountain that
Dialogue: 0,0:44:28.40,0:44:32.10,Default,,0000,0000,0000,,was more like a cylinder.
Dialogue: 0,0:44:32.10,0:44:35.32,Default,,0000,0000,0000,,And this is the train\Ngoing at a constant slope.
Dialogue: 0,0:44:35.32,0:44:42.59,Default,,0000,0000,0000,,And one of my colleagues,\Nactually, he's at Stanford,
Dialogue: 0,0:44:42.59,0:44:47.30,Default,,0000,0000,0000,,was telling me that he\Ngave his students in Calc 1
Dialogue: 0,0:44:47.30,0:44:51.86,Default,,0000,0000,0000,,to prove, formally prove,\Nthat the angle formed
Dialogue: 0,0:44:51.86,0:44:56.59,Default,,0000,0000,0000,,by the law of motion\Nby the velocity vector,
Dialogue: 0,0:44:56.59,0:45:01.99,Default,,0000,0000,0000,,with the horizontal plane\Npassing through the particle,
Dialogue: 0,0:45:01.99,0:45:04.05,Default,,0000,0000,0000,,is always a constant.
Dialogue: 0,0:45:04.05,0:45:07.16,Default,,0000,0000,0000,,I didn't think about doing\Nin now, but of course we can.
Dialogue: 0,0:45:07.16,0:45:08.52,Default,,0000,0000,0000,,We could do that.
Dialogue: 0,0:45:08.52,0:45:10.96,Default,,0000,0000,0000,,So maybe the next\Nthing would be, like,
Dialogue: 0,0:45:10.96,0:45:12.63,Default,,0000,0000,0000,,if you [INAUDIBLE]\Nan extra problem, can
Dialogue: 0,0:45:12.63,0:45:17.28,Default,,0000,0000,0000,,we show that angle between the\Nvelocity vector on the helix
Dialogue: 0,0:45:17.28,0:45:20.70,Default,,0000,0000,0000,,and the horizontal plane through\Nthat point is a constant.
Dialogue: 0,0:45:20.70,0:45:22.54,Default,,0000,0000,0000,,STUDENT: Wouldn't it\Njust be, because B of t
Dialogue: 0,0:45:22.54,0:45:23.94,Default,,0000,0000,0000,,is just a constant times t?
Dialogue: 0,0:45:23.94,0:45:24.81,Default,,0000,0000,0000,,PROFESSOR TODA: Yeah.
Dialogue: 0,0:45:24.81,0:45:25.59,Default,,0000,0000,0000,,We'll get to that.
Dialogue: 0,0:45:25.59,0:45:27.17,Default,,0000,0000,0000,,We'll get to that in a second.
Dialogue: 0,0:45:27.17,0:45:32.49,Default,,0000,0000,0000,,So he reminded me of an old\Nmovie from like 70 years ago,
Dialogue: 0,0:45:32.49,0:45:33.82,Default,,0000,0000,0000,,with Mickey Mouse and the train.
Dialogue: 0,0:45:33.82,0:45:38.65,Default,,0000,0000,0000,,And the train going\Nup at the same speed.
Dialogue: 0,0:45:38.65,0:45:41.16,Default,,0000,0000,0000,,You have to maintain\Nthe same speed.
Dialogue: 0,0:45:41.16,0:45:44.81,Default,,0000,0000,0000,,Because if you risk it\Nnot, then you sort of
Dialogue: 0,0:45:44.81,0:45:46.16,Default,,0000,0000,0000,,are getting trouble.
Dialogue: 0,0:45:46.16,0:45:47.76,Default,,0000,0000,0000,,So you never stop.
Dialogue: 0,0:45:47.76,0:45:49.29,Default,,0000,0000,0000,,If you stop you go back.
Dialogue: 0,0:45:49.29,0:45:50.74,Default,,0000,0000,0000,,So it's a regular curve.
Dialogue: 0,0:45:50.74,0:45:52.88,Default,,0000,0000,0000,,What I have here is\Nthat such a curve.
Dialogue: 0,0:45:52.88,0:45:54.79,Default,,0000,0000,0000,,Regular curve, never stop.
Dialogue: 0,0:45:54.79,0:45:56.80,Default,,0000,0000,0000,,Get up with a constant speed.
Dialogue: 0,0:45:56.80,0:45:58.83,Default,,0000,0000,0000,,Do you guys remember the\Nspeed from last time?
Dialogue: 0,0:45:58.83,0:46:01.08,Default,,0000,0000,0000,,We'll square root the a\Nsquared plus b squared.
Dialogue: 0,0:46:01.08,0:46:04.43,Default,,0000,0000,0000,,When we did the\Nvelocity thingie.
Dialogue: 0,0:46:04.43,0:46:10.73,Default,,0000,0000,0000,,And I get square root a\Nsquared plus b squared times t.
Dialogue: 0,0:46:10.73,0:46:19.04,Default,,0000,0000,0000,,Now, today I would like\Nto ask you one question.
Dialogue: 0,0:46:19.04,0:46:21.52,Default,,0000,0000,0000,,What if-- Ryan brought this up.
Dialogue: 0,0:46:21.52,0:46:22.46,Default,,0000,0000,0000,,It's very good.
Dialogue: 0,0:46:22.46,0:46:23.66,Default,,0000,0000,0000,,b is a constant.
Dialogue: 0,0:46:23.66,0:46:26.55,Default,,0000,0000,0000,,What if b would\Nnot be a constant,
Dialogue: 0,0:46:26.55,0:46:28.61,Default,,0000,0000,0000,,or maybe could be worse?
Dialogue: 0,0:46:28.61,0:46:32.71,Default,,0000,0000,0000,,For example, instead of having\Nanother linear function with t,
Dialogue: 0,0:46:32.71,0:46:36.18,Default,,0000,0000,0000,,but something that contains\Nhigher powers of t.
Dialogue: 0,0:46:36.18,0:46:39.36,Default,,0000,0000,0000,,
Dialogue: 0,0:46:39.36,0:46:43.41,Default,,0000,0000,0000,,Then you don't go at the\Nconstant speed anymore.
Dialogue: 0,0:46:43.41,0:46:45.37,Default,,0000,0000,0000,,You can say goodbye\Nto the cartoon.
Dialogue: 0,0:46:45.37,0:46:45.88,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:46:45.88,0:46:49.02,Default,,0000,0000,0000,,STUDENT: And then\Nit's [INAUDIBLE].
Dialogue: 0,0:46:49.02,0:46:50.10,Default,,0000,0000,0000,,One that goes [INAUDIBLE].
Dialogue: 0,0:46:50.10,0:46:51.60,Default,,0000,0000,0000,,PROFESSOR TODA: I\Nmean, it's still--
Dialogue: 0,0:46:51.60,0:46:54.83,Default,,0000,0000,0000,,STUDENT: s is not\Nmultiplied by a constant.
Dialogue: 0,0:46:54.83,0:46:57.02,Default,,0000,0000,0000,,The function between t and\Ns is not a constant one.
Dialogue: 0,0:46:57.02,0:46:59.60,Default,,0000,0000,0000,,PROFESSOR TODA: It's going to\Nbe a different parameterization,
Dialogue: 0,0:46:59.60,0:47:00.58,Default,,0000,0000,0000,,different speed.
Dialogue: 0,0:47:00.58,0:47:03.77,Default,,0000,0000,0000,,Sometimes-- OK, you\Nhave to understand.
Dialogue: 0,0:47:03.77,0:47:06.74,Default,,0000,0000,0000,,Let's say I have a cone.
Dialogue: 0,0:47:06.74,0:47:10.23,Default,,0000,0000,0000,,And I'm going slow\Nat first, and I
Dialogue: 0,0:47:10.23,0:47:11.98,Default,,0000,0000,0000,,go faster and faster\Nand faster and faster
Dialogue: 0,0:47:11.98,0:47:13.90,Default,,0000,0000,0000,,to the end of the cone.
Dialogue: 0,0:47:13.90,0:47:18.39,Default,,0000,0000,0000,,But then I have the\Nsame physical curve,
Dialogue: 0,0:47:18.39,0:47:21.04,Default,,0000,0000,0000,,and I parameterized\N[INAUDIBLE] the length.
Dialogue: 0,0:47:21.04,0:47:24.31,Default,,0000,0000,0000,,And I say, no, I'm a mechanic.
Dialogue: 0,0:47:24.31,0:47:26.84,Default,,0000,0000,0000,,Or I'm the engineer\Nof the strain.
Dialogue: 0,0:47:26.84,0:47:29.42,Default,,0000,0000,0000,,I can make the motion\Nhave a constant speed.
Dialogue: 0,0:47:29.42,0:47:33.13,Default,,0000,0000,0000,,So even if the helix\Nis no longer circular,
Dialogue: 0,0:47:33.13,0:47:36.56,Default,,0000,0000,0000,,and it's sort of a crazy helix\Ngoing on top of the mountain,
Dialogue: 0,0:47:36.56,0:47:39.33,Default,,0000,0000,0000,,as an engineer I\Ncan just say, oh no,
Dialogue: 0,0:47:39.33,0:47:42.15,Default,,0000,0000,0000,,I want cruise control\Nfor my little train.
Dialogue: 0,0:47:42.15,0:47:45.50,Default,,0000,0000,0000,,And I will go at the same speed.
Dialogue: 0,0:47:45.50,0:47:48.94,Default,,0000,0000,0000,,See, the problem is\Nthe slope a constant.
Dialogue: 0,0:47:48.94,0:47:51.27,Default,,0000,0000,0000,,And thinking of\Nwhat they did that
Dialogue: 0,0:47:51.27,0:47:53.07,Default,,0000,0000,0000,,stand for, because\Nit didn't stand
Dialogue: 0,0:47:53.07,0:47:54.88,Default,,0000,0000,0000,,for [INAUDIBLE] in honors.
Dialogue: 0,0:47:54.88,0:47:57.28,Default,,0000,0000,0000,,We can do it in honors as well.
Dialogue: 0,0:47:57.28,0:47:58.70,Default,,0000,0000,0000,,We'll do it in a second.
Dialogue: 0,0:47:58.70,0:48:04.95,Default,,0000,0000,0000,,Now, k obviously is what?
Dialogue: 0,0:48:04.95,0:48:08.46,Default,,0000,0000,0000,,Some of you have\Nvery good memory,
Dialogue: 0,0:48:08.46,0:48:13.25,Default,,0000,0000,0000,,and like the memory of a\Nmedical doctor, which is great.
Dialogue: 0,0:48:13.25,0:48:14.56,Default,,0000,0000,0000,,Some of you don't.
Dialogue: 0,0:48:14.56,0:48:18.77,Default,,0000,0000,0000,,But if you don't you just go\Nback and look at the notes.
Dialogue: 0,0:48:18.77,0:48:20.76,Default,,0000,0000,0000,,What I'm trying to\Ndo, but I don't know,
Dialogue: 0,0:48:20.76,0:48:22.96,Default,,0000,0000,0000,,it's also a matter\Nof money-- I don't
Dialogue: 0,0:48:22.96,0:48:26.04,Default,,0000,0000,0000,,want to use the math\Ndepartment copier-- I'd
Dialogue: 0,0:48:26.04,0:48:29.85,Default,,0000,0000,0000,,like to make a stack of notes.
Dialogue: 0,0:48:29.85,0:48:33.09,Default,,0000,0000,0000,,So that's why I'm collecting\Nthese notes, to bring them back
Dialogue: 0,0:48:33.09,0:48:33.97,Default,,0000,0000,0000,,to you.
Dialogue: 0,0:48:33.97,0:48:34.47,Default,,0000,0000,0000,,For free!
Dialogue: 0,0:48:34.47,0:48:36.47,Default,,0000,0000,0000,,I'm not going to\Nsell them to you.
Dialogue: 0,0:48:36.47,0:48:38.06,Default,,0000,0000,0000,,I'm [INAUDIBLE].
Dialogue: 0,0:48:38.06,0:48:41.52,Default,,0000,0000,0000,,So that you can have those\Nwith you whenever you want,
Dialogue: 0,0:48:41.52,0:48:45.48,Default,,0000,0000,0000,,or put them in a spiral,\Npunch holes in them,
Dialogue: 0,0:48:45.48,0:48:48.68,Default,,0000,0000,0000,,and have them for\Nreview at any time.
Dialogue: 0,0:48:48.68,0:48:51.45,Default,,0000,0000,0000,,Reminds me of what that\Nwas-- that was in the notes.
Dialogue: 0,0:48:51.45,0:48:54.69,Default,,0000,0000,0000,,a over a squared plus b squared.
Dialogue: 0,0:48:54.69,0:48:57.30,Default,,0000,0000,0000,,So who can tell me, a\Nand b really quickly,
Dialogue: 0,0:48:57.30,0:49:00.87,Default,,0000,0000,0000,,so we don't waste too\Nmuch time, Mr. a is--?
Dialogue: 0,0:49:00.87,0:49:05.73,Default,,0000,0000,0000,,
Dialogue: 0,0:49:05.73,0:49:07.02,Default,,0000,0000,0000,,STUDENT: So this is another way
Dialogue: 0,0:49:07.02,0:49:07.60,Default,,0000,0000,0000,,STUDENT: 2.
Dialogue: 0,0:49:07.60,0:49:08.35,Default,,0000,0000,0000,,PROFESSOR TODA: 2.
Dialogue: 0,0:49:08.35,0:49:13.04,Default,,0000,0000,0000,,STUDENT: So is this another\Nway of defining k in k of s?
Dialogue: 0,0:49:13.04,0:49:14.12,Default,,0000,0000,0000,,PROFESSOR TODA: Actually--
Dialogue: 0,0:49:14.12,0:49:16.90,Default,,0000,0000,0000,,STUDENT: That's the general\Ncurvature for [INAUDIBLE].
Dialogue: 0,0:49:16.90,0:49:21.04,Default,,0000,0000,0000,,PROFESSOR TODA: You know\Nwhat is the magic thing?
Dialogue: 0,0:49:21.04,0:49:23.10,Default,,0000,0000,0000,,Even if-- the curvature\Nis an invariant.
Dialogue: 0,0:49:23.10,0:49:26.53,Default,,0000,0000,0000,,It doesn't depend the\Nreparametrization.
Dialogue: 0,0:49:26.53,0:49:29.83,Default,,0000,0000,0000,,There is a way maybe I'm going\Nto teach you, although this
Dialogue: 0,0:49:29.83,0:49:32.16,Default,,0000,0000,0000,,is not in the book.
Dialogue: 0,0:49:32.16,0:49:35.87,Default,,0000,0000,0000,,What are the formulas\Ncorresponding
Dialogue: 0,0:49:35.87,0:49:41.58,Default,,0000,0000,0000,,to the [INAUDIBLE] t and v that\Ndepend on curvature and torsion
Dialogue: 0,0:49:41.58,0:49:44.00,Default,,0000,0000,0000,,and the speed along the curve.
Dialogue: 0,0:49:44.00,0:49:48.75,Default,,0000,0000,0000,,And if you analyze the notion\Nof curvature, [INAUDIBLE],
Dialogue: 0,0:49:48.75,0:49:52.23,Default,,0000,0000,0000,,no matter what your\Nparameter will be, t, s, tau,
Dialogue: 0,0:49:52.23,0:49:56.69,Default,,0000,0000,0000,,God knows what, k will\Nstill be the same number.
Dialogue: 0,0:49:56.69,0:49:59.30,Default,,0000,0000,0000,,So k is viewed as an\Ninvariant with respect
Dialogue: 0,0:49:59.30,0:50:01.42,Default,,0000,0000,0000,,to the parametrization.
Dialogue: 0,0:50:01.42,0:50:04.12,Default,,0000,0000,0000,,STUDENT: So then that a over\Na squared plus b squared,
Dialogue: 0,0:50:04.12,0:50:05.91,Default,,0000,0000,0000,,that's another way of finding k?
Dialogue: 0,0:50:05.91,0:50:07.12,Default,,0000,0000,0000,,PROFESSOR TODA: Say it again?
Dialogue: 0,0:50:07.12,0:50:09.28,Default,,0000,0000,0000,,STUDENT: So using a over\Na squared plus b squared
Dialogue: 0,0:50:09.28,0:50:10.83,Default,,0000,0000,0000,,is another way of finding k?
Dialogue: 0,0:50:10.83,0:50:11.62,Default,,0000,0000,0000,,PROFESSOR TODA: No.
Dialogue: 0,0:50:11.62,0:50:13.92,Default,,0000,0000,0000,,Somebody gave you k.
Dialogue: 0,0:50:13.92,0:50:17.44,Default,,0000,0000,0000,,And then you say, if it's\Na standard parametrization,
Dialogue: 0,0:50:17.44,0:50:25.29,Default,,0000,0000,0000,,and then I get 2/5,\Ncan I be sure a is 2?
Dialogue: 0,0:50:25.29,0:50:28.22,Default,,0000,0000,0000,,I'm sure a is 2 from nothing.
Dialogue: 0,0:50:28.22,0:50:32.86,Default,,0000,0000,0000,,This is what makes me aware\Nthat a is 2 the first place.
Dialogue: 0,0:50:32.86,0:50:36.64,Default,,0000,0000,0000,,Because its the radius\Nof the cylinder.
Dialogue: 0,0:50:36.64,0:50:39.29,Default,,0000,0000,0000,,This is x squared, x and y.
Dialogue: 0,0:50:39.29,0:50:41.86,Default,,0000,0000,0000,,You see, x squared plus\Ny squared is a squared.
Dialogue: 0,0:50:41.86,0:50:43.65,Default,,0000,0000,0000,,This is where I get a from.
Dialogue: 0,0:50:43.65,0:50:44.62,Default,,0000,0000,0000,,a is 2.
Dialogue: 0,0:50:44.62,0:50:47.14,Default,,0000,0000,0000,,I replace it in here\Nand I say, all righty,
Dialogue: 0,0:50:47.14,0:50:51.78,Default,,0000,0000,0000,,so I only have one\Nchoice. a is 2 and b is?
Dialogue: 0,0:50:51.78,0:50:52.61,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:50:52.61,0:50:57.26,Default,,0000,0000,0000,,
Dialogue: 0,0:50:57.26,0:51:00.47,Default,,0000,0000,0000,,PROFESSOR TODA: But can b\Nplus-- So what I'm saying,
Dialogue: 0,0:51:00.47,0:51:01.38,Default,,0000,0000,0000,,a is 2, right?
Dialogue: 0,0:51:01.38,0:51:04.39,Default,,0000,0000,0000,,We know that from this.
Dialogue: 0,0:51:04.39,0:51:08.44,Default,,0000,0000,0000,,If I block in here I have 4\Nand somebody says plus minus 1.
Dialogue: 0,0:51:08.44,0:51:09.52,Default,,0000,0000,0000,,No.
Dialogue: 0,0:51:09.52,0:51:11.00,Default,,0000,0000,0000,,b is always positive.
Dialogue: 0,0:51:11.00,0:51:13.38,Default,,0000,0000,0000,,So you remember the\Nlast time we discussed
Dialogue: 0,0:51:13.38,0:51:16.64,Default,,0000,0000,0000,,about the standard\Nparametrization.
Dialogue: 0,0:51:16.64,0:51:20.28,Default,,0000,0000,0000,,But somebody will say,\Nbut what if I put a minus?
Dialogue: 0,0:51:20.28,0:51:22.84,Default,,0000,0000,0000,,What if I'm going\Nto put a minus?
Dialogue: 0,0:51:22.84,0:51:24.15,Default,,0000,0000,0000,,That's an excellent question.
Dialogue: 0,0:51:24.15,0:51:27.07,Default,,0000,0000,0000,,What's going to happen\Nif you put minus t?
Dialogue: 0,0:51:27.07,0:51:28.01,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:51:28.01,0:51:29.01,Default,,0000,0000,0000,,PROFESSOR TODA: Exactly.
Dialogue: 0,0:51:29.01,0:51:31.26,Default,,0000,0000,0000,,In the opposite direction.
Dialogue: 0,0:51:31.26,0:51:35.55,Default,,0000,0000,0000,,Instead of going\Nup, you go down.
Dialogue: 0,0:51:35.55,0:51:37.43,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:51:37.43,0:51:41.10,Default,,0000,0000,0000,,Now, I'm gonna-- what else?
Dialogue: 0,0:51:41.10,0:51:43.27,Default,,0000,0000,0000,,Ah, I said, let's do this.
Dialogue: 0,0:51:43.27,0:51:47.99,Default,,0000,0000,0000,,Let's prove that the\Nangle is a constant,
Dialogue: 0,0:51:47.99,0:51:51.08,Default,,0000,0000,0000,,the angle that's\Nmade by the velocity
Dialogue: 0,0:51:51.08,0:51:56.22,Default,,0000,0000,0000,,vector of the train with the\Nhorizontal plane is a constant.
Dialogue: 0,0:51:56.22,0:51:57.84,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:51:57.84,0:51:58.34,Default,,0000,0000,0000,,Nah.
Dialogue: 0,0:51:58.34,0:51:58.84,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:51:58.84,0:52:03.93,Default,,0000,0000,0000,,STUDENT: Are we still going\Nto find R of t given only k?
Dialogue: 0,0:52:03.93,0:52:05.55,Default,,0000,0000,0000,,PROFESSOR TODA: But didn't we?
Dialogue: 0,0:52:05.55,0:52:07.30,Default,,0000,0000,0000,,We did.
Dialogue: 0,0:52:07.30,0:52:13.75,Default,,0000,0000,0000,,R of t was 2 cosine\Nt, 2 sine t, and t.
Dialogue: 0,0:52:13.75,0:52:16.28,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:52:16.28,0:52:17.47,Default,,0000,0000,0000,,OK, so we are done.
Dialogue: 0,0:52:17.47,0:52:18.64,Default,,0000,0000,0000,,What did I say?
Dialogue: 0,0:52:18.64,0:52:22.41,Default,,0000,0000,0000,,I said that let's\Nprove-- it's a proof.
Dialogue: 0,0:52:22.41,0:52:27.30,Default,,0000,0000,0000,,Let's prove that the angle made\Nby the velocity to the train--
Dialogue: 0,0:52:27.30,0:52:30.64,Default,,0000,0000,0000,,to the train?-- to the direction\Nof motion, which is the helix.
Dialogue: 0,0:52:30.64,0:52:37.44,Default,,0000,0000,0000,,And the horizontal\Nplane is a constant.
Dialogue: 0,0:52:37.44,0:52:38.43,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:52:38.43,0:52:39.91,Default,,0000,0000,0000,,How are we going to do that?
Dialogue: 0,0:52:39.91,0:52:42.87,Default,,0000,0000,0000,,Now I start waking up,\Nbecause I was very tired.
Dialogue: 0,0:52:42.87,0:52:44.26,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:52:44.26,0:52:45.34,Default,,0000,0000,0000,,PROFESSOR TODA: Excuse me.
Dialogue: 0,0:52:45.34,0:52:46.84,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:52:46.84,0:53:01.24,Default,,0000,0000,0000,,PROFESSOR TODA: So you see,\Nthe helix contains this point.
Dialogue: 0,0:53:01.24,0:53:03.92,Default,,0000,0000,0000,,And I'm looking at\Nthe velocity vector
Dialogue: 0,0:53:03.92,0:53:06.31,Default,,0000,0000,0000,,that is standard to the helix.
Dialogue: 0,0:53:06.31,0:53:09.32,Default,,0000,0000,0000,,And I'll call that R prime.
Dialogue: 0,0:53:09.32,0:53:10.98,Default,,0000,0000,0000,,And then you say,\Nyea, but how am I
Dialogue: 0,0:53:10.98,0:53:13.99,Default,,0000,0000,0000,,going to compute that angle?
Dialogue: 0,0:53:13.99,0:53:15.63,Default,,0000,0000,0000,,What is that angle?
Dialogue: 0,0:53:15.63,0:53:17.99,Default,,0000,0000,0000,,STUDENT: It's a function of b.
Dialogue: 0,0:53:17.99,0:53:20.82,Default,,0000,0000,0000,,
Dialogue: 0,0:53:20.82,0:53:21.98,Default,,0000,0000,0000,,PROFESSOR TODA: It will be.
Dialogue: 0,0:53:21.98,0:53:24.82,Default,,0000,0000,0000,,But we have to do it rigorously.
Dialogue: 0,0:53:24.82,0:53:27.92,Default,,0000,0000,0000,,So what's going to happen\Nfor me to draw that angle?
Dialogue: 0,0:53:27.92,0:53:30.09,Default,,0000,0000,0000,,First of all, I should\Ntake-- from the tip
Dialogue: 0,0:53:30.09,0:53:33.24,Default,,0000,0000,0000,,of the vector I should\Ndraw perpendicular
Dialogue: 0,0:53:33.24,0:53:36.24,Default,,0000,0000,0000,,to the horizontal plane\Npassing through the point.
Dialogue: 0,0:53:36.24,0:53:37.11,Default,,0000,0000,0000,,And I'll get P prime.
Dialogue: 0,0:53:37.11,0:53:37.69,Default,,0000,0000,0000,,God knows why.
Dialogue: 0,0:53:37.69,0:53:41.49,Default,,0000,0000,0000,,I don't know why, I don't know\Nwhy. [? Q. ?] And this is PR,
Dialogue: 0,0:53:41.49,0:53:42.91,Default,,0000,0000,0000,,and P-- not PR.
Dialogue: 0,0:53:42.91,0:53:46.99,Default,,0000,0000,0000,,PR is too much\N[INAUDIBLE] radius, M.
Dialogue: 0,0:53:46.99,0:53:50.84,Default,,0000,0000,0000,,OK, so then you would\Ntake PQ and then
Dialogue: 0,0:53:50.84,0:53:52.60,Default,,0000,0000,0000,,you would measure this angle.
Dialogue: 0,0:53:52.60,0:53:54.77,Default,,0000,0000,0000,,Well, you have to be a\Nlittle bit smarter than that,
Dialogue: 0,0:53:54.77,0:53:58.39,Default,,0000,0000,0000,,because you can\Nprove something else.
Dialogue: 0,0:53:58.39,0:54:02.93,Default,,0000,0000,0000,,This is the complement of\Nanother angle that you love.
Dialogue: 0,0:54:02.93,0:54:07.10,Default,,0000,0000,0000,,And using chapter 9 you can\Ndo that angle in no time.
Dialogue: 0,0:54:07.10,0:54:15.84,Default,,0000,0000,0000,,
Dialogue: 0,0:54:15.84,0:54:20.80,Default,,0000,0000,0000,,So this is the\Ncomplement of the angle
Dialogue: 0,0:54:20.80,0:54:23.50,Default,,0000,0000,0000,,formed by the velocity vector\Nof prime with the normal.
Dialogue: 0,0:54:23.50,0:54:26.68,Default,,0000,0000,0000,,
Dialogue: 0,0:54:26.68,0:54:29.72,Default,,0000,0000,0000,,But not the normal principle\Nnormal to the curve,
Dialogue: 0,0:54:29.72,0:54:32.34,Default,,0000,0000,0000,,but the normal to the plane.
Dialogue: 0,0:54:32.34,0:54:34.51,Default,,0000,0000,0000,,And what is the\Nnormal to the plane?
Dialogue: 0,0:54:34.51,0:54:38.96,Default,,0000,0000,0000,,Let's call the principal normal\Nn to the curve big N bar.
Dialogue: 0,0:54:38.96,0:54:42.11,Default,,0000,0000,0000,,So in order to avoid confusion,\NI'll write this little n.
Dialogue: 0,0:54:42.11,0:54:42.94,Default,,0000,0000,0000,,How about that?
Dialogue: 0,0:54:42.94,0:54:45.24,Default,,0000,0000,0000,,Do you guys know-- like\Nthey do in mechanics.
Dialogue: 0,0:54:45.24,0:54:48.36,Default,,0000,0000,0000,,If you have two normals,\Nthey call that 1n.
Dialogue: 0,0:54:48.36,0:54:51.20,Default,,0000,0000,0000,,1 is little n, and\Nstuff like that.
Dialogue: 0,0:54:51.20,0:54:52.98,Default,,0000,0000,0000,,So this is the complement.
Dialogue: 0,0:54:52.98,0:54:55.43,Default,,0000,0000,0000,,If I were able to prove\Nthat that complement
Dialogue: 0,0:54:55.43,0:54:59.60,Default,,0000,0000,0000,,is a constant-- this is the\NStanford [? property-- ?] then
Dialogue: 0,0:54:59.60,0:55:00.99,Default,,0000,0000,0000,,I will be happy.
Dialogue: 0,0:55:00.99,0:55:03.10,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:55:03.10,0:55:04.29,Default,,0000,0000,0000,,No, for god's sake.
Dialogue: 0,0:55:04.29,0:55:07.03,Default,,0000,0000,0000,,Who is little n?
Dialogue: 0,0:55:07.03,0:55:11.35,Default,,0000,0000,0000,,Little n would be-- is\Nthat the normal to a plane
Dialogue: 0,0:55:11.35,0:55:12.33,Default,,0000,0000,0000,,that you love?
Dialogue: 0,0:55:12.33,0:55:13.34,Default,,0000,0000,0000,,What is your plane?
Dialogue: 0,0:55:13.34,0:55:14.09,Default,,0000,0000,0000,,STUDENT: xy plane.
Dialogue: 0,0:55:14.09,0:55:16.52,Default,,0000,0000,0000,,PROFESSOR TODA: Your\Nplane is horizontal plane.
Dialogue: 0,0:55:16.52,0:55:17.32,Default,,0000,0000,0000,,STUDENT: xy.
Dialogue: 0,0:55:17.32,0:55:18.57,Default,,0000,0000,0000,,PROFESSOR TODA: Yes, xy plane.
Dialogue: 0,0:55:18.57,0:55:22.12,Default,,0000,0000,0000,,Or xy plane shifted,\Nshifted, shifted, shifted.
Dialogue: 0,0:55:22.12,0:55:23.18,Default,,0000,0000,0000,,That's the normal change?
Dialogue: 0,0:55:23.18,0:55:23.68,Default,,0000,0000,0000,,No.
Dialogue: 0,0:55:23.68,0:55:24.89,Default,,0000,0000,0000,,Who is the normal?
Dialogue: 0,0:55:24.89,0:55:26.17,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:55:26.17,0:55:27.34,Default,,0000,0000,0000,,PROFESSOR TODA: [INAUDIBLE].
Dialogue: 0,0:55:27.34,0:55:28.31,Default,,0000,0000,0000,,STUDENT: 0, 0, 1.
Dialogue: 0,0:55:28.31,0:55:29.31,Default,,0000,0000,0000,,PROFESSOR TODA: 0, 0, 1.
Dialogue: 0,0:55:29.31,0:55:29.81,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:55:29.81,0:55:32.11,Default,,0000,0000,0000,,When I put 0 I was [INAUDIBLE].
Dialogue: 0,0:55:32.11,0:55:33.72,Default,,0000,0000,0000,,So this is k.
Dialogue: 0,0:55:33.72,0:55:36.42,Default,,0000,0000,0000,,
Dialogue: 0,0:55:36.42,0:55:37.73,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:55:37.73,0:55:39.62,Default,,0000,0000,0000,,And what is our prime?
Dialogue: 0,0:55:39.62,0:55:42.30,Default,,0000,0000,0000,,I was-- that was\Na piece of cake.
Dialogue: 0,0:55:42.30,0:55:47.36,Default,,0000,0000,0000,,We did it last time minus a\Nsine t, a equals sine t and b.
Dialogue: 0,0:55:47.36,0:55:50.72,Default,,0000,0000,0000,,
Dialogue: 0,0:55:50.72,0:55:53.62,Default,,0000,0000,0000,,Let's find that angle.
Dialogue: 0,0:55:53.62,0:55:54.69,Default,,0000,0000,0000,,Well, I don't know.
Dialogue: 0,0:55:54.69,0:55:58.32,Default,,0000,0000,0000,,You have to teach me, because\Nyou have chapter 9 fresher
Dialogue: 0,0:55:58.32,0:56:01.59,Default,,0000,0000,0000,,in your memory than I have it.
Dialogue: 0,0:56:01.59,0:56:03.92,Default,,0000,0000,0000,,Are you taking attendance also?
Dialogue: 0,0:56:03.92,0:56:07.18,Default,,0000,0000,0000,,Are you writing your name down?
Dialogue: 0,0:56:07.18,0:56:08.26,Default,,0000,0000,0000,,Oh, no problem whatsoever.
Dialogue: 0,0:56:08.26,0:56:09.39,Default,,0000,0000,0000,,STUDENT: We didn't get it.
Dialogue: 0,0:56:09.39,0:56:10.81,Default,,0000,0000,0000,,PROFESSOR TODA:\NYou didn't get it.
Dialogue: 0,0:56:10.81,0:56:11.75,Default,,0000,0000,0000,,Circulate it.
Dialogue: 0,0:56:11.75,0:56:16.66,Default,,0000,0000,0000,,All right, so who is going\Nto help me with the angle?
Dialogue: 0,0:56:16.66,0:56:19.87,Default,,0000,0000,0000,,What is the angle between\Ntwo vectors, guys?
Dialogue: 0,0:56:19.87,0:56:24.07,Default,,0000,0000,0000,,That should be review from what\Nwe just covered in chapter 9.
Dialogue: 0,0:56:24.07,0:56:27.98,Default,,0000,0000,0000,,Let me call them\Nu and v. And who's
Dialogue: 0,0:56:27.98,0:56:29.92,Default,,0000,0000,0000,,going to tell me how\NI get that angle?
Dialogue: 0,0:56:29.92,0:56:31.96,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] is equal\Nto the inverse cosine of the dot
Dialogue: 0,0:56:31.96,0:56:33.29,Default,,0000,0000,0000,,product of [? the magnitude. ?]
Dialogue: 0,0:56:33.29,0:56:35.08,Default,,0000,0000,0000,,PROFESSOR TODA: Do you\Nlike me to write arc
Dialogue: 0,0:56:35.08,0:56:36.45,Default,,0000,0000,0000,,cosine or cosine [INAUDIBLE].
Dialogue: 0,0:56:36.45,0:56:37.76,Default,,0000,0000,0000,,Doesn't matter.
Dialogue: 0,0:56:37.76,0:56:39.85,Default,,0000,0000,0000,,Arc cosine of--
Dialogue: 0,0:56:39.85,0:56:40.96,Default,,0000,0000,0000,,STUDENT: The dot products.
Dialogue: 0,0:56:40.96,0:56:47.48,Default,,0000,0000,0000,,PROFESSOR TODA: The dot\Nproduct between u and v.
Dialogue: 0,0:56:47.48,0:56:48.94,Default,,0000,0000,0000,,STUDENT: Over magnitude.
Dialogue: 0,0:56:48.94,0:56:52.47,Default,,0000,0000,0000,,PROFESSOR TODA: Divided by the\Nproduct of their magnitudes.
Dialogue: 0,0:56:52.47,0:56:54.69,Default,,0000,0000,0000,,Look, I will change the\Norder, because you're not
Dialogue: 0,0:56:54.69,0:56:56.14,Default,,0000,0000,0000,,going to like it.
Dialogue: 0,0:56:56.14,0:56:56.91,Default,,0000,0000,0000,,Doesn't matter.
Dialogue: 0,0:56:56.91,0:56:57.68,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:56:57.68,0:57:03.45,Default,,0000,0000,0000,,So the angle phi between\Nmy favorite vectors
Dialogue: 0,0:57:03.45,0:57:08.46,Default,,0000,0000,0000,,here is going to be\Nsimply the dot product.
Dialogue: 0,0:57:08.46,0:57:09.57,Default,,0000,0000,0000,,That's a blessing.
Dialogue: 0,0:57:09.57,0:57:10.24,Default,,0000,0000,0000,,It's a constant.
Dialogue: 0,0:57:10.24,0:57:11.91,Default,,0000,0000,0000,,STUDENT: So you're\Ndoing the dot product
Dialogue: 0,0:57:11.91,0:57:13.42,Default,,0000,0000,0000,,between the normal [INAUDIBLE]?
Dialogue: 0,0:57:13.42,0:57:14.100,Default,,0000,0000,0000,,PROFESSOR TODA:\NBetween this and that.
Dialogue: 0,0:57:14.100,0:57:18.06,Default,,0000,0000,0000,,So this is u and this\Nis v. So the dot product
Dialogue: 0,0:57:18.06,0:57:22.34,Default,,0000,0000,0000,,would be 0 plus v.\NSo the dot product
Dialogue: 0,0:57:22.34,0:57:28.52,Default,,0000,0000,0000,,is arc cosine of v, which,\Nthank god, is a constant.
Dialogue: 0,0:57:28.52,0:57:30.31,Default,,0000,0000,0000,,I don't have to do\Nanything anymore.
Dialogue: 0,0:57:30.31,0:57:33.15,Default,,0000,0000,0000,,I'm done with the proof\Nbit, because arc cosine
Dialogue: 0,0:57:33.15,0:57:36.00,Default,,0000,0000,0000,,of a constant will\Nbe a constant.
Dialogue: 0,0:57:36.00,0:57:36.72,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:57:36.72,0:57:37.60,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:57:37.60,0:57:40.85,Default,,0000,0000,0000,,So I have v over what?
Dialogue: 0,0:57:40.85,0:57:45.09,Default,,0000,0000,0000,,What is the length\Nof this vector?
Dialogue: 0,0:57:45.09,0:57:46.75,Default,,0000,0000,0000,,1. [INAUDIBLE].
Dialogue: 0,0:57:46.75,0:57:50.51,Default,,0000,0000,0000,,What's the length\Nof that vector?
Dialogue: 0,0:57:50.51,0:57:55.90,Default,,0000,0000,0000,,Square root of a\Nsquared plus b squared.
Dialogue: 0,0:57:55.90,0:57:56.43,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:57:56.43,0:58:01.83,Default,,0000,0000,0000,,
Dialogue: 0,0:58:01.83,0:58:05.28,Default,,0000,0000,0000,,STUDENT: How did\Nyou [INAUDIBLE].
Dialogue: 0,0:58:05.28,0:58:07.65,Default,,0000,0000,0000,,PROFESSOR TODA: So now\Nlet me ask you one thing.
Dialogue: 0,0:58:07.65,0:58:11.36,Default,,0000,0000,0000,,
Dialogue: 0,0:58:11.36,0:58:13.91,Default,,0000,0000,0000,,What kind of function\Nis arc cosine?
Dialogue: 0,0:58:13.91,0:58:16.43,Default,,0000,0000,0000,,Of course I said arc cosine\Nof a constant is a constant.
Dialogue: 0,0:58:16.43,0:58:18.39,Default,,0000,0000,0000,,What kind of a\Nfunction is arc cosine?
Dialogue: 0,0:58:18.39,0:58:21.74,Default,,0000,0000,0000,,I'm doing review with you\Nbecause I think it's useful.
Dialogue: 0,0:58:21.74,0:58:26.07,Default,,0000,0000,0000,,Arc cosine is defined on\Nwhat with values in what?
Dialogue: 0,0:58:26.07,0:58:30.29,Default,,0000,0000,0000,,
Dialogue: 0,0:58:30.29,0:58:32.64,Default,,0000,0000,0000,,STUDENT: Repeat the question?
Dialogue: 0,0:58:32.64,0:58:33.89,Default,,0000,0000,0000,,PROFESSOR TODA: Arc cosine.
Dialogue: 0,0:58:33.89,0:58:36.10,Default,,0000,0000,0000,,Or cosine inverse,\Nlike Ryan prefers.
Dialogue: 0,0:58:36.10,0:58:38.13,Default,,0000,0000,0000,,Cosine inverse is\Nthe same thing.
Dialogue: 0,0:58:38.13,0:58:40.44,Default,,0000,0000,0000,,It's a function defined\Nby where to where?
Dialogue: 0,0:58:40.44,0:58:43.19,Default,,0000,0000,0000,,Cosine is defined\Nfrom where to where?
Dialogue: 0,0:58:43.19,0:58:46.14,Default,,0000,0000,0000,,From R to minus 1.
Dialogue: 0,0:58:46.14,0:58:47.74,Default,,0000,0000,0000,,It's a cosine of t.
Dialogue: 0,0:58:47.74,0:58:49.69,Default,,0000,0000,0000,,t could be any real number.
Dialogue: 0,0:58:49.69,0:58:51.80,Default,,0000,0000,0000,,The range is minus 1, 1.
Dialogue: 0,0:58:51.80,0:58:53.24,Default,,0000,0000,0000,,Close the interval.
Dialogue: 0,0:58:53.24,0:58:54.95,Default,,0000,0000,0000,,STUDENT: So it's-- so\NI just wonder why--
Dialogue: 0,0:58:54.95,0:58:57.01,Default,,0000,0000,0000,,PROFESSOR TODA: Minus\N1 to 1, close interval.
Dialogue: 0,0:58:57.01,0:58:58.32,Default,,0000,0000,0000,,But pay attention, please.
Dialogue: 0,0:58:58.32,0:59:03.04,Default,,0000,0000,0000,,Because it cannot go back to R.\NIt has to be a 1 to 1 function.
Dialogue: 0,0:59:03.04,0:59:05.96,Default,,0000,0000,0000,,You cannot have an inverse\Nfunction if you don't take
Dialogue: 0,0:59:05.96,0:59:09.24,Default,,0000,0000,0000,,a restriction of a\Nfunction to be 1 to 1.
Dialogue: 0,0:59:09.24,0:59:11.60,Default,,0000,0000,0000,,And we took that\Nrestriction of a function.
Dialogue: 0,0:59:11.60,0:59:14.89,Default,,0000,0000,0000,,And do you remember what it was?
Dialogue: 0,0:59:14.89,0:59:15.84,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:59:15.84,0:59:17.64,Default,,0000,0000,0000,,PROFESSOR TODA: 0 to pi.
Dialogue: 0,0:59:17.64,0:59:19.71,Default,,0000,0000,0000,,Now, on this one\NI'm really happy.
Dialogue: 0,0:59:19.71,0:59:23.16,Default,,0000,0000,0000,,Because I asked\Nseveral people-- people
Dialogue: 0,0:59:23.16,0:59:27.13,Default,,0000,0000,0000,,come to my office to get\Nall sorts of transcripts,
Dialogue: 0,0:59:27.13,0:59:27.63,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,0:59:27.63,0:59:30.60,Default,,0000,0000,0000,,And in trigonometry\NI asked one student,
Dialogue: 0,0:59:30.60,0:59:31.91,Default,,0000,0000,0000,,so you took trigonometry.
Dialogue: 0,0:59:31.91,0:59:32.91,Default,,0000,0000,0000,,So do you remember that?
Dialogue: 0,0:59:32.91,0:59:34.36,Default,,0000,0000,0000,,He didn't remember that.
Dialogue: 0,0:59:34.36,0:59:35.30,Default,,0000,0000,0000,,So I'm glad you do.
Dialogue: 0,0:59:35.30,0:59:40.13,Default,,0000,0000,0000,,How about when I had\Nthe sine inverse?
Dialogue: 0,0:59:40.13,0:59:44.76,Default,,0000,0000,0000,,How was my restriction so that\Nwould be a 1 to 1 function?
Dialogue: 0,0:59:44.76,0:59:46.90,Default,,0000,0000,0000,,It's got to go\Nfrom minus 1 to 1.
Dialogue: 0,0:59:46.90,0:59:48.18,Default,,0000,0000,0000,,What is the range?
Dialogue: 0,0:59:48.18,0:59:49.02,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:59:49.02,0:59:51.30,Default,,0000,0000,0000,,PROFESSOR TODA: Minus pi over 2.
Dialogue: 0,0:59:51.30,0:59:53.25,Default,,0000,0000,0000,,You guys know your trig.
Dialogue: 0,0:59:53.25,0:59:53.75,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:59:53.75,0:59:55.63,Default,,0000,0000,0000,,That's a very good thing.
Dialogue: 0,0:59:55.63,0:59:59.44,Default,,0000,0000,0000,,You were in high school\Nwhen you learned that?
Dialogue: 0,0:59:59.44,1:00:00.43,Default,,0000,0000,0000,,Here at Lubbock High?
Dialogue: 0,1:00:00.43,1:00:01.15,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,1:00:01.15,1:00:02.07,Default,,0000,0000,0000,,PROFESSOR TODA: Great.
Dialogue: 0,1:00:02.07,1:00:03.56,Default,,0000,0000,0000,,Good job, Lubbock High.
Dialogue: 0,1:00:03.56,1:00:06.23,Default,,0000,0000,0000,,But many students, I caught\Nthem, who wanted credit
Dialogue: 0,1:00:06.23,1:00:08.35,Default,,0000,0000,0000,,for trig who didn't know that.
Dialogue: 0,1:00:08.35,1:00:09.68,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:00:09.68,1:00:19.87,Default,,0000,0000,0000,,So since arc cosine is a\Nfunction that is of 0, pi,
Dialogue: 0,1:00:19.87,1:00:25.23,Default,,0000,0000,0000,,for example, what if my--\Nlet me give you an example.
Dialogue: 0,1:00:25.23,1:00:26.91,Default,,0000,0000,0000,,What was last time, guys?
Dialogue: 0,1:00:26.91,1:00:30.80,Default,,0000,0000,0000,,a was 1. b was 1.
Dialogue: 0,1:00:30.80,1:00:32.01,Default,,0000,0000,0000,,For one example.
Dialogue: 0,1:00:32.01,1:00:33.81,Default,,0000,0000,0000,,In that case, 1 with 5b.
Dialogue: 0,1:00:33.81,1:00:36.33,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] ask you\Nfor the example you just did?
Dialogue: 0,1:00:36.33,1:00:37.54,Default,,0000,0000,0000,,PROFESSOR TODA: No last time.
Dialogue: 0,1:00:37.54,1:00:39.56,Default,,0000,0000,0000,,STUDENT: A was 3 and b was--
Dialogue: 0,1:00:39.56,1:00:44.26,Default,,0000,0000,0000,,PROFESSOR TODA: So what would\Nthat be, in this case 5?
Dialogue: 0,1:00:44.26,1:00:46.68,Default,,0000,0000,0000,,STUDENT: That would be\Nb over the square root--
Dialogue: 0,1:00:46.68,1:00:47.56,Default,,0000,0000,0000,,STUDENT: 3 over pi.
Dialogue: 0,1:00:47.56,1:00:49.98,Default,,0000,0000,0000,,
Dialogue: 0,1:00:49.98,1:00:52.36,Default,,0000,0000,0000,,PROFESSOR TODA: a is 1 and b\Nis 1, like we did last time.
Dialogue: 0,1:00:52.36,1:00:55.05,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\N2, which is--
Dialogue: 0,1:00:55.05,1:00:57.06,Default,,0000,0000,0000,,PROFESSOR TODA: Plug\Nin 1 is a, b is 1.
Dialogue: 0,1:00:57.06,1:00:57.60,Default,,0000,0000,0000,,What is this?
Dialogue: 0,1:00:57.60,1:00:59.42,Default,,0000,0000,0000,,STUDENT: It's just pi over 4.
Dialogue: 0,1:00:59.42,1:01:00.50,Default,,0000,0000,0000,,PROFESSOR TODA: Pi over 4.
Dialogue: 0,1:01:00.50,1:01:06.80,Default,,0000,0000,0000,,So pi will be our cosine, of\N1 over square root 2, which
Dialogue: 0,1:01:06.80,1:01:12.09,Default,,0000,0000,0000,,is 45 degree angle, which is--\Nyou said pi over 4, right?
Dialogue: 0,1:01:12.09,1:01:14.54,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,1:01:14.54,1:01:19.80,Default,,0000,0000,0000,,So exactly, you would\Nhave that over here.
Dialogue: 0,1:01:19.80,1:01:22.58,Default,,0000,0000,0000,,This is where the\Ncosine [INAUDIBLE].
Dialogue: 0,1:01:22.58,1:01:28.22,Default,,0000,0000,0000,,Now you see, guys, the way we\Nhave, the way I assume a and b,
Dialogue: 0,1:01:28.22,1:01:30.98,Default,,0000,0000,0000,,the way anybody-- the\Nbook also introduces
Dialogue: 0,1:01:30.98,1:01:33.35,Default,,0000,0000,0000,,a and b to be positive numbers.
Dialogue: 0,1:01:33.35,1:01:37.23,Default,,0000,0000,0000,,Can you tell me what kind\Nof angle phi will be,
Dialogue: 0,1:01:37.23,1:01:39.90,Default,,0000,0000,0000,,not only restricted to 0 pi?
Dialogue: 0,1:01:39.90,1:01:41.36,Default,,0000,0000,0000,,Well, a is positive.
Dialogue: 0,1:01:41.36,1:01:42.48,Default,,0000,0000,0000,,b is positive.
Dialogue: 0,1:01:42.48,1:01:44.36,Default,,0000,0000,0000,,a doesn't matter.
Dialogue: 0,1:01:44.36,1:01:46.67,Default,,0000,0000,0000,,The whole thing\Nwill be positive.
Dialogue: 0,1:01:46.67,1:01:50.51,Default,,0000,0000,0000,,Arc cosine of a\Npositive number--
Dialogue: 0,1:01:50.51,1:01:52.01,Default,,0000,0000,0000,,STUDENT: Between\N0 and pi over 2.
Dialogue: 0,1:01:52.01,1:01:53.01,Default,,0000,0000,0000,,PROFESSOR TODA: That is.
Dialogue: 0,1:01:53.01,1:01:56.33,Default,,0000,0000,0000,,Yeah, so it has to be\Nbetween 0 and pi over 2.
Dialogue: 0,1:01:56.33,1:01:57.95,Default,,0000,0000,0000,,So it's going to be\Nonly this quadrant.
Dialogue: 0,1:01:57.95,1:01:59.64,Default,,0000,0000,0000,,Does that make sense?
Dialogue: 0,1:01:59.64,1:02:03.39,Default,,0000,0000,0000,,Yes, think with the\Nimagination of your eyes,
Dialogue: 0,1:02:03.39,1:02:05.22,Default,,0000,0000,0000,,or the eyes of your imagination.
Dialogue: 0,1:02:05.22,1:02:06.43,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:02:06.43,1:02:08.36,Default,,0000,0000,0000,,You have a cylinder.
Dialogue: 0,1:02:08.36,1:02:10.27,Default,,0000,0000,0000,,And you are moving\Nalong that cylinder.
Dialogue: 0,1:02:10.27,1:02:12.16,Default,,0000,0000,0000,,And this is how you turn.
Dialogue: 0,1:02:12.16,1:02:14.40,Default,,0000,0000,0000,,You turn with that little train.
Dialogue: 0,1:02:14.40,1:02:16.58,Default,,0000,0000,0000,,Du-du-du-du-du, you go up.
Dialogue: 0,1:02:16.58,1:02:19.91,Default,,0000,0000,0000,,When you turn the\Nvelocity vector and you
Dialogue: 0,1:02:19.91,1:02:23.12,Default,,0000,0000,0000,,look at the-- mm.
Dialogue: 0,1:02:23.12,1:02:23.96,Default,,0000,0000,0000,,STUDENT: The normal.
Dialogue: 0,1:02:23.96,1:02:24.86,Default,,0000,0000,0000,,PROFESSOR TODA: The normal!
Dialogue: 0,1:02:24.86,1:02:25.36,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,1:02:25.36,1:02:30.24,Default,,0000,0000,0000,,The z axis, you always have an\Nangle between 0 and pi over 2.
Dialogue: 0,1:02:30.24,1:02:31.71,Default,,0000,0000,0000,,So it makes sense.
Dialogue: 0,1:02:31.71,1:02:34.16,Default,,0000,0000,0000,,I'm going to go ahead and\Nerase the whole thing.
Dialogue: 0,1:02:34.16,1:02:41.02,Default,,0000,0000,0000,,
Dialogue: 0,1:02:41.02,1:02:47.72,Default,,0000,0000,0000,,So we reviewed, more or less, s\Nof t, integration, derivation,
Dialogue: 0,1:02:47.72,1:02:52.02,Default,,0000,0000,0000,,moving from position vector\Nto velocity to acceleration
Dialogue: 0,1:02:52.02,1:02:56.26,Default,,0000,0000,0000,,and back, acceleration to\Nvelocity to position vector,
Dialogue: 0,1:02:56.26,1:02:58.50,Default,,0000,0000,0000,,the meaning of arclength.
Dialogue: 0,1:02:58.50,1:03:00.89,Default,,0000,0000,0000,,There are some things I\Nwould like to tell you,
Dialogue: 0,1:03:00.89,1:03:07.83,Default,,0000,0000,0000,,because Ryan asked me a few more\Nquestions about the curvature.
Dialogue: 0,1:03:07.83,1:03:11.56,Default,,0000,0000,0000,,The curvature\Nformula depends very
Dialogue: 0,1:03:11.56,1:03:16.83,Default,,0000,0000,0000,,much on the type of formula\Nyou used for the curve.
Dialogue: 0,1:03:16.83,1:03:18.80,Default,,0000,0000,0000,,So you say, wait,\Nwait, wait, Magdelena,
Dialogue: 0,1:03:18.80,1:03:21.29,Default,,0000,0000,0000,,you told us-- you\Nare confusing us.
Dialogue: 0,1:03:21.29,1:03:23.71,Default,,0000,0000,0000,,You told us that the\Ncurvature is uniquely
Dialogue: 0,1:03:23.71,1:03:33.74,Default,,0000,0000,0000,,defined as the magnitude\Nof the acceleration vector
Dialogue: 0,1:03:33.74,1:03:36.80,Default,,0000,0000,0000,,when the law of motion\Nis an arclength.
Dialogue: 0,1:03:36.80,1:03:38.85,Default,,0000,0000,0000,,And that is correct.
Dialogue: 0,1:03:38.85,1:03:43.19,Default,,0000,0000,0000,,So suppose my original law of\Nmotion was R of t [INAUDIBLE]
Dialogue: 0,1:03:43.19,1:03:47.75,Default,,0000,0000,0000,,time, any time, t,\Nany time parameter.
Dialogue: 0,1:03:47.75,1:03:49.37,Default,,0000,0000,0000,,I'm making a face.
Dialogue: 0,1:03:49.37,1:03:53.29,Default,,0000,0000,0000,,But then from that we switch\Nto something beautiful,
Dialogue: 0,1:03:53.29,1:03:56.44,Default,,0000,0000,0000,,which is called the\Narclength parametrization.
Dialogue: 0,1:03:56.44,1:03:58.28,Default,,0000,0000,0000,,Why am I so happy?
Dialogue: 0,1:03:58.28,1:04:04.97,Default,,0000,0000,0000,,Because in this parametrization\Nthe magnitude of the speed
Dialogue: 0,1:04:04.97,1:04:07.12,Default,,0000,0000,0000,,is 1.
Dialogue: 0,1:04:07.12,1:04:17.70,Default,,0000,0000,0000,,And I define k to\Nbe the magnitude
Dialogue: 0,1:04:17.70,1:04:19.87,Default,,0000,0000,0000,,of R double prime of s, right?
Dialogue: 0,1:04:19.87,1:04:22.43,Default,,0000,0000,0000,,The acceleration only in\Nthe arclength [? time ?]
Dialogue: 0,1:04:22.43,1:04:23.43,Default,,0000,0000,0000,,parameterization.
Dialogue: 0,1:04:23.43,1:04:24.92,Default,,0000,0000,0000,,And then this was\Nthe definition.
Dialogue: 0,1:04:24.92,1:04:30.41,Default,,0000,0000,0000,,
Dialogue: 0,1:04:30.41,1:04:36.55,Default,,0000,0000,0000,,A. Can you prove-- what?
Dialogue: 0,1:04:36.55,1:04:40.19,Default,,0000,0000,0000,,Can you prove the\Nfollowing formula?
Dialogue: 0,1:04:40.19,1:04:52.20,Default,,0000,0000,0000,,
Dialogue: 0,1:04:52.20,1:04:58.51,Default,,0000,0000,0000,,T prime of s equals\Nk times N of s.
Dialogue: 0,1:04:58.51,1:05:02.55,Default,,0000,0000,0000,,This is famous for people\Nwho do-- not for everybody.
Dialogue: 0,1:05:02.55,1:05:05.53,Default,,0000,0000,0000,,But imagine you have\Nan engineer who does
Dialogue: 0,1:05:05.53,1:05:08.43,Default,,0000,0000,0000,,research of the laws of motion.
Dialogue: 0,1:05:08.43,1:05:13.13,Default,,0000,0000,0000,,Maybe he works for\Nthe railways and he's
Dialogue: 0,1:05:13.13,1:05:17.17,Default,,0000,0000,0000,,looking at skew\Ncurves, or he is one
Dialogue: 0,1:05:17.17,1:05:20.48,Default,,0000,0000,0000,,of those people who\Nproject the ski slopes,
Dialogue: 0,1:05:20.48,1:05:25.36,Default,,0000,0000,0000,,or all sorts of winter sports\Nslope or something, that
Dialogue: 0,1:05:25.36,1:05:29.15,Default,,0000,0000,0000,,involve a lot of\Ncurvatures and torsions.
Dialogue: 0,1:05:29.15,1:05:31.24,Default,,0000,0000,0000,,That guy has to know\Nthe Frenet formula.
Dialogue: 0,1:05:31.24,1:05:34.26,Default,,0000,0000,0000,,So this is the famous\Nfirst Frenet formula.
Dialogue: 0,1:05:34.26,1:05:40.14,Default,,0000,0000,0000,,
Dialogue: 0,1:05:40.14,1:05:46.69,Default,,0000,0000,0000,,Frenet was a mathematician\Nwho gave the name to the TNB
Dialogue: 0,1:05:46.69,1:05:48.48,Default,,0000,0000,0000,,vectors, the trihedron.
Dialogue: 0,1:05:48.48,1:05:49.97,Default,,0000,0000,0000,,You have the T was what?
Dialogue: 0,1:05:49.97,1:05:52.96,Default,,0000,0000,0000,,The T was the tangent\N[INAUDIBLE] vector.
Dialogue: 0,1:05:52.96,1:05:58.18,Default,,0000,0000,0000,,The N was the\Nprincipal unit normal.
Dialogue: 0,1:05:58.18,1:06:00.93,Default,,0000,0000,0000,,In those videos that I'm\Nwatching that I also sent you--
Dialogue: 0,1:06:00.93,1:06:02.40,Default,,0000,0000,0000,,I like most of them.
Dialogue: 0,1:06:02.40,1:06:05.66,Default,,0000,0000,0000,,I like the Khan Academy\Nmore than everything.
Dialogue: 0,1:06:05.66,1:06:09.10,Default,,0000,0000,0000,,Also I like the one that\Nwas made by Dr. [? Gock ?]
Dialogue: 0,1:06:09.10,1:06:12.54,Default,,0000,0000,0000,,But Dr. [? Gock ?] made a\Nlittle bit of a mistake.
Dialogue: 0,1:06:12.54,1:06:13.92,Default,,0000,0000,0000,,A conceptual mistake.
Dialogue: 0,1:06:13.92,1:06:17.01,Default,,0000,0000,0000,,We all make mistakes by\Nmisprinting or misreading
Dialogue: 0,1:06:17.01,1:06:18.37,Default,,0000,0000,0000,,or goofy mistake.
Dialogue: 0,1:06:18.37,1:06:20.69,Default,,0000,0000,0000,,But he said this is\Nthe normal vector.
Dialogue: 0,1:06:20.69,1:06:22.93,Default,,0000,0000,0000,,This is not-- it's the\Nprinciple normal vectors.
Dialogue: 0,1:06:22.93,1:06:24.73,Default,,0000,0000,0000,,There are many normals.
Dialogue: 0,1:06:24.73,1:06:26.63,Default,,0000,0000,0000,,There is only one\Ntangent direction,
Dialogue: 0,1:06:26.63,1:06:29.01,Default,,0000,0000,0000,,but in terms of normals\Nthere are many that
Dialogue: 0,1:06:29.01,1:06:30.91,Default,,0000,0000,0000,,are-- all of these are normals.
Dialogue: 0,1:06:30.91,1:06:34.94,Default,,0000,0000,0000,,All the perpendicular in\Nthe plane-- [INAUDIBLE]
Dialogue: 0,1:06:34.94,1:06:39.78,Default,,0000,0000,0000,,so this is my law of motion,\NT. All this plane is normal.
Dialogue: 0,1:06:39.78,1:06:41.96,Default,,0000,0000,0000,,So any of these\Nvectors is a normal.
Dialogue: 0,1:06:41.96,1:06:44.99,Default,,0000,0000,0000,,The one we choose and\Ndefined as T prime
Dialogue: 0,1:06:44.99,1:06:47.22,Default,,0000,0000,0000,,over T prime [INAUDIBLE]\Nabsolute values
Dialogue: 0,1:06:47.22,1:06:48.99,Default,,0000,0000,0000,,called the principal normal.
Dialogue: 0,1:06:48.99,1:06:51.35,Default,,0000,0000,0000,,It's like the principal\Nof a high school.
Dialogue: 0,1:06:51.35,1:06:53.23,Default,,0000,0000,0000,,He is important.
Dialogue: 0,1:06:53.23,1:06:58.35,Default,,0000,0000,0000,,So T and B-- B goes\Ndown, or goes-- down.
Dialogue: 0,1:06:58.35,1:07:04.54,Default,,0000,0000,0000,,Well, yeah, because B is T cross\NN. So when you find the Frenet
Dialogue: 0,1:07:04.54,1:07:10.44,Default,,0000,0000,0000,,Trihedron, TNB, it's like that.
Dialogue: 0,1:07:10.44,1:07:15.68,Default,,0000,0000,0000,,T, N, and B. What's special,\Nwhy do we call it the frame,
Dialogue: 0,1:07:15.68,1:07:18.46,Default,,0000,0000,0000,,is that every\N[? payer ?] of vectors
Dialogue: 0,1:07:18.46,1:07:20.09,Default,,0000,0000,0000,,are mutually orthogonal.
Dialogue: 0,1:07:20.09,1:07:22.27,Default,,0000,0000,0000,,And they are all unit vectors.
Dialogue: 0,1:07:22.27,1:07:25.91,Default,,0000,0000,0000,,This is the famous Frenet frame.
Dialogue: 0,1:07:25.91,1:07:27.58,Default,,0000,0000,0000,,Now, Mr. Frenet was a smart guy.
Dialogue: 0,1:07:27.58,1:07:32.33,Default,,0000,0000,0000,,He found-- I don't know whether\Nhe was adopting mathematics
Dialogue: 0,1:07:32.33,1:07:33.05,Default,,0000,0000,0000,,or not.
Dialogue: 0,1:07:33.05,1:07:34.29,Default,,0000,0000,0000,,Doesn't matter.
Dialogue: 0,1:07:34.29,1:07:37.97,Default,,0000,0000,0000,,He found a bunch of formulas,\Nof which this is the first one.
Dialogue: 0,1:07:37.97,1:07:42.26,Default,,0000,0000,0000,,And it's called a\Nfirst Frenet formula.
Dialogue: 0,1:07:42.26,1:07:44.23,Default,,0000,0000,0000,,That's one thing\NI want to ask you.
Dialogue: 0,1:07:44.23,1:07:47.17,Default,,0000,0000,0000,,And then I'm going to give you\Nmore formulas for curvatures,
Dialogue: 0,1:07:47.17,1:07:50.46,Default,,0000,0000,0000,,depending on how you\Ndefine your curve.
Dialogue: 0,1:07:50.46,1:08:08.83,Default,,0000,0000,0000,,So for example, base B\Nbased on the definition one
Dialogue: 0,1:08:08.83,1:08:18.87,Default,,0000,0000,0000,,can prove that for a curve\Nthat is not parametrizing
Dialogue: 0,1:08:18.87,1:08:22.87,Default,,0000,0000,0000,,arclength-- you say, ugh,\Nforget about parametrization
Dialogue: 0,1:08:22.87,1:08:23.59,Default,,0000,0000,0000,,in arclength.
Dialogue: 0,1:08:23.59,1:08:26.84,Default,,0000,0000,0000,,This time you're\Nassuming, I want to know!
Dialogue: 0,1:08:26.84,1:08:29.41,Default,,0000,0000,0000,,I'm coming to this\Nbecause Ryan asked.
Dialogue: 0,1:08:29.41,1:08:32.38,Default,,0000,0000,0000,,I want to know, what is\Nthe formula directly?
Dialogue: 0,1:08:32.38,1:08:34.44,Default,,0000,0000,0000,,Is there a direct\Nformula that comes
Dialogue: 0,1:08:34.44,1:08:38.53,Default,,0000,0000,0000,,from here for the curvature?
Dialogue: 0,1:08:38.53,1:08:41.31,Default,,0000,0000,0000,,Yeah, but it's a lot\Nmore complicated.
Dialogue: 0,1:08:41.31,1:08:45.26,Default,,0000,0000,0000,,When I was a freshman, maybe\Na freshman or a sophomore,
Dialogue: 0,1:08:45.26,1:08:48.09,Default,,0000,0000,0000,,I don't remember, when\NI was asked to memorize
Dialogue: 0,1:08:48.09,1:08:52.69,Default,,0000,0000,0000,,that, I did not memorize it.
Dialogue: 0,1:08:52.69,1:08:56.65,Default,,0000,0000,0000,,Then when I started working\Nas a faculty member,
Dialogue: 0,1:08:56.65,1:09:01.81,Default,,0000,0000,0000,,I saw that I am supposed\Nto ask it from my students.
Dialogue: 0,1:09:01.81,1:09:05.58,Default,,0000,0000,0000,,So this is going to be\NR prime plus product
Dialogue: 0,1:09:05.58,1:09:12.06,Default,,0000,0000,0000,,R double prime in magnitude\Nover R prime cubed.
Dialogue: 0,1:09:12.06,1:09:14.55,Default,,0000,0000,0000,,So how am I supposed\Nto remember that?
Dialogue: 0,1:09:14.55,1:09:15.66,Default,,0000,0000,0000,,It's not so easy.
Dialogue: 0,1:09:15.66,1:09:17.80,Default,,0000,0000,0000,,Are you cold there?
Dialogue: 0,1:09:17.80,1:09:18.65,Default,,0000,0000,0000,,It's cold there.
Dialogue: 0,1:09:18.65,1:09:22.59,Default,,0000,0000,0000,,I don't know how\Nthese roofs are made.
Dialogue: 0,1:09:22.59,1:09:24.67,Default,,0000,0000,0000,,Velocity times acceleration.
Dialogue: 0,1:09:24.67,1:09:26.62,Default,,0000,0000,0000,,This is what I try\Nto teach myself.
Dialogue: 0,1:09:26.62,1:09:29.81,Default,,0000,0000,0000,,I was old already, 26 or 27.
Dialogue: 0,1:09:29.81,1:09:32.98,Default,,0000,0000,0000,,Velocity times\Nacceleration, cross product,
Dialogue: 0,1:09:32.98,1:09:35.76,Default,,0000,0000,0000,,take the magnitude,\Ndivide by the speed, cube.
Dialogue: 0,1:09:35.76,1:09:36.81,Default,,0000,0000,0000,,Oh my god.
Dialogue: 0,1:09:36.81,1:09:41.34,Default,,0000,0000,0000,,So I was supposed to know\Nthat when I was 18 or 19.
Dialogue: 0,1:09:41.34,1:09:44.51,Default,,0000,0000,0000,,Now, I was teaching majors\Nof mechanical engineering.
Dialogue: 0,1:09:44.51,1:09:45.84,Default,,0000,0000,0000,,They knew that by heart.
Dialogue: 0,1:09:45.84,1:09:48.21,Default,,0000,0000,0000,,I didn't, so I had to learn it.
Dialogue: 0,1:09:48.21,1:09:51.48,Default,,0000,0000,0000,,So if one is too\Nlazy or it's simply
Dialogue: 0,1:09:51.48,1:09:54.92,Default,,0000,0000,0000,,inconvenient to try to\Nreparametrize from R of T
Dialogue: 0,1:09:54.92,1:10:00.79,Default,,0000,0000,0000,,being arclength parametrization\NR of s and do that thing here,
Dialogue: 0,1:10:00.79,1:10:05.30,Default,,0000,0000,0000,,one can just plug in and\Nfind the curvature like that.
Dialogue: 0,1:10:05.30,1:10:08.45,Default,,0000,0000,0000,,For example, guys,\Nas Ryan asked,
Dialogue: 0,1:10:08.45,1:10:13.29,Default,,0000,0000,0000,,if I have A cosine, [INAUDIBLE],\Nand I do this with respect
Dialogue: 0,1:10:13.29,1:10:16.74,Default,,0000,0000,0000,,to T, can I get k\Nwithout-- k will not
Dialogue: 0,1:10:16.74,1:10:18.95,Default,,0000,0000,0000,,depend on T or s or tau.
Dialogue: 0,1:10:18.95,1:10:20.86,Default,,0000,0000,0000,,It will always be the same.
Dialogue: 0,1:10:20.86,1:10:23.40,Default,,0000,0000,0000,,I will still get A\Nover A squared plus B
Dialogue: 0,1:10:23.40,1:10:25.26,Default,,0000,0000,0000,,squared, no matter what.
Dialogue: 0,1:10:25.26,1:10:28.93,Default,,0000,0000,0000,,So even if I use this\Nformula for my helix,
Dialogue: 0,1:10:28.93,1:10:30.95,Default,,0000,0000,0000,,I'm going to get the same thing.
Dialogue: 0,1:10:30.95,1:10:33.06,Default,,0000,0000,0000,,I'll get A over A\Nsquared plus B squared,
Dialogue: 0,1:10:33.06,1:10:35.39,Default,,0000,0000,0000,,because curvature\Nis an invariant.
Dialogue: 0,1:10:35.39,1:10:38.51,Default,,0000,0000,0000,,There is another invariant\Nthat's-- the other invariant,
Dialogue: 0,1:10:38.51,1:10:40.55,Default,,0000,0000,0000,,of course, in space\Nis called torsion.
Dialogue: 0,1:10:40.55,1:10:43.68,Default,,0000,0000,0000,,We want to talk a little\Nbit about that later.
Dialogue: 0,1:10:43.68,1:10:48.78,Default,,0000,0000,0000,,So is this hard?
Dialogue: 0,1:10:48.78,1:10:49.28,Default,,0000,0000,0000,,No.
Dialogue: 0,1:10:49.28,1:10:50.45,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,1:10:50.45,1:10:54.93,Default,,0000,0000,0000,,And you guys should be able\Nto help me on that, hopefully.
Dialogue: 0,1:10:54.93,1:10:56.60,Default,,0000,0000,0000,,How do we prove that?
Dialogue: 0,1:10:56.60,1:10:58.48,Default,,0000,0000,0000,,STUDENT: N is G\Nprime [INAUDIBLE].
Dialogue: 0,1:10:58.48,1:11:01.95,Default,,0000,0000,0000,,
Dialogue: 0,1:11:01.95,1:11:03.45,Default,,0000,0000,0000,,PROFESSOR TODA:\NThat's right, proof.
Dialogue: 0,1:11:03.45,1:11:06.08,Default,,0000,0000,0000,,And that's a very good\Nstart, wouldn't you say?
Dialogue: 0,1:11:06.08,1:11:09.09,Default,,0000,0000,0000,,So what were the definitions?
Dialogue: 0,1:11:09.09,1:11:14.35,Default,,0000,0000,0000,,Let me start from\Nthe definition of T.
Dialogue: 0,1:11:14.35,1:11:17.18,Default,,0000,0000,0000,,That's going to be-- I\Nam in hard planes, right?
Dialogue: 0,1:11:17.18,1:11:21.05,Default,,0000,0000,0000,,So you say, wait, why do\Nyou write it as a quotient?
Dialogue: 0,1:11:21.05,1:11:22.43,Default,,0000,0000,0000,,You're being silly.
Dialogue: 0,1:11:22.43,1:11:24.53,Default,,0000,0000,0000,,You are in arclength, Magdalena.
Dialogue: 0,1:11:24.53,1:11:25.47,Default,,0000,0000,0000,,I am.
Dialogue: 0,1:11:25.47,1:11:26.34,Default,,0000,0000,0000,,I am.
Dialogue: 0,1:11:26.34,1:11:29.86,Default,,0000,0000,0000,,I just pretend that\NI cannot see that.
Dialogue: 0,1:11:29.86,1:11:32.16,Default,,0000,0000,0000,,So if I'm in\Narclength, that means
Dialogue: 0,1:11:32.16,1:11:35.87,Default,,0000,0000,0000,,that the denominator is 1.
Dialogue: 0,1:11:35.87,1:11:37.32,Default,,0000,0000,0000,,So I'm being silly.
Dialogue: 0,1:11:37.32,1:11:44.38,Default,,0000,0000,0000,,So R prime of s is\NT. Say it again.
Dialogue: 0,1:11:44.38,1:11:49.28,Default,,0000,0000,0000,,R prime of s is T. OK.
Dialogue: 0,1:11:49.28,1:11:53.72,Default,,0000,0000,0000,,Now, did we know that\NT and N are orthogonal?
Dialogue: 0,1:11:53.72,1:12:00.53,Default,,0000,0000,0000,,
Dialogue: 0,1:12:00.53,1:12:04.35,Default,,0000,0000,0000,,How did we know that T\Nand N were orthogonal?
Dialogue: 0,1:12:04.35,1:12:07.52,Default,,0000,0000,0000,,We proved that last\Ntime, actually.
Dialogue: 0,1:12:07.52,1:12:11.00,Default,,0000,0000,0000,,T and N are orthogonal.
Dialogue: 0,1:12:11.00,1:12:12.99,Default,,0000,0000,0000,,How do I write\Nthat? [INAUDIBLE].
Dialogue: 0,1:12:12.99,1:12:15.97,Default,,0000,0000,0000,,
Dialogue: 0,1:12:15.97,1:12:21.99,Default,,0000,0000,0000,,Meaning that T is\Nperpendicular to N, right?
Dialogue: 0,1:12:21.99,1:12:24.11,Default,,0000,0000,0000,,From the definition.
Dialogue: 0,1:12:24.11,1:12:26.00,Default,,0000,0000,0000,,You said it right, Sandra.
Dialogue: 0,1:12:26.00,1:12:28.00,Default,,0000,0000,0000,,But why is it from\Nthe definition
Dialogue: 0,1:12:28.00,1:12:30.90,Default,,0000,0000,0000,,that I can jump to\Nconclusions and say, oh,
Dialogue: 0,1:12:30.90,1:12:36.12,Default,,0000,0000,0000,,since I have T prime here, then\Nthis is perpendicular to T?
Dialogue: 0,1:12:36.12,1:12:37.44,Default,,0000,0000,0000,,Well, we did that last time.
Dialogue: 0,1:12:37.44,1:12:39.24,Default,,0000,0000,0000,,STUDENT: Two parallel vectors.
Dialogue: 0,1:12:39.24,1:12:41.11,Default,,0000,0000,0000,,PROFESSOR TODA: We did\Nit-- how did we do it?
Dialogue: 0,1:12:41.11,1:12:42.25,Default,,0000,0000,0000,,We did this last.
Dialogue: 0,1:12:42.25,1:12:45.80,Default,,0000,0000,0000,,We said T dot T equals 1.
Dialogue: 0,1:12:45.80,1:12:47.96,Default,,0000,0000,0000,,Prime the whole thing.
Dialogue: 0,1:12:47.96,1:12:54.27,Default,,0000,0000,0000,,T prime times T plus T times T\Nprime, T dot T prime will be 0.
Dialogue: 0,1:12:54.27,1:12:57.03,Default,,0000,0000,0000,,So T and T prime are\Nperpendicular always.
Dialogue: 0,1:12:57.03,1:12:58.15,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:12:58.15,1:13:03.03,Default,,0000,0000,0000,,OK, so the whole thing is a\Ncolinear vector to T prime.
Dialogue: 0,1:13:03.03,1:13:05.02,Default,,0000,0000,0000,,It's just T prime\Ntimes the scalar.
Dialogue: 0,1:13:05.02,1:13:08.32,Default,,0000,0000,0000,,So he must be\Nperpendicular to T.
Dialogue: 0,1:13:08.32,1:13:10.72,Default,,0000,0000,0000,,So T and N are perpendicular.
Dialogue: 0,1:13:10.72,1:13:14.68,Default,,0000,0000,0000,,So I do have the\Ndirection of motion.
Dialogue: 0,1:13:14.68,1:13:19.48,Default,,0000,0000,0000,,I know that I must\Nhave some scalar here.
Dialogue: 0,1:13:19.48,1:13:22.80,Default,,0000,0000,0000,,
Dialogue: 0,1:13:22.80,1:13:26.63,Default,,0000,0000,0000,,How do I prove that this\Nscalar is the curvature?
Dialogue: 0,1:13:26.63,1:13:30.51,Default,,0000,0000,0000,,
Dialogue: 0,1:13:30.51,1:13:35.100,Default,,0000,0000,0000,,So if I have-- if they\Nare colinear-- why are
Dialogue: 0,1:13:35.100,1:13:36.74,Default,,0000,0000,0000,,they colinear?
Dialogue: 0,1:13:36.74,1:13:42.20,Default,,0000,0000,0000,,T perpendicular to T\Nprime implies that T prime
Dialogue: 0,1:13:42.20,1:13:46.02,Default,,0000,0000,0000,,is colinear to N. Say it again.
Dialogue: 0,1:13:46.02,1:13:49.86,Default,,0000,0000,0000,,If T and T prime are\Nperpendicular to one another,
Dialogue: 0,1:13:49.86,1:13:53.26,Default,,0000,0000,0000,,that means T prime is\Ncalling it to the normal.
Dialogue: 0,1:13:53.26,1:13:58.47,Default,,0000,0000,0000,,So here I may have\Nalph-- no alpha.
Dialogue: 0,1:13:58.47,1:14:00.26,Default,,0000,0000,0000,,I don't know!
Dialogue: 0,1:14:00.26,1:14:03.70,Default,,0000,0000,0000,,Alpha over [INAUDIBLE]\Nsounds like a curve.
Dialogue: 0,1:14:03.70,1:14:04.62,Default,,0000,0000,0000,,Give me some function.
Dialogue: 0,1:14:04.62,1:14:08.86,Default,,0000,0000,0000,,
Dialogue: 0,1:14:08.86,1:14:09.53,Default,,0000,0000,0000,,STUDENT: u of s?
Dialogue: 0,1:14:09.53,1:14:11.05,Default,,0000,0000,0000,,PROFESSOR TODA: Gamma of s.
Dialogue: 0,1:14:11.05,1:14:15.36,Default,,0000,0000,0000,,u of s, I don't know.
Dialogue: 0,1:14:15.36,1:14:17.41,Default,,0000,0000,0000,,So how did I conclude that?
Dialogue: 0,1:14:17.41,1:14:19.50,Default,,0000,0000,0000,,From T perpendicular to T prime.
Dialogue: 0,1:14:19.50,1:14:22.21,Default,,0000,0000,0000,,Now from here on, you\Nhave to tell me why
Dialogue: 0,1:14:22.21,1:14:29.43,Default,,0000,0000,0000,,gamma must be exactly kappa.
Dialogue: 0,1:14:29.43,1:14:33.71,Default,,0000,0000,0000,,Well, let's take\NT prime from here.
Dialogue: 0,1:14:33.71,1:14:38.09,Default,,0000,0000,0000,,T prime from here\Nwill give me what?
Dialogue: 0,1:14:38.09,1:14:40.73,Default,,0000,0000,0000,,T prime is our prime prime.
Dialogue: 0,1:14:40.73,1:14:42.30,Default,,0000,0000,0000,,Say what?
Dialogue: 0,1:14:42.30,1:14:43.20,Default,,0000,0000,0000,,Our prime prime.
Dialogue: 0,1:14:43.20,1:14:44.66,Default,,0000,0000,0000,,What is our prime prime?
Dialogue: 0,1:14:44.66,1:14:46.56,Default,,0000,0000,0000,,Our [? problem ?] prime of s.
Dialogue: 0,1:14:46.56,1:14:48.58,Default,,0000,0000,0000,,STUDENT: You have one\Ntoo many primes inside.
Dialogue: 0,1:14:48.58,1:14:49.66,Default,,0000,0000,0000,,PROFESSOR TODA: Oh my god.
Dialogue: 0,1:14:49.66,1:14:50.16,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,1:14:50.16,1:14:52.77,Default,,0000,0000,0000,,
Dialogue: 0,1:14:52.77,1:14:54.12,Default,,0000,0000,0000,,So R prime prime.
Dialogue: 0,1:14:54.12,1:14:58.43,Default,,0000,0000,0000,,So T prime in\Nabsolute value will
Dialogue: 0,1:14:58.43,1:15:02.65,Default,,0000,0000,0000,,be exactly R double prime of s.
Dialogue: 0,1:15:02.65,1:15:04.99,Default,,0000,0000,0000,,Oh, OK.
Dialogue: 0,1:15:04.99,1:15:10.13,Default,,0000,0000,0000,,Note that from here also T\Nprime of s in absolute value,
Dialogue: 0,1:15:10.13,1:15:13.84,Default,,0000,0000,0000,,in magnitude, I'm sorry,\Nhas to be gamma of s.
Dialogue: 0,1:15:13.84,1:15:14.82,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,1:15:14.82,1:15:17.47,Default,,0000,0000,0000,,Because the magnitude of N is 1.
Dialogue: 0,1:15:17.47,1:15:20.93,Default,,0000,0000,0000,,N is unique vector\Nby definition.
Dialogue: 0,1:15:20.93,1:15:24.87,Default,,0000,0000,0000,,So these two guys\Nhave to coincide.
Dialogue: 0,1:15:24.87,1:15:27.13,Default,,0000,0000,0000,,So R double prime,\Nthe best thing
Dialogue: 0,1:15:27.13,1:15:28.59,Default,,0000,0000,0000,,that I need to do,\Nit must coincide
Dialogue: 0,1:15:28.59,1:15:30.50,Default,,0000,0000,0000,,with the scalar gamma of s.
Dialogue: 0,1:15:30.50,1:15:32.84,Default,,0000,0000,0000,,So who is the\Nmysterious gamma of s?
Dialogue: 0,1:15:32.84,1:15:36.34,Default,,0000,0000,0000,,He has no chance\Nbut being this guy.
Dialogue: 0,1:15:36.34,1:15:38.66,Default,,0000,0000,0000,,But this guy has a name.
Dialogue: 0,1:15:38.66,1:15:41.92,Default,,0000,0000,0000,,This guy, he's the curvature\N[? cap ?] of s by definition.
Dialogue: 0,1:15:41.92,1:15:45.74,Default,,0000,0000,0000,,
Dialogue: 0,1:15:45.74,1:15:49.13,Default,,0000,0000,0000,,Remember, Ryan, this\Nis the definition.
Dialogue: 0,1:15:49.13,1:15:51.55,Default,,0000,0000,0000,,So by definition the\Ncurvature was the magnitude
Dialogue: 0,1:15:51.55,1:15:55.01,Default,,0000,0000,0000,,of the acceleration\Nin arclength.
Dialogue: 0,1:15:55.01,1:15:55.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:15:55.91,1:15:58.46,Default,,0000,0000,0000,,Both of these guys are\NT prime in magnitude.
Dialogue: 0,1:15:58.46,1:16:01.77,Default,,0000,0000,0000,,So they must be equal\Nfrom here and here.
Dialogue: 0,1:16:01.77,1:16:04.70,Default,,0000,0000,0000,,It implies that my\Ngamma must be kappa.
Dialogue: 0,1:16:04.70,1:16:07.78,Default,,0000,0000,0000,,And I prove the formula.
Dialogue: 0,1:16:07.78,1:16:09.13,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:16:09.13,1:16:10.91,Default,,0000,0000,0000,,How do you say\Nsomething is proved?
Dialogue: 0,1:16:10.91,1:16:12.29,Default,,0000,0000,0000,,Because this is what we wanted.
Dialogue: 0,1:16:12.29,1:16:16.24,Default,,0000,0000,0000,,We wanted to replace this\Ngeneric scalar function
Dialogue: 0,1:16:16.24,1:16:20.08,Default,,0000,0000,0000,,to prove that this is\Njust the curvature.
Dialogue: 0,1:16:20.08,1:16:20.58,Default,,0000,0000,0000,,QED.
Dialogue: 0,1:16:20.58,1:16:24.42,Default,,0000,0000,0000,,
Dialogue: 0,1:16:24.42,1:16:26.96,Default,,0000,0000,0000,,That's exactly what\Nwe wanted to prove.
Dialogue: 0,1:16:26.96,1:16:29.05,Default,,0000,0000,0000,,Now, whatever scalar\Nfunction you have here,
Dialogue: 0,1:16:29.05,1:16:30.17,Default,,0000,0000,0000,,that must be the curvature.
Dialogue: 0,1:16:30.17,1:16:34.46,Default,,0000,0000,0000,,
Dialogue: 0,1:16:34.46,1:16:36.42,Default,,0000,0000,0000,,Very smart guy, this Mr. Frenet.
Dialogue: 0,1:16:36.42,1:16:39.72,Default,,0000,0000,0000,,
Dialogue: 0,1:16:39.72,1:16:40.99,Default,,0000,0000,0000,,I'm now going to take a break.
Dialogue: 0,1:16:40.99,1:16:43.83,Default,,0000,0000,0000,,If you want to go use the\Nbathroom really quickly,
Dialogue: 0,1:16:43.83,1:16:45.02,Default,,0000,0000,0000,,feel free to do it.
Dialogue: 0,1:16:45.02,1:16:47.59,Default,,0000,0000,0000,,
Dialogue: 0,1:16:47.59,1:16:49.47,Default,,0000,0000,0000,,I'm just going to\Nclean the board,
Dialogue: 0,1:16:49.47,1:16:51.69,Default,,0000,0000,0000,,and I'll keep going\Nin a few minutes.
Dialogue: 0,1:16:51.69,1:17:50.50,Default,,0000,0000,0000,,
Dialogue: 0,1:17:50.50,1:17:51.33,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:17:51.33,1:17:55.80,Default,,0000,0000,0000,,
Dialogue: 0,1:17:55.80,1:17:57.30,Default,,0000,0000,0000,,PROFESSOR TODA: I\Nwill do it-- well,
Dialogue: 0,1:17:57.30,1:18:01.27,Default,,0000,0000,0000,,actually I want to do a\Ndifferent example, simple one,
Dialogue: 0,1:18:01.27,1:18:06.24,Default,,0000,0000,0000,,which is a plain curve, and show\Nthat the curvature has a very
Dialogue: 0,1:18:06.24,1:18:11.04,Default,,0000,0000,0000,,pretty formula that you\Ncould [INAUDIBLE] memorize,
Dialogue: 0,1:18:11.04,1:18:13.52,Default,,0000,0000,0000,,that in essence is the same.
Dialogue: 0,1:18:13.52,1:18:17.48,Default,,0000,0000,0000,,But it depends on\Ny equals f of x.
Dialogue: 0,1:18:17.48,1:18:19.47,Default,,0000,0000,0000,,[INAUDIBLE] So if\Nsomebody gives you
Dialogue: 0,1:18:19.47,1:18:22.44,Default,,0000,0000,0000,,a plane called y\Nequals f of x, can you
Dialogue: 0,1:18:22.44,1:18:25.42,Default,,0000,0000,0000,,write that curvature\N[INAUDIBLE] function of f?
Dialogue: 0,1:18:25.42,1:18:26.91,Default,,0000,0000,0000,,And you can.
Dialogue: 0,1:18:26.91,1:18:30.54,Default,,0000,0000,0000,,And again, I was deep in\Nthat when I was 18 or 19
Dialogue: 0,1:18:30.54,1:18:31.87,Default,,0000,0000,0000,,as a freshman.
Dialogue: 0,1:18:31.87,1:18:35.84,Default,,0000,0000,0000,,But unfortunately for me I\Ndidn't learn it at that time.
Dialogue: 0,1:18:35.84,1:18:41.34,Default,,0000,0000,0000,,And several years later when\NI started teaching engineers,
Dialogue: 0,1:18:41.34,1:18:43.78,Default,,0000,0000,0000,,well, they are\Nmostly mechanical.
Dialogue: 0,1:18:43.78,1:18:46.77,Default,,0000,0000,0000,,And mechanical\Nengineering [INAUDIBLE].
Dialogue: 0,1:18:46.77,1:18:50.26,Default,,0000,0000,0000,,They knew those, and they needed\Nthose in every research paper.
Dialogue: 0,1:18:50.26,1:18:54.26,Default,,0000,0000,0000,,So I had to learn it\Ntogether with them.
Dialogue: 0,1:18:54.26,1:18:57.75,Default,,0000,0000,0000,,I'll worry about [INAUDIBLE].
Dialogue: 0,1:18:57.75,1:19:01.24,Default,,0000,0000,0000,,STUDENT: Can you do a really\Nugly one, like [INAUDIBLE]?
Dialogue: 0,1:19:01.24,1:19:04.73,Default,,0000,0000,0000,,PROFESSOR TODA: I can\Ndo some ugly ones.
Dialogue: 0,1:19:04.73,1:20:37.24,Default,,0000,0000,0000,,
Dialogue: 0,1:20:37.24,1:20:47.67,Default,,0000,0000,0000,,And once you know the\Ngeneral parametrization,
Dialogue: 0,1:20:47.67,1:20:51.91,Default,,0000,0000,0000,,it will give you a curvature.
Dialogue: 0,1:20:51.91,1:20:53.33,Default,,0000,0000,0000,,Now I'm testing your memory.
Dialogue: 0,1:20:53.33,1:20:55.23,Default,,0000,0000,0000,,Let's see what you remember.
Dialogue: 0,1:20:55.23,1:20:59.85,Default,,0000,0000,0000,,Um-- don't look at the notes.
Dialogue: 0,1:20:59.85,1:21:03.27,Default,,0000,0000,0000,,A positive function,\Nabsolute-- actually,
Dialogue: 0,1:21:03.27,1:21:06.68,Default,,0000,0000,0000,,magnitude of what vector?
Dialogue: 0,1:21:06.68,1:21:07.58,Default,,0000,0000,0000,,STUDENT: R prime.
Dialogue: 0,1:21:07.58,1:21:17.06,Default,,0000,0000,0000,,PROFESSOR TODA: R prime velocity\Nplus acceleration speed cubed.
Dialogue: 0,1:21:17.06,1:21:18.54,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:21:18.54,1:21:19.04,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:21:19.04,1:21:24.07,Default,,0000,0000,0000,,Now, can we take advantage\Nof what we just learned
Dialogue: 0,1:21:24.07,1:21:30.06,Default,,0000,0000,0000,,and find-- you find\Nwith me, of course, not
Dialogue: 0,1:21:30.06,1:21:33.65,Default,,0000,0000,0000,,as professor and student,\Nbut like a group of students
Dialogue: 0,1:21:33.65,1:21:35.07,Default,,0000,0000,0000,,together.
Dialogue: 0,1:21:35.07,1:21:46.29,Default,,0000,0000,0000,,Let's find a simple\Nformula corresponding
Dialogue: 0,1:21:46.29,1:21:52.20,Default,,0000,0000,0000,,to the curvature\Nof a plane curve.
Dialogue: 0,1:21:52.20,1:21:59.60,Default,,0000,0000,0000,,
Dialogue: 0,1:21:59.60,1:22:05.28,Default,,0000,0000,0000,,And the plane curve\Ncould be [INAUDIBLE]
Dialogue: 0,1:22:05.28,1:22:09.02,Default,,0000,0000,0000,,in two different ways,\Njust because I want
Dialogue: 0,1:22:09.02,1:22:14.51,Default,,0000,0000,0000,,you to practice more on that.
Dialogue: 0,1:22:14.51,1:22:18.36,Default,,0000,0000,0000,,Either given as a general\Nparametrization-- guys,
Dialogue: 0,1:22:18.36,1:22:20.09,Default,,0000,0000,0000,,what is the general\Nparametrization
Dialogue: 0,1:22:20.09,1:22:24.51,Default,,0000,0000,0000,,I'm talking about\Nfor a plane curve?
Dialogue: 0,1:22:24.51,1:22:26.40,Default,,0000,0000,0000,,x of t, y of t, right?
Dialogue: 0,1:22:26.40,1:22:28.66,Default,,0000,0000,0000,,x equals x of t.
Dialogue: 0,1:22:28.66,1:22:29.87,Default,,0000,0000,0000,,y equals y of t.
Dialogue: 0,1:22:29.87,1:22:34.40,Default,,0000,0000,0000,,So one should not have\Nto do that all the time,
Dialogue: 0,1:22:34.40,1:22:37.31,Default,,0000,0000,0000,,not have to do that for a\Nsimplification like a playing
Dialogue: 0,1:22:37.31,1:22:38.19,Default,,0000,0000,0000,,card.
Dialogue: 0,1:22:38.19,1:22:41.71,Default,,0000,0000,0000,,We have to find another\Nformula that's pretty, right?
Dialogue: 0,1:22:41.71,1:22:43.21,Default,,0000,0000,0000,,Well, maybe it's not as pretty.
Dialogue: 0,1:22:43.21,1:22:45.25,Default,,0000,0000,0000,,But when is it really pretty?
Dialogue: 0,1:22:45.25,1:22:49.09,Default,,0000,0000,0000,,I bet it's going to be really\Npretty if you have a plane
Dialogue: 0,1:22:49.09,1:22:54.61,Default,,0000,0000,0000,,curve even as you're used\Nto in an explicit form--
Dialogue: 0,1:22:54.61,1:22:56.60,Default,,0000,0000,0000,,I keep going.
Dialogue: 0,1:22:56.60,1:22:59.91,Default,,0000,0000,0000,,No stop. [INAUDIBLE].
Dialogue: 0,1:22:59.91,1:23:01.11,Default,,0000,0000,0000,,I think it's better.
Dialogue: 0,1:23:01.11,1:23:03.45,Default,,0000,0000,0000,,We make better use\Nof time this way.
Dialogue: 0,1:23:03.45,1:23:06.89,Default,,0000,0000,0000,,Or y equals f of x.
Dialogue: 0,1:23:06.89,1:23:12.60,Default,,0000,0000,0000,,
Dialogue: 0,1:23:12.60,1:23:17.95,Default,,0000,0000,0000,,This is an explicit way to\Nwrite the equation of a curve.
Dialogue: 0,1:23:17.95,1:23:20.66,Default,,0000,0000,0000,,
Dialogue: 0,1:23:20.66,1:23:23.33,Default,,0000,0000,0000,,OK, so what do we need to do?
Dialogue: 0,1:23:23.33,1:23:26.43,Default,,0000,0000,0000,,That should be really easy.
Dialogue: 0,1:23:26.43,1:23:32.96,Default,,0000,0000,0000,,R of t being the first case of\Nour general parametrization,
Dialogue: 0,1:23:32.96,1:23:40.72,Default,,0000,0000,0000,,x equals x of t, y equals y of\Nt will be-- who tells me, guys,
Dialogue: 0,1:23:40.72,1:23:43.47,Default,,0000,0000,0000,,that-- this is in your hands.
Dialogue: 0,1:23:43.47,1:23:47.70,Default,,0000,0000,0000,,Now you convinced me\Nthat, for whatever reason,
Dialogue: 0,1:23:47.70,1:23:49.88,Default,,0000,0000,0000,,you [INAUDIBLE].
Dialogue: 0,1:23:49.88,1:23:51.80,Default,,0000,0000,0000,,You became friends\Nwith these curves.
Dialogue: 0,1:23:51.80,1:23:52.76,Default,,0000,0000,0000,,I don't know when.
Dialogue: 0,1:23:52.76,1:23:54.68,Default,,0000,0000,0000,,I guess in the process\Nof doing homework.
Dialogue: 0,1:23:54.68,1:23:55.65,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,1:23:55.65,1:24:00.48,Default,,0000,0000,0000,,I think you did not quite like\Nthem before or the last week.
Dialogue: 0,1:24:00.48,1:24:02.47,Default,,0000,0000,0000,,But I think you're\Nfriends with them now.
Dialogue: 0,1:24:02.47,1:24:06.56,Default,,0000,0000,0000,,x of t, y of t.
Dialogue: 0,1:24:06.56,1:24:07.99,Default,,0000,0000,0000,,Let people talk.
Dialogue: 0,1:24:07.99,1:24:12.77,Default,,0000,0000,0000,,
Dialogue: 0,1:24:12.77,1:24:13.74,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:24:13.74,1:24:15.67,Default,,0000,0000,0000,,PROFESSOR TODA: So.
Dialogue: 0,1:24:15.67,1:24:16.17,Default,,0000,0000,0000,,Great.
Dialogue: 0,1:24:16.17,1:24:21.11,Default,,0000,0000,0000,,And then R prime of t will be\Nx prime of t, y prime of t,
Dialogue: 0,1:24:21.11,1:24:21.76,Default,,0000,0000,0000,,and 0.
Dialogue: 0,1:24:21.76,1:24:24.49,Default,,0000,0000,0000,,I assume this to\Nbe always non-zero.
Dialogue: 0,1:24:24.49,1:24:26.24,Default,,0000,0000,0000,,I have a regular curve.
Dialogue: 0,1:24:26.24,1:24:30.68,Default,,0000,0000,0000,,R double prime will be--\Nx double prime where
Dialogue: 0,1:24:30.68,1:24:34.15,Default,,0000,0000,0000,,double prime-- we\Ndid the review today
Dialogue: 0,1:24:34.15,1:24:36.45,Default,,0000,0000,0000,,of the lasting acceleration.
Dialogue: 0,1:24:36.45,1:24:39.63,Default,,0000,0000,0000,,Now, your friends over\Nhere, are they nice or mean?
Dialogue: 0,1:24:39.63,1:24:42.60,Default,,0000,0000,0000,,I hope they are not so mean.
Dialogue: 0,1:24:42.60,1:24:45.81,Default,,0000,0000,0000,,The cross product is\Na friendly fellow.
Dialogue: 0,1:24:45.81,1:24:48.97,Default,,0000,0000,0000,,You have i, j, k, and\Nthen the second row
Dialogue: 0,1:24:48.97,1:24:50.64,Default,,0000,0000,0000,,would be x prime, y prime, 0.
Dialogue: 0,1:24:50.64,1:24:54.85,Default,,0000,0000,0000,,The last row would be x double\Nprime, y double prime, 0.
Dialogue: 0,1:24:54.85,1:24:58.55,Default,,0000,0000,0000,,And it's a piece of cake.
Dialogue: 0,1:24:58.55,1:25:02.15,Default,,0000,0000,0000,,
Dialogue: 0,1:25:02.15,1:25:03.52,Default,,0000,0000,0000,,OK, piece of cake,\Npiece of cake.
Dialogue: 0,1:25:03.52,1:25:08.96,Default,,0000,0000,0000,,But I want to know\Nwhat the answer is.
Dialogue: 0,1:25:08.96,1:25:15.63,Default,,0000,0000,0000,,So you have exactly 15 seconds\Nto answer this question.
Dialogue: 0,1:25:15.63,1:25:23.30,Default,,0000,0000,0000,,Who is R prime plus R double\Nprime as a [? coordinate. ?]
Dialogue: 0,1:25:23.30,1:25:25.27,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:25:25.27,1:25:28.71,Default,,0000,0000,0000,,
Dialogue: 0,1:25:28.71,1:25:29.70,Default,,0000,0000,0000,,PROFESSOR TODA: Good.
Dialogue: 0,1:25:29.70,1:25:35.37,Default,,0000,0000,0000,,x prime, y double prime minus x\Ndouble prime, y prime times k.
Dialogue: 0,1:25:35.37,1:25:37.72,Default,,0000,0000,0000,,And it doesn't matter\Nwhen I take the magnitude,
Dialogue: 0,1:25:37.72,1:25:40.60,Default,,0000,0000,0000,,because magnitude of k is 1.
Dialogue: 0,1:25:40.60,1:25:42.38,Default,,0000,0000,0000,,So I discovered some.
Dialogue: 0,1:25:42.38,1:25:46.70,Default,,0000,0000,0000,,This is how mathematicians like\Nto discover new formulas based
Dialogue: 0,1:25:46.70,1:25:48.73,Default,,0000,0000,0000,,on the formulas they\N[? knew. ?] They
Dialogue: 0,1:25:48.73,1:25:50.09,Default,,0000,0000,0000,,have a lot of satisfaction.
Dialogue: 0,1:25:50.09,1:25:51.02,Default,,0000,0000,0000,,Look what I got.
Dialogue: 0,1:25:51.02,1:25:56.63,Default,,0000,0000,0000,,Of course, they in general have\Nmore complicated things to do,
Dialogue: 0,1:25:56.63,1:25:58.95,Default,,0000,0000,0000,,and they have to\Ncheck and recheck.
Dialogue: 0,1:25:58.95,1:26:06.30,Default,,0000,0000,0000,,But every piece of a\Ncomputation is a challenge.
Dialogue: 0,1:26:06.30,1:26:10.31,Default,,0000,0000,0000,,And that gives\Npeople satisfaction.
Dialogue: 0,1:26:10.31,1:26:14.90,Default,,0000,0000,0000,,And when they make a mistake, it\Nbrings a lot of tears as well.
Dialogue: 0,1:26:14.90,1:26:21.19,Default,,0000,0000,0000,,So what-- could be written\Non the bottom, what's
Dialogue: 0,1:26:21.19,1:26:24.68,Default,,0000,0000,0000,,the speed cubed?
Dialogue: 0,1:26:24.68,1:26:26.74,Default,,0000,0000,0000,,Speed is coming from this guy.
Dialogue: 0,1:26:26.74,1:26:32.06,Default,,0000,0000,0000,,So the speed of the velocity,\Nthe magnitude of the velocity
Dialogue: 0,1:26:32.06,1:26:32.99,Default,,0000,0000,0000,,is the speed.
Dialogue: 0,1:26:32.99,1:26:35.10,Default,,0000,0000,0000,,And that-- going\Nto give you square.
Dialogue: 0,1:26:35.10,1:26:37.01,Default,,0000,0000,0000,,I'm not going to write\Ndown [INAUDIBLE].
Dialogue: 0,1:26:37.01,1:26:39.14,Default,,0000,0000,0000,,Square root of x squared,\Nx prime squared times
Dialogue: 0,1:26:39.14,1:26:42.74,Default,,0000,0000,0000,,y prime squared,\Nand I cube that.
Dialogue: 0,1:26:42.74,1:26:46.37,Default,,0000,0000,0000,,Many people, and I saw\Nthat in engineering, they
Dialogue: 0,1:26:46.37,1:26:49.76,Default,,0000,0000,0000,,don't like to put that\Nsquare root anymore.
Dialogue: 0,1:26:49.76,1:26:53.95,Default,,0000,0000,0000,,And they just write x prime\Nsquared plus y prime squared
Dialogue: 0,1:26:53.95,1:26:55.11,Default,,0000,0000,0000,,to the what power?
Dialogue: 0,1:26:55.11,1:26:55.69,Default,,0000,0000,0000,,STUDENT: 3/2.
Dialogue: 0,1:26:55.69,1:26:56.54,Default,,0000,0000,0000,,PROFESSOR TODA: 3/2.
Dialogue: 0,1:26:56.54,1:27:01.65,Default,,0000,0000,0000,,So this is very useful\Nfor engineering styles,
Dialogue: 0,1:27:01.65,1:27:05.11,Default,,0000,0000,0000,,when you have to deal\Nwith plane curves, motions
Dialogue: 0,1:27:05.11,1:27:08.56,Default,,0000,0000,0000,,in plane curves.
Dialogue: 0,1:27:08.56,1:27:13.77,Default,,0000,0000,0000,,But now what do you\Nhave in the case,
Dialogue: 0,1:27:13.77,1:27:19.22,Default,,0000,0000,0000,,in the happy case, when\Nyou have y equals f of x?
Dialogue: 0,1:27:19.22,1:27:21.34,Default,,0000,0000,0000,,I'm going to do\Nthat in a second.
Dialogue: 0,1:27:21.34,1:27:26.46,Default,,0000,0000,0000,,
Dialogue: 0,1:27:26.46,1:27:29.14,Default,,0000,0000,0000,,I want to keep this\Nformula on the board.
Dialogue: 0,1:27:29.14,1:27:38.11,Default,,0000,0000,0000,,
Dialogue: 0,1:27:38.11,1:27:40.60,Default,,0000,0000,0000,,What's the simplest\Nparametrization?
Dialogue: 0,1:27:40.60,1:27:43.09,Default,,0000,0000,0000,,Because that's why we\Nneed it, to look over
Dialogue: 0,1:27:43.09,1:27:46.58,Default,,0000,0000,0000,,parametrizations\Nagain and again.
Dialogue: 0,1:27:46.58,1:27:52.19,Default,,0000,0000,0000,,R of t for this plane\Ncurve will be-- what is t?
Dialogue: 0,1:27:52.19,1:27:53.75,Default,,0000,0000,0000,,x is t, right?
Dialogue: 0,1:27:53.75,1:27:56.00,Default,,0000,0000,0000,,x is t, y is f of t.
Dialogue: 0,1:27:56.00,1:27:57.05,Default,,0000,0000,0000,,Piece of cake.
Dialogue: 0,1:27:57.05,1:28:00.31,Default,,0000,0000,0000,,So you have t and f of t.
Dialogue: 0,1:28:00.31,1:28:03.36,Default,,0000,0000,0000,,And how many of you watched\Nthe videos that I sent you?
Dialogue: 0,1:28:03.36,1:28:06.39,Default,,0000,0000,0000,,
Dialogue: 0,1:28:06.39,1:28:08.89,Default,,0000,0000,0000,,Do you prefer Khan\NAcademy, or do you
Dialogue: 0,1:28:08.89,1:28:12.88,Default,,0000,0000,0000,,prefer the guys, [INAUDIBLE]\Nguys who are lecturing?
Dialogue: 0,1:28:12.88,1:28:15.58,Default,,0000,0000,0000,,The professors who are lecturing\Nin front of a board or in front
Dialogue: 0,1:28:15.58,1:28:17.96,Default,,0000,0000,0000,,of a-- what is that?
Dialogue: 0,1:28:17.96,1:28:21.35,Default,,0000,0000,0000,,A projector screen?
Dialogue: 0,1:28:21.35,1:28:22.62,Default,,0000,0000,0000,,I like all of them.
Dialogue: 0,1:28:22.62,1:28:25.14,Default,,0000,0000,0000,,I think they're very good.
Dialogue: 0,1:28:25.14,1:28:27.57,Default,,0000,0000,0000,,I think you can learn\Na lot from three
Dialogue: 0,1:28:27.57,1:28:29.53,Default,,0000,0000,0000,,or four different\Ninstructors at the same time.
Dialogue: 0,1:28:29.53,1:28:32.01,Default,,0000,0000,0000,,That's ideal.
Dialogue: 0,1:28:32.01,1:28:35.59,Default,,0000,0000,0000,,I guess that you have\Nthis chance only now
Dialogue: 0,1:28:35.59,1:28:36.98,Default,,0000,0000,0000,,in the past few years.
Dialogue: 0,1:28:36.98,1:28:41.30,Default,,0000,0000,0000,,Because 20 years ago, if you're\Ndidn't like your instructor
Dialogue: 0,1:28:41.30,1:28:45.79,Default,,0000,0000,0000,,or just you couldn't stand\Nthem, you had no other chance.
Dialogue: 0,1:28:45.79,1:28:48.03,Default,,0000,0000,0000,,There was no\NYouTube, no internet,
Dialogue: 0,1:28:48.03,1:28:50.97,Default,,0000,0000,0000,,no way to learn from others.
Dialogue: 0,1:28:50.97,1:29:00.22,Default,,0000,0000,0000,,R prime of t would\Nbe 1 f prime of t.
Dialogue: 0,1:29:00.22,1:29:02.84,Default,,0000,0000,0000,,But instead of t I'll\Nout x, because x is t.
Dialogue: 0,1:29:02.84,1:29:03.93,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,1:29:03.93,1:29:07.47,Default,,0000,0000,0000,,R double prime of t would\Nbe 0, f double prime of x.
Dialogue: 0,1:29:07.47,1:29:12.43,Default,,0000,0000,0000,,So I feel that, hey, I know\Nwhat's going to come up.
Dialogue: 0,1:29:12.43,1:29:15.39,Default,,0000,0000,0000,,And I'm ready.
Dialogue: 0,1:29:15.39,1:29:17.68,Default,,0000,0000,0000,,Well, we are ready\Nto write it down.
Dialogue: 0,1:29:17.68,1:29:20.19,Default,,0000,0000,0000,,This is going to be Mr. x prime.
Dialogue: 0,1:29:20.19,1:29:22.79,Default,,0000,0000,0000,,This is going to be\Nreplacing Mr. y prime.
Dialogue: 0,1:29:22.79,1:29:25.78,Default,,0000,0000,0000,,This is going to replace\NMr. a double prime.
Dialogue: 0,1:29:25.78,1:29:29.35,Default,,0000,0000,0000,,This is going to be replacing\NMr. y double prime of x.
Dialogue: 0,1:29:29.35,1:29:31.12,Default,,0000,0000,0000,,Oh, OK, all right.
Dialogue: 0,1:29:31.12,1:29:38.84,Default,,0000,0000,0000,,So k, our old friend from\Nhere will become what?
Dialogue: 0,1:29:38.84,1:29:42.49,Default,,0000,0000,0000,,And I'd better shut up,\Nbecause I'm talking too much.
Dialogue: 0,1:29:42.49,1:29:45.02,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Ndouble prime [INAUDIBLE].
Dialogue: 0,1:29:45.02,1:29:48.09,Default,,0000,0000,0000,,PROFESSOR TODA: That is\Nthe absolute value, mm-hmm.
Dialogue: 0,1:29:48.09,1:29:54.04,Default,,0000,0000,0000,,[? n ?] double prime\Nof x, and nothing else.
Dialogue: 0,1:29:54.04,1:29:54.54,Default,,0000,0000,0000,,Right, guys?
Dialogue: 0,1:29:54.54,1:29:55.63,Default,,0000,0000,0000,,Are you with me?
Dialogue: 0,1:29:55.63,1:29:56.96,Default,,0000,0000,0000,,Divided by--
Dialogue: 0,1:29:56.96,1:29:57.85,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:29:57.85,1:29:59.56,Default,,0000,0000,0000,,PROFESSOR TODA: Should\NI add square root?
Dialogue: 0,1:29:59.56,1:30:00.89,Default,,0000,0000,0000,,I love square roots.
Dialogue: 0,1:30:00.89,1:30:01.88,Default,,0000,0000,0000,,I'm crazy about them.
Dialogue: 0,1:30:01.88,1:30:11.95,Default,,0000,0000,0000,,So you go 1 plus f\Nprime squared cubed.
Dialogue: 0,1:30:11.95,1:30:16.41,Default,,0000,0000,0000,,So that's going to\Nbe-- any questions?
Dialogue: 0,1:30:16.41,1:30:18.09,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:30:18.09,1:30:21.30,Default,,0000,0000,0000,,That's going to be the\Nformula that I'm going
Dialogue: 0,1:30:21.30,1:30:22.42,Default,,0000,0000,0000,,to use in the next example.
Dialogue: 0,1:30:22.42,1:30:25.54,Default,,0000,0000,0000,,
Dialogue: 0,1:30:25.54,1:30:29.82,Default,,0000,0000,0000,,In case somebody\Nwants to know-- I got
Dialogue: 0,1:30:29.82,1:30:32.01,Default,,0000,0000,0000,,this question from one of you.
Dialogue: 0,1:30:32.01,1:30:35.06,Default,,0000,0000,0000,,Suppose we get a\Nparametrization of a circle
Dialogue: 0,1:30:35.06,1:30:37.41,Default,,0000,0000,0000,,in the midterm or the final.
Dialogue: 0,1:30:37.41,1:30:44.34,Default,,0000,0000,0000,,Somebody says, I have x\Nof t, just like we did it
Dialogue: 0,1:30:44.34,1:30:47.56,Default,,0000,0000,0000,,today, a cosine t plus 0.
Dialogue: 0,1:30:47.56,1:30:52.38,Default,,0000,0000,0000,,And y of t equals\Na sine t plus y 0.
Dialogue: 0,1:30:52.38,1:30:53.99,Default,,0000,0000,0000,,What is this, guys?
Dialogue: 0,1:30:53.99,1:31:04.50,Default,,0000,0000,0000,,This is a circle, a center\Nat 0, y, 0, and radius a.
Dialogue: 0,1:31:04.50,1:31:08.63,Default,,0000,0000,0000,,
Dialogue: 0,1:31:08.63,1:31:14.92,Default,,0000,0000,0000,,Can use a better formula-- that\Nanticipated my action today--
Dialogue: 0,1:31:14.92,1:31:18.54,Default,,0000,0000,0000,,to actually prove that k\Nis going to be [? 1/a? ?]
Dialogue: 0,1:31:18.54,1:31:19.26,Default,,0000,0000,0000,,Precisely.
Dialogue: 0,1:31:19.26,1:31:20.55,Default,,0000,0000,0000,,Can we do that in the exam?
Dialogue: 0,1:31:20.55,1:31:23.12,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:31:23.12,1:31:25.49,Default,,0000,0000,0000,,So while I told\Nyou a long time ago
Dialogue: 0,1:31:25.49,1:31:28.59,Default,,0000,0000,0000,,that engineers and\Nmathematicians observed
Dialogue: 0,1:31:28.59,1:31:31.00,Default,,0000,0000,0000,,hundreds of years\Nago-- actually,
Dialogue: 0,1:31:31.00,1:31:32.77,Default,,0000,0000,0000,,somebody said, no,\Nyou're not right.
Dialogue: 0,1:31:32.77,1:31:35.01,Default,,0000,0000,0000,,The Egyptians already saw that.
Dialogue: 0,1:31:35.01,1:31:38.24,Default,,0000,0000,0000,,They had the notion of\Ninverse proportionality
Dialogue: 0,1:31:38.24,1:31:42.39,Default,,0000,0000,0000,,in Egypt, which makes sense\Nif you look at the pyramids.
Dialogue: 0,1:31:42.39,1:31:47.75,Default,,0000,0000,0000,,So one look at the radius,\Nit says if the radius is 2,
Dialogue: 0,1:31:47.75,1:31:50.64,Default,,0000,0000,0000,,then the curvature\Nis not very bent.
Dialogue: 0,1:31:50.64,1:31:52.76,Default,,0000,0000,0000,,So the curvature's inverse\Nproportion [INAUDIBLE]
Dialogue: 0,1:31:52.76,1:31:53.48,Default,,0000,0000,0000,,the radius.
Dialogue: 0,1:31:53.48,1:31:57.48,Default,,0000,0000,0000,,So if this is 2, we said\Nthe curvature's 1/2.
Dialogue: 0,1:31:57.48,1:32:01.68,Default,,0000,0000,0000,,If you take a big\Ncircle, the bigger
Dialogue: 0,1:32:01.68,1:32:04.06,Default,,0000,0000,0000,,the radius, the\Nsmaller the bending
Dialogue: 0,1:32:04.06,1:32:07.64,Default,,0000,0000,0000,,of the arc of the circle,\Nthe smaller of the curvature.
Dialogue: 0,1:32:07.64,1:32:10.67,Default,,0000,0000,0000,,Apparently the ancient\Nworld knew that already.
Dialogue: 0,1:32:10.67,1:32:12.20,Default,,0000,0000,0000,,They Egyptians knew that.
Dialogue: 0,1:32:12.20,1:32:13.14,Default,,0000,0000,0000,,The Greeks knew that.
Dialogue: 0,1:32:13.14,1:32:15.43,Default,,0000,0000,0000,,But I think they\Nnever formalized it--
Dialogue: 0,1:32:15.43,1:32:16.62,Default,,0000,0000,0000,,not that I know.
Dialogue: 0,1:32:16.62,1:32:19.29,Default,,0000,0000,0000,,
Dialogue: 0,1:32:19.29,1:32:24.53,Default,,0000,0000,0000,,So if you are asked to\Ndo this in any exam,
Dialogue: 0,1:32:24.53,1:32:26.94,Default,,0000,0000,0000,,do you think that\Nwould be a problem?
Dialogue: 0,1:32:26.94,1:32:28.15,Default,,0000,0000,0000,,Of course we would do review.
Dialogue: 0,1:32:28.15,1:32:31.73,Default,,0000,0000,0000,,Because people are going to\Nforget this formula, or even
Dialogue: 0,1:32:31.73,1:32:33.37,Default,,0000,0000,0000,,the definition.
Dialogue: 0,1:32:33.37,1:32:36.01,Default,,0000,0000,0000,,You can compute k\Nfor this formula.
Dialogue: 0,1:32:36.01,1:32:39.33,Default,,0000,0000,0000,,And we are going to\Nget k to the 1/a.
Dialogue: 0,1:32:39.33,1:32:41.34,Default,,0000,0000,0000,,This is a piece\Nof cake, actually.
Dialogue: 0,1:32:41.34,1:32:44.11,Default,,0000,0000,0000,,You may not believe me, but\Nonce you plug in the equations
Dialogue: 0,1:32:44.11,1:32:46.29,Default,,0000,0000,0000,,it's very easy.
Dialogue: 0,1:32:46.29,1:32:48.42,Default,,0000,0000,0000,,Or you can do it\Nfrom the definition
Dialogue: 0,1:32:48.42,1:32:50.64,Default,,0000,0000,0000,,that gives you k of s.
Dialogue: 0,1:32:50.64,1:32:52.56,Default,,0000,0000,0000,,You'll reparametrize\Nthis in arclength.
Dialogue: 0,1:32:52.56,1:32:54.71,Default,,0000,0000,0000,,You can do that as well.
Dialogue: 0,1:32:54.71,1:32:57.60,Default,,0000,0000,0000,,And you still get 1/a.
Dialogue: 0,1:32:57.60,1:32:59.88,Default,,0000,0000,0000,,The question that\NI got by email,
Dialogue: 0,1:32:59.88,1:33:01.44,Default,,0000,0000,0000,,and I get a lot of email.
Dialogue: 0,1:33:01.44,1:33:04.66,Default,,0000,0000,0000,,I told you, that\Nkeeps me busy a lot,
Dialogue: 0,1:33:04.66,1:33:08.49,Default,,0000,0000,0000,,about 200 emails every day.
Dialogue: 0,1:33:08.49,1:33:10.68,Default,,0000,0000,0000,,I really like the emails\NI get from students,
Dialogue: 0,1:33:10.68,1:33:13.62,Default,,0000,0000,0000,,because I get emails from\Nall sorts of sources--
Dialogue: 0,1:33:13.62,1:33:15.34,Default,,0000,0000,0000,,Got some spam also.
Dialogue: 0,1:33:15.34,1:33:21.78,Default,,0000,0000,0000,,Anyway, what I'm trying to say,\NI got this question last time
Dialogue: 0,1:33:21.78,1:33:24.68,Default,,0000,0000,0000,,saying, if on the midterm\Nwe get such a question,
Dialogue: 0,1:33:24.68,1:33:28.07,Default,,0000,0000,0000,,can we say simply, curvature's\N1/a, a is the radius.
Dialogue: 0,1:33:28.07,1:33:30.82,Default,,0000,0000,0000,,Is that enough?
Dialogue: 0,1:33:30.82,1:33:34.25,Default,,0000,0000,0000,,Depends on how the\Nproblem was formulated.
Dialogue: 0,1:33:34.25,1:33:39.36,Default,,0000,0000,0000,,Most likely I'm going to make\Nit through that or show that.
Dialogue: 0,1:33:39.36,1:33:43.45,Default,,0000,0000,0000,,Even if you state something,\Nlike, yes, it's 1/a,
Dialogue: 0,1:33:43.45,1:33:46.27,Default,,0000,0000,0000,,with a little argument,\Nit's inverse proportional
Dialogue: 0,1:33:46.27,1:33:50.38,Default,,0000,0000,0000,,to the radius, I will\Nstill give partial credit.
Dialogue: 0,1:33:50.38,1:33:53.65,Default,,0000,0000,0000,,For any argument that\Nis valid, especially
Dialogue: 0,1:33:53.65,1:33:56.39,Default,,0000,0000,0000,,if it's based on\Nempirical observation,
Dialogue: 0,1:33:56.39,1:33:58.92,Default,,0000,0000,0000,,I do give some extra\Ncredit, even if you didn't
Dialogue: 0,1:33:58.92,1:34:02.60,Default,,0000,0000,0000,,use the specific formula.
Dialogue: 0,1:34:02.60,1:34:04.91,Default,,0000,0000,0000,,Let's see one example.
Dialogue: 0,1:34:04.91,1:34:07.62,Default,,0000,0000,0000,,Let's take y equals e to the x.
Dialogue: 0,1:34:07.62,1:34:11.87,Default,,0000,0000,0000,,
Dialogue: 0,1:34:11.87,1:34:15.73,Default,,0000,0000,0000,,No, let's take e\Nto the negative x.
Dialogue: 0,1:34:15.73,1:34:16.69,Default,,0000,0000,0000,,Doesn't matter.
Dialogue: 0,1:34:16.69,1:34:20.56,Default,,0000,0000,0000,,
Dialogue: 0,1:34:20.56,1:34:26.38,Default,,0000,0000,0000,,y equals e to the negative x.
Dialogue: 0,1:34:26.38,1:34:30.83,Default,,0000,0000,0000,,And let's make x\Nbetween 0 and 1.
Dialogue: 0,1:34:30.83,1:34:35.29,Default,,0000,0000,0000,,
Dialogue: 0,1:34:35.29,1:34:36.97,Default,,0000,0000,0000,,I'll say, write the curvature.
Dialogue: 0,1:34:36.97,1:34:40.61,Default,,0000,0000,0000,,
Dialogue: 0,1:34:40.61,1:34:45.25,Default,,0000,0000,0000,,Write the equation or the\Nformula of the curvature.
Dialogue: 0,1:34:45.25,1:34:50.23,Default,,0000,0000,0000,,
Dialogue: 0,1:34:50.23,1:34:54.70,Default,,0000,0000,0000,,And I know it's 2 o'clock\Nand I am answering questions.
Dialogue: 0,1:34:54.70,1:34:58.07,Default,,0000,0000,0000,,This was a question that one of\Nyou had during the short break
Dialogue: 0,1:34:58.07,1:34:59.30,Default,,0000,0000,0000,,we took.
Dialogue: 0,1:34:59.30,1:35:00.48,Default,,0000,0000,0000,,Can we do such a problem?
Dialogue: 0,1:35:00.48,1:35:01.54,Default,,0000,0000,0000,,Like she said.
Dialogue: 0,1:35:01.54,1:35:02.96,Default,,0000,0000,0000,,Yes, I [INAUDIBLE]\Nto the negative
Dialogue: 0,1:35:02.96,1:35:05.52,Default,,0000,0000,0000,,x because I want\Nto catch somebody
Dialogue: 0,1:35:05.52,1:35:06.79,Default,,0000,0000,0000,,not knowing the derivative.
Dialogue: 0,1:35:06.79,1:35:08.99,Default,,0000,0000,0000,,I don't know why I'm doing this.
Dialogue: 0,1:35:08.99,1:35:10.51,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:35:10.51,1:35:15.48,Default,,0000,0000,0000,,So if I were to draw that, OK,\Ntry and draw that, but not now.
Dialogue: 0,1:35:15.48,1:35:18.62,Default,,0000,0000,0000,,Now, what formula\Nare you going to use?
Dialogue: 0,1:35:18.62,1:35:21.86,Default,,0000,0000,0000,,Of course, you could\Ndo this in many ways.
Dialogue: 0,1:35:21.86,1:35:24.51,Default,,0000,0000,0000,,All those formulas are\Nequivalent for the curvature.
Dialogue: 0,1:35:24.51,1:35:27.36,Default,,0000,0000,0000,,What's the simplest\Nway to do it?
Dialogue: 0,1:35:27.36,1:35:30.02,Default,,0000,0000,0000,,Do y prime.
Dialogue: 0,1:35:30.02,1:35:33.77,Default,,0000,0000,0000,,Minus it to the minus x.
Dialogue: 0,1:35:33.77,1:35:36.79,Default,,0000,0000,0000,,Note here in this problem that\Neven if you mess up and forget
Dialogue: 0,1:35:36.79,1:35:39.95,Default,,0000,0000,0000,,the minus sign, you still\Nget the final answer correct.
Dialogue: 0,1:35:39.95,1:35:46.05,Default,,0000,0000,0000,,But I may subtract a few points\Nif I see something nonsensical.
Dialogue: 0,1:35:46.05,1:35:47.30,Default,,0000,0000,0000,,y double prime equals--
Dialogue: 0,1:35:47.30,1:35:48.87,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:35:48.87,1:35:51.66,Default,,0000,0000,0000,,--plus e to the minus x.
Dialogue: 0,1:35:51.66,1:35:56.74,Default,,0000,0000,0000,,And what is the\Ncurvature k of t?
Dialogue: 0,1:35:56.74,1:35:59.20,Default,,0000,0000,0000,,STUDENT: y prime over--
Dialogue: 0,1:35:59.20,1:36:01.34,Default,,0000,0000,0000,,PROFESSOR TODA: Oh, I\Ndidn't say one more thing.
Dialogue: 0,1:36:01.34,1:36:04.62,Default,,0000,0000,0000,,I want the curvature, but\NI also want the curvature
Dialogue: 0,1:36:04.62,1:36:08.49,Default,,0000,0000,0000,,in three separate moments,\Nin the beginning, in the end,
Dialogue: 0,1:36:08.49,1:36:09.36,Default,,0000,0000,0000,,and in the middle.
Dialogue: 0,1:36:09.36,1:36:11.43,Default,,0000,0000,0000,,STUDENT: Don't we\Nneed to parametrize it
Dialogue: 0,1:36:11.43,1:36:15.25,Default,,0000,0000,0000,,so we can [INAUDIBLE]\Nx prime [INAUDIBLE]?
Dialogue: 0,1:36:15.25,1:36:16.74,Default,,0000,0000,0000,,PROFESSOR TODA: No.
Dialogue: 0,1:36:16.74,1:36:18.34,Default,,0000,0000,0000,,Did I erase it?
Dialogue: 0,1:36:18.34,1:36:19.33,Default,,0000,0000,0000,,STUDENT: Yeah, you did.
Dialogue: 0,1:36:19.33,1:36:20.71,Default,,0000,0000,0000,,PROFESSOR TODA: [INAUDIBLE].
Dialogue: 0,1:36:20.71,1:36:24.36,Default,,0000,0000,0000,,And one of my colleagues\Nsaid, Magda, you are smart,
Dialogue: 0,1:36:24.36,1:36:28.00,Default,,0000,0000,0000,,but you are like one\Nof those people who,
Dialogue: 0,1:36:28.00,1:36:29.73,Default,,0000,0000,0000,,in the anecdotes\Nabout math professors,
Dialogue: 0,1:36:29.73,1:36:31.52,Default,,0000,0000,0000,,gets out of their office\Nand starts walking
Dialogue: 0,1:36:31.52,1:36:32.93,Default,,0000,0000,0000,,and stops a student.
Dialogue: 0,1:36:32.93,1:36:34.84,Default,,0000,0000,0000,,Was I going this\Nway or that way?
Dialogue: 0,1:36:34.84,1:36:36.07,Default,,0000,0000,0000,,And that's me.
Dialogue: 0,1:36:36.07,1:36:37.65,Default,,0000,0000,0000,,And I'm sorry about that.
Dialogue: 0,1:36:37.65,1:36:41.93,Default,,0000,0000,0000,,I should not have erased that.
Dialogue: 0,1:36:41.93,1:36:44.00,Default,,0000,0000,0000,,I'm going to go\Nahead and rewrite it,
Dialogue: 0,1:36:44.00,1:36:48.10,Default,,0000,0000,0000,,because I'm a goofball.
Dialogue: 0,1:36:48.10,1:36:55.50,Default,,0000,0000,0000,,So the one that I wanted to use\Nk of x will be f double prime.
Dialogue: 0,1:36:55.50,1:36:56.98,Default,,0000,0000,0000,,STUDENT: And cubed.
Dialogue: 0,1:36:56.98,1:36:58.45,Default,,0000,0000,0000,,PROFESSOR TODA: Cubed!
Dialogue: 0,1:36:58.45,1:36:59.44,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,1:36:59.44,1:37:02.92,Default,,0000,0000,0000,,
Dialogue: 0,1:37:02.92,1:37:09.64,Default,,0000,0000,0000,,So that 3/2, remember it,\N[INAUDIBLE] 3/2 [INAUDIBLE]
Dialogue: 0,1:37:09.64,1:37:10.99,Default,,0000,0000,0000,,square root cubed.
Dialogue: 0,1:37:10.99,1:37:13.75,Default,,0000,0000,0000,,Now, for this one, is it hard?
Dialogue: 0,1:37:13.75,1:37:15.08,Default,,0000,0000,0000,,No.
Dialogue: 0,1:37:15.08,1:37:16.19,Default,,0000,0000,0000,,That's a piece of cake.
Dialogue: 0,1:37:16.19,1:37:18.61,Default,,0000,0000,0000,,I said I like it in\Ngeneral, but I also
Dialogue: 0,1:37:18.61,1:37:22.91,Default,,0000,0000,0000,,like it-- find the curvature\Nof this curve in the beginning.
Dialogue: 0,1:37:22.91,1:37:24.10,Default,,0000,0000,0000,,You travel on me.
Dialogue: 0,1:37:24.10,1:37:27.59,Default,,0000,0000,0000,,From time 0 to 1\No'clock, whatever.
Dialogue: 0,1:37:27.59,1:37:28.43,Default,,0000,0000,0000,,One second.
Dialogue: 0,1:37:28.43,1:37:32.77,Default,,0000,0000,0000,,That's saying this is in seconds\Nto make it more physical.
Dialogue: 0,1:37:32.77,1:37:39.95,Default,,0000,0000,0000,,I want the k at 0, I want k\Nat 1/2, and I want k at 1.
Dialogue: 0,1:37:39.95,1:37:42.43,Default,,0000,0000,0000,,And I'd like you to\Ncompare those values.
Dialogue: 0,1:37:42.43,1:37:46.27,Default,,0000,0000,0000,,
Dialogue: 0,1:37:46.27,1:37:49.12,Default,,0000,0000,0000,,And I'll give you one\Nmore task after that.
Dialogue: 0,1:37:49.12,1:37:50.58,Default,,0000,0000,0000,,But let me start working.
Dialogue: 0,1:37:50.58,1:37:53.09,Default,,0000,0000,0000,,So you say you help me on that.
Dialogue: 0,1:37:53.09,1:37:55.10,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:37:55.10,1:38:02.52,Default,,0000,0000,0000,,Minus x over square\Nroot of 1 plus--
Dialogue: 0,1:38:02.52,1:38:04.38,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:38:04.38,1:38:05.31,Default,,0000,0000,0000,,PROFESSOR TODA: Right.
Dialogue: 0,1:38:05.31,1:38:08.19,Default,,0000,0000,0000,,So can I write this differently,\Na little bit differently?
Dialogue: 0,1:38:08.19,1:38:09.94,Default,,0000,0000,0000,,Like what?
Dialogue: 0,1:38:09.94,1:38:12.02,Default,,0000,0000,0000,,I don't want to square\Neach of the minus 2x.
Dialogue: 0,1:38:12.02,1:38:14.35,Default,,0000,0000,0000,,Can I do that?
Dialogue: 0,1:38:14.35,1:38:19.54,Default,,0000,0000,0000,,And then the whole thing\NI can say to the 3/2
Dialogue: 0,1:38:19.54,1:38:24.53,Default,,0000,0000,0000,,or I can use the square root,\Nwhichever is your favorite.
Dialogue: 0,1:38:24.53,1:38:28.99,Default,,0000,0000,0000,,Now, what is k of 0?
Dialogue: 0,1:38:28.99,1:38:29.49,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:38:29.49,1:38:32.99,Default,,0000,0000,0000,,Or 1.
Dialogue: 0,1:38:32.99,1:38:34.46,Default,,0000,0000,0000,,PROFESSOR TODA: Really?
Dialogue: 0,1:38:34.46,1:38:36.23,Default,,0000,0000,0000,,STUDENT: 1/2.
Dialogue: 0,1:38:36.23,1:38:36.96,Default,,0000,0000,0000,,3/2.
Dialogue: 0,1:38:36.96,1:38:38.71,Default,,0000,0000,0000,,PROFESSOR TODA: So\Nlet's take this slowly.
Dialogue: 0,1:38:38.71,1:38:43.58,Default,,0000,0000,0000,,Because we can all make\Nmistakes, goofy mistakes.
Dialogue: 0,1:38:43.58,1:38:45.00,Default,,0000,0000,0000,,That doesn't mean\Nwe're not smart.
Dialogue: 0,1:38:45.00,1:38:46.77,Default,,0000,0000,0000,,We're very smart, right?
Dialogue: 0,1:38:46.77,1:38:51.21,Default,,0000,0000,0000,,But it's just a matter of\Nbook-keeping and paying
Dialogue: 0,1:38:51.21,1:38:53.01,Default,,0000,0000,0000,,attention, being attentive.
Dialogue: 0,1:38:53.01,1:38:55.42,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:38:55.42,1:39:00.24,Default,,0000,0000,0000,,When I take 0 and replace--\Nthis is drying fast.
Dialogue: 0,1:39:00.24,1:39:02.86,Default,,0000,0000,0000,,I'm trying to draw it.
Dialogue: 0,1:39:02.86,1:39:10.00,Default,,0000,0000,0000,,I have 1 over 1\Nplus 1 to the 3/2.
Dialogue: 0,1:39:10.00,1:39:15.16,Default,,0000,0000,0000,,I have a student in one exam\Nwho was just-- I don't know.
Dialogue: 0,1:39:15.16,1:39:16.58,Default,,0000,0000,0000,,He was rushing.
Dialogue: 0,1:39:16.58,1:39:20.95,Default,,0000,0000,0000,,He didn't realize that\Nhe had to take it slowly.
Dialogue: 0,1:39:20.95,1:39:22.79,Default,,0000,0000,0000,,He was extremely smart, though.
Dialogue: 0,1:39:22.79,1:39:29.76,Default,,0000,0000,0000,,1 over-- you have\Nthat 1 plus 1 is 2.
Dialogue: 0,1:39:29.76,1:39:33.71,Default,,0000,0000,0000,,2 to the 1/2 would be\Nsquare root of 2 cubed.
Dialogue: 0,1:39:33.71,1:39:35.89,Default,,0000,0000,0000,,It would be exactly\N2 square root of 2.
Dialogue: 0,1:39:35.89,1:39:39.75,Default,,0000,0000,0000,,And more you can write\Nthis as rationalized.
Dialogue: 0,1:39:39.75,1:39:42.05,Default,,0000,0000,0000,,Now, I have a question for you.
Dialogue: 0,1:39:42.05,1:39:43.01,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:39:43.01,1:39:47.10,Default,,0000,0000,0000,,I'm When we were kids, if you\Nremember-- you are too young.
Dialogue: 0,1:39:47.10,1:39:48.14,Default,,0000,0000,0000,,Maybe you don't remember.
Dialogue: 0,1:39:48.14,1:39:52.71,Default,,0000,0000,0000,,But I remember when I was a kid,\Nmy teacher would always ask me,
Dialogue: 0,1:39:52.71,1:39:53.98,Default,,0000,0000,0000,,rationalize your answer.
Dialogue: 0,1:39:53.98,1:39:56.92,Default,,0000,0000,0000,,Rationalize your answer.
Dialogue: 0,1:39:56.92,1:40:00.36,Default,,0000,0000,0000,,Put the rational number\Nin the denominator.
Dialogue: 0,1:40:00.36,1:40:02.74,Default,,0000,0000,0000,,Why do you think that was?
Dialogue: 0,1:40:02.74,1:40:05.16,Default,,0000,0000,0000,,For hundreds of years\Npeople did that.
Dialogue: 0,1:40:05.16,1:40:07.43,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:40:07.43,1:40:11.90,Default,,0000,0000,0000,,PROFESSOR TODA: Because they\Ndidn't have a calculator.
Dialogue: 0,1:40:11.90,1:40:16.30,Default,,0000,0000,0000,,So we used to, even I used to be\Nable to get the square root out
Dialogue: 0,1:40:16.30,1:40:17.52,Default,,0000,0000,0000,,by hand.
Dialogue: 0,1:40:17.52,1:40:20.92,Default,,0000,0000,0000,,Has anybody taught you how to\Ncompute square root by hand?
Dialogue: 0,1:40:20.92,1:40:21.65,Default,,0000,0000,0000,,You know that.
Dialogue: 0,1:40:21.65,1:40:22.38,Default,,0000,0000,0000,,Who taught you?
Dialogue: 0,1:40:22.38,1:40:23.35,Default,,0000,0000,0000,,STUDENT: I don't remember it.
Dialogue: 0,1:40:23.35,1:40:24.94,Default,,0000,0000,0000,,My seventh grade\Nteacher taught us.
Dialogue: 0,1:40:24.94,1:40:26.44,Default,,0000,0000,0000,,PROFESSOR TODA:\NThere is a technique
Dialogue: 0,1:40:26.44,1:40:29.45,Default,,0000,0000,0000,,of taking groups of twos\Nand then fitting the--
Dialogue: 0,1:40:29.45,1:40:31.02,Default,,0000,0000,0000,,and they still teach that.
Dialogue: 0,1:40:31.02,1:40:33.25,Default,,0000,0000,0000,,I was amazed, they\Nstill teach that
Dialogue: 0,1:40:33.25,1:40:35.46,Default,,0000,0000,0000,,in half of the Asian countries.
Dialogue: 0,1:40:35.46,1:40:39.36,Default,,0000,0000,0000,,And it's hard, but kids\Nin fifth and sixth grade
Dialogue: 0,1:40:39.36,1:40:45.19,Default,,0000,0000,0000,,have that practice, which some\Nof us learned and forgot about.
Dialogue: 0,1:40:45.19,1:40:50.03,Default,,0000,0000,0000,,So imagine that how people would\Nhave done this, and of course,
Dialogue: 0,1:40:50.03,1:40:51.07,Default,,0000,0000,0000,,square root of 2 is easy.
Dialogue: 0,1:40:51.07,1:40:53.55,Default,,0000,0000,0000,,1.4142, blah blah blah.
Dialogue: 0,1:40:53.55,1:40:54.25,Default,,0000,0000,0000,,Divide by 2.
Dialogue: 0,1:40:54.25,1:40:56.35,Default,,0000,0000,0000,,You can do it by hand.
Dialogue: 0,1:40:56.35,1:40:57.86,Default,,0000,0000,0000,,At least a good approximation.
Dialogue: 0,1:40:57.86,1:41:01.97,Default,,0000,0000,0000,,But imagine having a nasty\Nsquare root there to compute,
Dialogue: 0,1:41:01.97,1:41:05.85,Default,,0000,0000,0000,,and then you would divide\Nby that natural number.
Dialogue: 0,1:41:05.85,1:41:09.14,Default,,0000,0000,0000,,You have to rely on your\Nown computation to do it.
Dialogue: 0,1:41:09.14,1:41:11.19,Default,,0000,0000,0000,,There were no calculators.
Dialogue: 0,1:41:11.19,1:41:14.10,Default,,0000,0000,0000,,How about k of 1?
Dialogue: 0,1:41:14.10,1:41:14.60,Default,,0000,0000,0000,,How is that?
Dialogue: 0,1:41:14.60,1:41:15.56,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:41:15.56,1:41:20.38,Default,,0000,0000,0000,,
Dialogue: 0,1:41:20.38,1:41:23.76,Default,,0000,0000,0000,,e to the minus 1.
Dialogue: 0,1:41:23.76,1:41:26.54,Default,,0000,0000,0000,,That's a little bit\Nharder to compute, right?
Dialogue: 0,1:41:26.54,1:41:28.61,Default,,0000,0000,0000,,1 plus [INAUDIBLE].
Dialogue: 0,1:41:28.61,1:41:31.76,Default,,0000,0000,0000,,What is that going to be?
Dialogue: 0,1:41:31.76,1:41:34.04,Default,,0000,0000,0000,,Minus 2.
Dialogue: 0,1:41:34.04,1:41:37.14,Default,,0000,0000,0000,,Replace it by 1 to the 3/2.
Dialogue: 0,1:41:37.14,1:41:41.93,Default,,0000,0000,0000,,I would like you to go\Nhome and do the following.
Dialogue: 0,1:41:41.93,1:41:45.81,Default,,0000,0000,0000,,[INAUDIBLE]-- Not now, not now.
Dialogue: 0,1:41:45.81,1:41:48.31,Default,,0000,0000,0000,,We stay a little\Nbit longer together.
Dialogue: 0,1:41:48.31,1:41:51.90,Default,,0000,0000,0000,,k of 0, k of 1/2, and k of 1.
Dialogue: 0,1:41:51.90,1:41:53.03,Default,,0000,0000,0000,,Which one is bigger?
Dialogue: 0,1:41:53.03,1:41:59.93,Default,,0000,0000,0000,,
Dialogue: 0,1:41:59.93,1:42:03.40,Default,,0000,0000,0000,,And one last question about\Nthat, how much extra credit
Dialogue: 0,1:42:03.40,1:42:04.22,Default,,0000,0000,0000,,should I give you?
Dialogue: 0,1:42:04.22,1:42:05.87,Default,,0000,0000,0000,,One point?
Dialogue: 0,1:42:05.87,1:42:08.42,Default,,0000,0000,0000,,One point if you turn this in.
Dialogue: 0,1:42:08.42,1:42:11.20,Default,,0000,0000,0000,,Um, yeah.
Dialogue: 0,1:42:11.20,1:42:13.38,Default,,0000,0000,0000,,Four, [? maybe ?] two points.
Dialogue: 0,1:42:13.38,1:42:19.43,Default,,0000,0000,0000,,Compare all these\Nthree values, and find
Dialogue: 0,1:42:19.43,1:42:32.79,Default,,0000,0000,0000,,the maximum and the\Nminimum of kappa of t,
Dialogue: 0,1:42:32.79,1:42:37.53,Default,,0000,0000,0000,,kappa of x, for\Nthe interval where
Dialogue: 0,1:42:37.53,1:42:46.69,Default,,0000,0000,0000,,x is in the interval 0, 1.
Dialogue: 0,1:42:46.69,1:42:48.87,Default,,0000,0000,0000,,0, closed 1.
Dialogue: 0,1:42:48.87,1:42:49.94,Default,,0000,0000,0000,,Close it.
Dialogue: 0,1:42:49.94,1:42:53.93,Default,,0000,0000,0000,,Now, don't ask me,\Nbecause it's extra credit.
Dialogue: 0,1:42:53.93,1:42:58.74,Default,,0000,0000,0000,,One question was, by email,\Ncan I ask my tutor to help me?
Dialogue: 0,1:42:58.74,1:43:02.17,Default,,0000,0000,0000,,As long as your tutor doesn't\Nwrite down your solution,
Dialogue: 0,1:43:02.17,1:43:03.72,Default,,0000,0000,0000,,you are in good shape.
Dialogue: 0,1:43:03.72,1:43:07.02,Default,,0000,0000,0000,,Your tutor should help you\Nunderstand some constants,
Dialogue: 0,1:43:07.02,1:43:08.29,Default,,0000,0000,0000,,spend time with you.
Dialogue: 0,1:43:08.29,1:43:12.77,Default,,0000,0000,0000,,But they should not write\Nyour assignment themselves.
Dialogue: 0,1:43:12.77,1:43:13.27,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:43:13.27,1:43:16.54,Default,,0000,0000,0000,,So it's not a big deal.
Dialogue: 0,1:43:16.54,1:43:22.03,Default,,0000,0000,0000,,Not I want to tell you one\Nsecret that I normally don't
Dialogue: 0,1:43:22.03,1:43:26.73,Default,,0000,0000,0000,,tell my Calculus 3 students.
Dialogue: 0,1:43:26.73,1:43:29.37,Default,,0000,0000,0000,,But the more I get\Nto know you, the more
Dialogue: 0,1:43:29.37,1:43:34.02,Default,,0000,0000,0000,,I realize that you are worth\Nme telling you about that.
Dialogue: 0,1:43:34.02,1:43:35.42,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:43:35.42,1:43:38.08,Default,,0000,0000,0000,,PROFESSOR TODA: No.
Dialogue: 0,1:43:38.08,1:43:41.90,Default,,0000,0000,0000,,There is a beautiful\Ntheory that engineers
Dialogue: 0,1:43:41.90,1:43:48.14,Default,,0000,0000,0000,,use when they start the motions\Nof curves and parametrizations
Dialogue: 0,1:43:48.14,1:43:51.06,Default,,0000,0000,0000,,in space.
Dialogue: 0,1:43:51.06,1:43:53.42,Default,,0000,0000,0000,,And that includes\Nthe Frenet formulas.
Dialogue: 0,1:43:53.42,1:43:56.05,Default,,0000,0000,0000,,
Dialogue: 0,1:43:56.05,1:43:58.51,Default,,0000,0000,0000,,And you already\Nknow the first one.
Dialogue: 0,1:43:58.51,1:44:04.52,Default,,0000,0000,0000,,And I was debating, I was just\Nreviewing what I taught you,
Dialogue: 0,1:44:04.52,1:44:07.32,Default,,0000,0000,0000,,and I was happy with\Nwhat I taught you.
Dialogue: 0,1:44:07.32,1:44:10.58,Default,,0000,0000,0000,,And I said, they know\Nabout position vector.
Dialogue: 0,1:44:10.58,1:44:13.10,Default,,0000,0000,0000,,They know about\Nvelocity, acceleration.
Dialogue: 0,1:44:13.10,1:44:15.93,Default,,0000,0000,0000,,They know how to get back\Nand forth from one another.
Dialogue: 0,1:44:15.93,1:44:16.76,Default,,0000,0000,0000,,They know our claim.
Dialogue: 0,1:44:16.76,1:44:18.47,Default,,0000,0000,0000,,They know how to\N[? reparameterize our ?]
Dialogue: 0,1:44:18.47,1:44:19.94,Default,,0000,0000,0000,,claims.
Dialogue: 0,1:44:19.94,1:44:25.04,Default,,0000,0000,0000,,They know the [INAUDIBLE]\Nand B. They know already
Dialogue: 0,1:44:25.04,1:44:27.01,Default,,0000,0000,0000,,the first Frenet formula.
Dialogue: 0,1:44:27.01,1:44:28.01,Default,,0000,0000,0000,,They know the curvature.
Dialogue: 0,1:44:28.01,1:44:29.99,Default,,0000,0000,0000,,What else can I teach them?
Dialogue: 0,1:44:29.99,1:44:34.07,Default,,0000,0000,0000,,I want to show you--\None of you asked me,
Dialogue: 0,1:44:34.07,1:44:38.28,Default,,0000,0000,0000,,is this all that we should know?
Dialogue: 0,1:44:38.28,1:44:41.88,Default,,0000,0000,0000,,This is all that a regular\Nstudent should know in Calculus
Dialogue: 0,1:44:41.88,1:44:43.75,Default,,0000,0000,0000,,3, but there is more.
Dialogue: 0,1:44:43.75,1:44:44.87,Default,,0000,0000,0000,,And you are honor students.
Dialogue: 0,1:44:44.87,1:44:49.93,Default,,0000,0000,0000,,And I want to show you some\Nbeautiful equations here.
Dialogue: 0,1:44:49.93,1:44:54.93,Default,,0000,0000,0000,,So do you remember that\Nif I introduce r of s
Dialogue: 0,1:44:54.93,1:45:03.90,Default,,0000,0000,0000,,as a curving arclength,\Nthat is a regular curve.
Dialogue: 0,1:45:03.90,1:45:11.05,Default,,0000,0000,0000,,I said there is a certain famous\Nformula that is T prime of s
Dialogue: 0,1:45:11.05,1:45:13.90,Default,,0000,0000,0000,,called-- leave space.
Dialogue: 0,1:45:13.90,1:45:15.32,Default,,0000,0000,0000,,Leave a little bit of space.
Dialogue: 0,1:45:15.32,1:45:15.94,Default,,0000,0000,0000,,You'll see why.
Dialogue: 0,1:45:15.94,1:45:17.85,Default,,0000,0000,0000,,It's a surprise.
Dialogue: 0,1:45:17.85,1:45:22.95,Default,,0000,0000,0000,,k times-- why\Ndon't I say k of s?
Dialogue: 0,1:45:22.95,1:45:26.22,Default,,0000,0000,0000,,Because I want to point\Nout that k is an invariant.
Dialogue: 0,1:45:26.22,1:45:28.58,Default,,0000,0000,0000,,Even if you have\Nanother parameter,
Dialogue: 0,1:45:28.58,1:45:29.90,Default,,0000,0000,0000,,would be the same function.
Dialogue: 0,1:45:29.90,1:45:38.56,Default,,0000,0000,0000,,But yes, as a function of s,\Nwould be k times N bar, bar.
Dialogue: 0,1:45:38.56,1:45:40.91,Default,,0000,0000,0000,,More bars because\Nthey are free vectors.
Dialogue: 0,1:45:40.91,1:45:42.92,Default,,0000,0000,0000,,They are not bound\Nto a certain point.
Dialogue: 0,1:45:42.92,1:45:44.55,Default,,0000,0000,0000,,They're not married\Nto a certain point.
Dialogue: 0,1:45:44.55,1:45:49.18,Default,,0000,0000,0000,,They are free to shift\Nby parallelism in space.
Dialogue: 0,1:45:49.18,1:45:54.25,Default,,0000,0000,0000,,However, I'm going to review\Nthem as bound at the point
Dialogue: 0,1:45:54.25,1:45:55.20,Default,,0000,0000,0000,,where they are.
Dialogue: 0,1:45:55.20,1:45:58.11,Default,,0000,0000,0000,,So they-- no way they\Nare married to the point
Dialogue: 0,1:45:58.11,1:46:03.64,Default,,0000,0000,0000,,that they belong to.
Dialogue: 0,1:46:03.64,1:46:07.20,Default,,0000,0000,0000,,Maybe the [? bend ?]\Nwill change.
Dialogue: 0,1:46:07.20,1:46:09.39,Default,,0000,0000,0000,,I don't know how it's\Ngoing to change like crazy.
Dialogue: 0,1:46:09.39,1:46:18.17,Default,,0000,0000,0000,,
Dialogue: 0,1:46:18.17,1:46:19.25,Default,,0000,0000,0000,,Something like that.
Dialogue: 0,1:46:19.25,1:46:26.82,Default,,0000,0000,0000,,At every point you have a T, an\NN, and it's a 90 degree angle.
Dialogue: 0,1:46:26.82,1:46:30.90,Default,,0000,0000,0000,,Then you have the binormal,\Nwhich makes a 90 degree
Dialogue: 0,1:46:30.90,1:46:33.28,Default,,0000,0000,0000,,angle-- [INAUDIBLE].
Dialogue: 0,1:46:33.28,1:46:36.86,Default,,0000,0000,0000,,So the way you should\Nimagine these corners
Dialogue: 0,1:46:36.86,1:46:39.36,Default,,0000,0000,0000,,would be something\Nlike that, right?
Dialogue: 0,1:46:39.36,1:46:40.86,Default,,0000,0000,0000,,90-90-90.
Dialogue: 0,1:46:40.86,1:46:43.36,Default,,0000,0000,0000,,It's just hard to draw them.
Dialogue: 0,1:46:43.36,1:46:51.73,Default,,0000,0000,0000,,Between the vectors you have--\NIf you draw T and N, am I
Dialogue: 0,1:46:51.73,1:46:53.43,Default,,0000,0000,0000,,right, that is coming out?
Dialogue: 0,1:46:53.43,1:46:54.11,Default,,0000,0000,0000,,No.
Dialogue: 0,1:46:54.11,1:46:56.05,Default,,0000,0000,0000,,I have to switch them.
Dialogue: 0,1:46:56.05,1:46:57.65,Default,,0000,0000,0000,,T and N. Now, am I right?
Dialogue: 0,1:46:57.65,1:46:59.34,Default,,0000,0000,0000,,Now I'm thinking of\Nthe [? faucet. ?]
Dialogue: 0,1:46:59.34,1:47:02.44,Default,,0000,0000,0000,,If I move T-- yeah,\Nnow it's coming out.
Dialogue: 0,1:47:02.44,1:47:08.12,Default,,0000,0000,0000,,So this is not getting\Ninto the formula.
Dialogue: 0,1:47:08.12,1:47:09.49,Default,,0000,0000,0000,,So this is the first formula.
Dialogue: 0,1:47:09.49,1:47:10.35,Default,,0000,0000,0000,,You say, so what?
Dialogue: 0,1:47:10.35,1:47:11.18,Default,,0000,0000,0000,,You've taught that.
Dialogue: 0,1:47:11.18,1:47:12.45,Default,,0000,0000,0000,,We proved it together.
Dialogue: 0,1:47:12.45,1:47:14.07,Default,,0000,0000,0000,,What do you want from us?
Dialogue: 0,1:47:14.07,1:47:17.56,Default,,0000,0000,0000,,I want to teach you\Ntwo more formulas.
Dialogue: 0,1:47:17.56,1:47:18.54,Default,,0000,0000,0000,,N prime.
Dialogue: 0,1:47:18.54,1:47:21.97,Default,,0000,0000,0000,,
Dialogue: 0,1:47:21.97,1:47:24.42,Default,,0000,0000,0000,,And I'd like you to\Nleave more space here.
Dialogue: 0,1:47:24.42,1:47:27.36,Default,,0000,0000,0000,,
Dialogue: 0,1:47:27.36,1:47:30.75,Default,,0000,0000,0000,,So you have like an empty field\Nhere and an empty field here
Dialogue: 0,1:47:30.75,1:47:32.02,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,1:47:32.02,1:47:35.82,Default,,0000,0000,0000,,If you were to compute\NT prime, the magic thing
Dialogue: 0,1:47:35.82,1:47:40.45,Default,,0000,0000,0000,,is that T prime is a vector.
Dialogue: 0,1:47:40.45,1:47:41.38,Default,,0000,0000,0000,,N prime is a vector.
Dialogue: 0,1:47:41.38,1:47:42.77,Default,,0000,0000,0000,,B prime is a vector.
Dialogue: 0,1:47:42.77,1:47:44.43,Default,,0000,0000,0000,,They're all vectors.
Dialogue: 0,1:47:44.43,1:47:48.63,Default,,0000,0000,0000,,They are the derivatives\Nof the vectors T and NB.
Dialogue: 0,1:47:48.63,1:47:50.97,Default,,0000,0000,0000,,And you say, why would I\Ncare about the derivatives
Dialogue: 0,1:47:50.97,1:47:52.21,Default,,0000,0000,0000,,of the vectors T and NB?
Dialogue: 0,1:47:52.21,1:47:54.31,Default,,0000,0000,0000,,I'll tell you in a second.
Dialogue: 0,1:47:54.31,1:47:58.07,Default,,0000,0000,0000,,So if you were to\Ncompute in prime,
Dialogue: 0,1:47:58.07,1:48:00.05,Default,,0000,0000,0000,,you're going to get here.
Dialogue: 0,1:48:00.05,1:48:04.36,Default,,0000,0000,0000,,Minus k of s times T of s.
Dialogue: 0,1:48:04.36,1:48:07.22,Default,,0000,0000,0000,,Leave room.
Dialogue: 0,1:48:07.22,1:48:09.57,Default,,0000,0000,0000,,Leave room, because there\Nis no component that
Dialogue: 0,1:48:09.57,1:48:13.76,Default,,0000,0000,0000,,depends on N. No such component\Nthat that depends on N.
Dialogue: 0,1:48:13.76,1:48:14.68,Default,,0000,0000,0000,,This is [INAUDIBLE].
Dialogue: 0,1:48:14.68,1:48:17.26,Default,,0000,0000,0000,,There is nothing in\NN. And then in the end
Dialogue: 0,1:48:17.26,1:48:28.58,Default,,0000,0000,0000,,you'll say, plus tau of s\Ntimes B. There is missing--
Dialogue: 0,1:48:28.58,1:48:30.35,Default,,0000,0000,0000,,something is.
Dialogue: 0,1:48:30.35,1:48:32.99,Default,,0000,0000,0000,,And finally, if\Nyou take B prime,
Dialogue: 0,1:48:32.99,1:48:34.95,Default,,0000,0000,0000,,there is nothing\Nhere, nothing here.
Dialogue: 0,1:48:34.95,1:48:42.86,Default,,0000,0000,0000,,In the middle you have\Nminus tau of s times N of s.
Dialogue: 0,1:48:42.86,1:48:45.70,Default,,0000,0000,0000,,
Dialogue: 0,1:48:45.70,1:48:49.73,Default,,0000,0000,0000,,And now you know that nobody\Nelse but you knows that.
Dialogue: 0,1:48:49.73,1:48:54.06,Default,,0000,0000,0000,,The other regular sections\Ndon't know these formulas.
Dialogue: 0,1:48:54.06,1:48:57.42,Default,,0000,0000,0000,,
Dialogue: 0,1:48:57.42,1:49:02.22,Default,,0000,0000,0000,,What do you observe about\Nthis bunch of equations?
Dialogue: 0,1:49:02.22,1:49:04.16,Default,,0000,0000,0000,,Say, oh, wait a minute.
Dialogue: 0,1:49:04.16,1:49:06.52,Default,,0000,0000,0000,,First of all, why did\Nyou put it like that?
Dialogue: 0,1:49:06.52,1:49:07.84,Default,,0000,0000,0000,,Looks like a cross.
Dialogue: 0,1:49:07.84,1:49:09.16,Default,,0000,0000,0000,,It is a cross.
Dialogue: 0,1:49:09.16,1:49:12.83,Default,,0000,0000,0000,,It is like one is shaped in the\Nname of the Father, of the Son,
Dialogue: 0,1:49:12.83,1:49:13.73,Default,,0000,0000,0000,,and so on.
Dialogue: 0,1:49:13.73,1:49:17.04,Default,,0000,0000,0000,,So does it have anything\Nto do with religion?
Dialogue: 0,1:49:17.04,1:49:17.54,Default,,0000,0000,0000,,No.
Dialogue: 0,1:49:17.54,1:49:23.26,Default,,0000,0000,0000,,But it's going to help you\Nmemorize better the equations.
Dialogue: 0,1:49:23.26,1:49:27.00,Default,,0000,0000,0000,,These are the famous\NFrenet equations.
Dialogue: 0,1:49:27.00,1:49:30.31,Default,,0000,0000,0000,,
Dialogue: 0,1:49:30.31,1:49:33.98,Default,,0000,0000,0000,,You only saw the first one.
Dialogue: 0,1:49:33.98,1:49:35.06,Default,,0000,0000,0000,,What do they represent?
Dialogue: 0,1:49:35.06,1:49:38.23,Default,,0000,0000,0000,,
Dialogue: 0,1:49:38.23,1:49:40.09,Default,,0000,0000,0000,,If somebody asks you, what is k?
Dialogue: 0,1:49:40.09,1:49:42.91,Default,,0000,0000,0000,,What it is k of s?
Dialogue: 0,1:49:42.91,1:49:44.13,Default,,0000,0000,0000,,What's the curvature?
Dialogue: 0,1:49:44.13,1:49:44.88,Default,,0000,0000,0000,,You go to a party.
Dialogue: 0,1:49:44.88,1:49:46.82,Default,,0000,0000,0000,,There are only nerds.
Dialogue: 0,1:49:46.82,1:49:47.44,Default,,0000,0000,0000,,It's you.
Dialogue: 0,1:49:47.44,1:49:50.37,Default,,0000,0000,0000,,Some people taking advanced\Ncalculus or some people
Dialogue: 0,1:49:50.37,1:49:54.79,Default,,0000,0000,0000,,from Physics, and they say, OK,\Nhave you heard of the Frenet
Dialogue: 0,1:49:54.79,1:49:56.92,Default,,0000,0000,0000,,motion, Frenet\Nformulas, and you say,
Dialogue: 0,1:49:56.92,1:49:58.76,Default,,0000,0000,0000,,I know everything about it.
Dialogue: 0,1:49:58.76,1:50:02.31,Default,,0000,0000,0000,,What if they ask you, what\Nis the curvature of k?
Dialogue: 0,1:50:02.31,1:50:07.64,Default,,0000,0000,0000,,You say, curvature measures\Nhow a curve is bent.
Dialogue: 0,1:50:07.64,1:50:11.82,Default,,0000,0000,0000,,And they say, yeah, but the\NFrenet formula tells you
Dialogue: 0,1:50:11.82,1:50:13.61,Default,,0000,0000,0000,,more about that.
Dialogue: 0,1:50:13.61,1:50:17.72,Default,,0000,0000,0000,,Not only k shows you\Nhow bent the curve is.
Dialogue: 0,1:50:17.72,1:50:27.08,Default,,0000,0000,0000,,But k is a measure of\Nhow fast T changes.
Dialogue: 0,1:50:27.08,1:50:28.24,Default,,0000,0000,0000,,And he sees why.
Dialogue: 0,1:50:28.24,1:50:31.03,Default,,0000,0000,0000,,Practically, if you take\Nthe [INAUDIBLE] to the bat,
Dialogue: 0,1:50:31.03,1:50:37.31,Default,,0000,0000,0000,,this is the speed of T. So how\Nfast the teaching will change.
Dialogue: 0,1:50:37.31,1:50:39.89,Default,,0000,0000,0000,,That will be magnitude,\Nwill be just k.
Dialogue: 0,1:50:39.89,1:50:42.44,Default,,0000,0000,0000,,Because magnitude of N is 1.
Dialogue: 0,1:50:42.44,1:50:48.82,Default,,0000,0000,0000,,So note that k of s is\Nthe length of T prime.
Dialogue: 0,1:50:48.82,1:51:04.39,Default,,0000,0000,0000,,This measures the change\Nin T. So how fast T varies.
Dialogue: 0,1:51:04.39,1:51:08.61,Default,,0000,0000,0000,,
Dialogue: 0,1:51:08.61,1:51:11.32,Default,,0000,0000,0000,,What does the torsion represent?
Dialogue: 0,1:51:11.32,1:51:16.69,Default,,0000,0000,0000,,Well, how fast the\Nbinormal varies.
Dialogue: 0,1:51:16.69,1:51:20.60,Default,,0000,0000,0000,,But if you want to\Nthink of a helix,
Dialogue: 0,1:51:20.60,1:51:25.64,Default,,0000,0000,0000,,and it's a little\Nbit hard to imagine,
Dialogue: 0,1:51:25.64,1:51:30.16,Default,,0000,0000,0000,,the curvature measures how\Nbent a certain curve is.
Dialogue: 0,1:51:30.16,1:51:33.80,Default,,0000,0000,0000,,And it measures how\Nbent a plane curve is.
Dialogue: 0,1:51:33.80,1:51:38.72,Default,,0000,0000,0000,,For example, for the circle you\Nhave radius a, 1/a, and so on.
Dialogue: 0,1:51:38.72,1:51:40.87,Default,,0000,0000,0000,,But there must be\Nalso a function that
Dialogue: 0,1:51:40.87,1:51:46.09,Default,,0000,0000,0000,,shows you how a curve twists.
Dialogue: 0,1:51:46.09,1:51:50.06,Default,,0000,0000,0000,,Because you have not\Njust a plane curve where
Dialogue: 0,1:51:50.06,1:51:52.37,Default,,0000,0000,0000,,you care about curvature only.
Dialogue: 0,1:51:52.37,1:51:58.57,Default,,0000,0000,0000,,But in the space curve you\Ncare how the curves twist.
Dialogue: 0,1:51:58.57,1:52:03.19,Default,,0000,0000,0000,,How fast do they move\Naway from a certain plane?
Dialogue: 0,1:52:03.19,1:52:10.96,Default,,0000,0000,0000,,Now, if I were to draw-- is\Nit hard to memorize these?
Dialogue: 0,1:52:10.96,1:52:11.46,Default,,0000,0000,0000,,No.
Dialogue: 0,1:52:11.46,1:52:14.06,Default,,0000,0000,0000,,I memorized them easily\Nbased on the fact
Dialogue: 0,1:52:14.06,1:52:19.85,Default,,0000,0000,0000,,that everything looks\Nlike a decomposition
Dialogue: 0,1:52:19.85,1:52:23.92,Default,,0000,0000,0000,,of a vector in terms of\NT, N, and B. So in my mind
Dialogue: 0,1:52:23.92,1:52:28.47,Default,,0000,0000,0000,,it was like, I take any vector\NI want, B. And this is T,
Dialogue: 0,1:52:28.47,1:52:32.70,Default,,0000,0000,0000,,this is N, and this is B.\NJust the weight was IJK.
Dialogue: 0,1:52:32.70,1:52:36.68,Default,,0000,0000,0000,,Instead if I, I have T. Instead\Nof J, I have N. Instead of K,
Dialogue: 0,1:52:36.68,1:52:40.04,Default,,0000,0000,0000,,I have B. They are\Nstill unit vectors.
Dialogue: 0,1:52:40.04,1:52:42.61,Default,,0000,0000,0000,,So locally at the\Npoint I have this frame
Dialogue: 0,1:52:42.61,1:52:44.23,Default,,0000,0000,0000,,and I have any vector.
Dialogue: 0,1:52:44.23,1:52:46.95,Default,,0000,0000,0000,,This vector-- I'm a physicist.
Dialogue: 0,1:52:46.95,1:52:50.94,Default,,0000,0000,0000,,So let's say I'm going to\Nrepresent that as v1 times
Dialogue: 0,1:52:50.94,1:52:54.05,Default,,0000,0000,0000,,the T plus v2\Ntimes-- instead of J,
Dialogue: 0,1:52:54.05,1:52:57.79,Default,,0000,0000,0000,,we'll use that N plus\NB3 times-- that's
Dialogue: 0,1:52:57.79,1:52:59.98,Default,,0000,0000,0000,,the last element of the bases.
Dialogue: 0,1:52:59.98,1:53:03.60,Default,,0000,0000,0000,,Instead of k I have v.\NSo it's the same here.
Dialogue: 0,1:53:03.60,1:53:06.36,Default,,0000,0000,0000,,You try to pick a\Nvector and decompose
Dialogue: 0,1:53:06.36,1:53:09.95,Default,,0000,0000,0000,,that in terms of T, N, and B.\NWill I put that on the final?
Dialogue: 0,1:53:09.95,1:53:10.72,Default,,0000,0000,0000,,No.
Dialogue: 0,1:53:10.72,1:53:12.92,Default,,0000,0000,0000,,But I would like you to\Nremember it, especially
Dialogue: 0,1:53:12.92,1:53:17.07,Default,,0000,0000,0000,,if you are an engineering\Nmajor or physics major,
Dialogue: 0,1:53:17.07,1:53:19.69,Default,,0000,0000,0000,,that there is this\Nkind of Frenet frame.
Dialogue: 0,1:53:19.69,1:53:26.01,Default,,0000,0000,0000,,For those of you who are taking\Na-- for differential equations,
Dialogue: 0,1:53:26.01,1:53:28.76,Default,,0000,0000,0000,,you already do some matrices\Nand built-in systems
Dialogue: 0,1:53:28.76,1:53:31.59,Default,,0000,0000,0000,,of equations, systems of\Ndifferential equations.
Dialogue: 0,1:53:31.59,1:53:33.22,Default,,0000,0000,0000,,I'm not going to get there.
Dialogue: 0,1:53:33.22,1:53:38.07,Default,,0000,0000,0000,,But suppose you don't know\Ndifferential equations,
Dialogue: 0,1:53:38.07,1:53:41.73,Default,,0000,0000,0000,,but you know a little\Nbit of linear algebra.
Dialogue: 0,1:53:41.73,1:53:44.95,Default,,0000,0000,0000,,And I know you know how\Nto multiply matrices.
Dialogue: 0,1:53:44.95,1:53:47.12,Default,,0000,0000,0000,,You know how I know\Nyou multiply matrices,
Dialogue: 0,1:53:47.12,1:53:49.54,Default,,0000,0000,0000,,no matter how much\Nmathematics you learn.
Dialogue: 0,1:53:49.54,1:53:52.67,Default,,0000,0000,0000,,And most of you, you are not in\Ngeneral algebra this semester.
Dialogue: 0,1:53:52.67,1:53:55.07,Default,,0000,0000,0000,,Only two of you are\Nin general algebra.
Dialogue: 0,1:53:55.07,1:54:03.25,Default,,0000,0000,0000,,When I took a C++ course,\Nthe first homework I got was
Dialogue: 0,1:54:03.25,1:54:06.53,Default,,0000,0000,0000,,to program a matrix\Nmultiplication.
Dialogue: 0,1:54:06.53,1:54:07.73,Default,,0000,0000,0000,,I have to give in matrices.
Dialogue: 0,1:54:07.73,1:54:10.90,Default,,0000,0000,0000,,I have to program that in C++.
Dialogue: 0,1:54:10.90,1:54:14.60,Default,,0000,0000,0000,,And freshmen knew that.
Dialogue: 0,1:54:14.60,1:54:20.44,Default,,0000,0000,0000,,So that means you know how\Nto write this as a matrix
Dialogue: 0,1:54:20.44,1:54:21.41,Default,,0000,0000,0000,,multiplication.
Dialogue: 0,1:54:21.41,1:54:23.05,Default,,0000,0000,0000,,Can anybody help me?
Dialogue: 0,1:54:23.05,1:54:25.88,Default,,0000,0000,0000,,So T, N, B is the magic triple.
Dialogue: 0,1:54:25.88,1:54:28.98,Default,,0000,0000,0000,,T, N, B's the magic corner.
Dialogue: 0,1:54:28.98,1:54:32.00,Default,,0000,0000,0000,,T, N, and B are the Three\NMusketeers who are all
Dialogue: 0,1:54:32.00,1:54:34.33,Default,,0000,0000,0000,,orthogonal to one another.
Dialogue: 0,1:54:34.33,1:54:37.98,Default,,0000,0000,0000,,And then I do derivative\Nwith respect to s.
Dialogue: 0,1:54:37.98,1:54:42.29,Default,,0000,0000,0000,,If I want to be\Nelegant, I'll put d/ds.
Dialogue: 0,1:54:42.29,1:54:44.28,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:54:44.28,1:54:47.33,Default,,0000,0000,0000,,How am I going to\Nfill in this matrix?
Dialogue: 0,1:54:47.33,1:54:50.65,Default,,0000,0000,0000,,So somebody who wants to know\Nabout differential equations,
Dialogue: 0,1:54:50.65,1:54:51.61,Default,,0000,0000,0000,,this would be a--
Dialogue: 0,1:54:51.61,1:54:52.79,Default,,0000,0000,0000,,STUDENT: 0, k, 0.
Dialogue: 0,1:54:52.79,1:54:53.87,Default,,0000,0000,0000,,PROFESSOR TODA: Very good.
Dialogue: 0,1:54:53.87,1:55:04.77,Default,,0000,0000,0000,,0, k, 0, minus k 0\Ntau, 0 minus tau 0.
Dialogue: 0,1:55:04.77,1:55:07.36,Default,,0000,0000,0000,,This is called the\Nskew symmetric matrix.
Dialogue: 0,1:55:07.36,1:55:11.81,Default,,0000,0000,0000,,
Dialogue: 0,1:55:11.81,1:55:14.74,Default,,0000,0000,0000,,Such matrices are very\Nimportant in robotics.
Dialogue: 0,1:55:14.74,1:55:17.43,Default,,0000,0000,0000,,If you've ever been\Nto a robotics team,
Dialogue: 0,1:55:17.43,1:55:20.04,Default,,0000,0000,0000,,like one of those\Nprojects, you should
Dialogue: 0,1:55:20.04,1:55:22.99,Default,,0000,0000,0000,,know that when we study\Nmotions of-- let's say
Dialogue: 0,1:55:22.99,1:55:26.62,Default,,0000,0000,0000,,that my arm performs\Ntwo rotations in a row.
Dialogue: 0,1:55:26.62,1:55:30.50,Default,,0000,0000,0000,,All these motions\Nare described based
Dialogue: 0,1:55:30.50,1:55:35.32,Default,,0000,0000,0000,,on some groups of rotations.
Dialogue: 0,1:55:35.32,1:55:39.95,Default,,0000,0000,0000,,And if I go into details,\Nit's going to be really hard.
Dialogue: 0,1:55:39.95,1:55:45.58,Default,,0000,0000,0000,,But practically\Nin such a setting
Dialogue: 0,1:55:45.58,1:55:49.80,Default,,0000,0000,0000,,we have to deal with matrices\Nthat either have determined
Dialogue: 0,1:55:49.80,1:55:53.52,Default,,0000,0000,0000,,one, like all rotations\Nactually have,
Dialogue: 0,1:55:53.52,1:55:58.30,Default,,0000,0000,0000,,or have some other\Nproperties, like this guy.
Dialogue: 0,1:55:58.30,1:56:00.41,Default,,0000,0000,0000,,What's the determinant\Nof this guy?
Dialogue: 0,1:56:00.41,1:56:02.01,Default,,0000,0000,0000,,What do you guys think?
Dialogue: 0,1:56:02.01,1:56:02.75,Default,,0000,0000,0000,,Just look at it.
Dialogue: 0,1:56:02.75,1:56:03.25,Default,,0000,0000,0000,,STUDENT: 0?
Dialogue: 0,1:56:03.25,1:56:04.00,Default,,0000,0000,0000,,PROFESSOR TODA: 0.
Dialogue: 0,1:56:04.00,1:56:05.66,Default,,0000,0000,0000,,It has determinant 0.
Dialogue: 0,1:56:05.66,1:56:08.47,Default,,0000,0000,0000,,And moreover, it\Nlooks in the mirror.
Dialogue: 0,1:56:08.47,1:56:11.20,Default,,0000,0000,0000,,So this comes from\Na group of motion,
Dialogue: 0,1:56:11.20,1:56:14.69,Default,,0000,0000,0000,,which is little s over 3,\Nthe linear algebra, actually.
Dialogue: 0,1:56:14.69,1:56:17.19,Default,,0000,0000,0000,,So when k is looking\Nin the mirror,
Dialogue: 0,1:56:17.19,1:56:20.82,Default,,0000,0000,0000,,it becomes minus k tau,\Nis becoming minus tau.
Dialogue: 0,1:56:20.82,1:56:24.19,Default,,0000,0000,0000,,It is antisymmetric\Nor skew symmetric.
Dialogue: 0,1:56:24.19,1:56:27.01,Default,,0000,0000,0000,,Skew symmetric or\Nantisymmetric is the same.
Dialogue: 0,1:56:27.01,1:56:29.96,Default,,0000,0000,0000,,STUDENT: Antisymmetric,\Nskew symmetric matrix.
Dialogue: 0,1:56:29.96,1:56:31.94,Default,,0000,0000,0000,,PROFESSOR TODA: Skew\Nsymmetric or antisymmetric
Dialogue: 0,1:56:31.94,1:56:33.37,Default,,0000,0000,0000,,is exactly the same thing.
Dialogue: 0,1:56:33.37,1:56:34.29,Default,,0000,0000,0000,,They are synonyms.
Dialogue: 0,1:56:34.29,1:56:37.06,Default,,0000,0000,0000,,
Dialogue: 0,1:56:37.06,1:56:40.13,Default,,0000,0000,0000,,So it looks in the mirror\Nand picks up the minus sign,
Dialogue: 0,1:56:40.13,1:56:41.82,Default,,0000,0000,0000,,has 0 in the bag.
Dialogue: 0,1:56:41.82,1:56:43.14,Default,,0000,0000,0000,,What am I going to put here?
Dialogue: 0,1:56:43.14,1:56:44.46,Default,,0000,0000,0000,,You already got the idea.
Dialogue: 0,1:56:44.46,1:56:47.06,Default,,0000,0000,0000,,So when Ryan gave\Nme this, he meant
Dialogue: 0,1:56:47.06,1:56:50.46,Default,,0000,0000,0000,,that he knew what I'm going\Nto put here, as a vector,
Dialogue: 0,1:56:50.46,1:56:54.02,Default,,0000,0000,0000,,as a column vector.
Dialogue: 0,1:56:54.02,1:56:54.94,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:56:54.94,1:56:56.02,Default,,0000,0000,0000,,PROFESSOR TODA: No, no no.
Dialogue: 0,1:56:56.02,1:56:57.04,Default,,0000,0000,0000,,How do I multiply?
Dialogue: 0,1:56:57.04,1:56:58.51,Default,,0000,0000,0000,,TNB, right?
Dialogue: 0,1:56:58.51,1:57:01.30,Default,,0000,0000,0000,,So guys, how do you\Nmultiply matrices?
Dialogue: 0,1:57:01.30,1:57:05.42,Default,,0000,0000,0000,,You go first row\Nand first column.
Dialogue: 0,1:57:05.42,1:57:06.40,Default,,0000,0000,0000,,So you go like this.
Dialogue: 0,1:57:06.40,1:57:13.63,Default,,0000,0000,0000,,0 times T plus k times 10\Nplus 0 times B. Here it is.
Dialogue: 0,1:57:13.63,1:57:15.17,Default,,0000,0000,0000,,So I'm teaching you\Na little bit more
Dialogue: 0,1:57:15.17,1:57:18.62,Default,,0000,0000,0000,,than-- if you are going to\Nstick with linear algebra
Dialogue: 0,1:57:18.62,1:57:21.25,Default,,0000,0000,0000,,and stick with\Ndifferential equations,
Dialogue: 0,1:57:21.25,1:57:25.44,Default,,0000,0000,0000,,this is a good introduction\Nto more of those mathematics.
Dialogue: 0,1:57:25.44,1:57:26.11,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:57:26.11,1:57:28.06,Default,,0000,0000,0000,,STUDENT: Why don't\Nyou use Cramer's rule?
Dialogue: 0,1:57:28.06,1:57:28.85,Default,,0000,0000,0000,,PROFESSOR TODA: Uh?
Dialogue: 0,1:57:28.85,1:57:31.27,Default,,0000,0000,0000,,STUDENT: Why don't you\Nuse the Cramer's rule?
Dialogue: 0,1:57:31.27,1:57:32.72,Default,,0000,0000,0000,,PROFESSOR TODA:\NThe Cramer's rule?
Dialogue: 0,1:57:32.72,1:57:34.84,Default,,0000,0000,0000,,STUDENT: Yeah. [INAUDIBLE].
Dialogue: 0,1:57:34.84,1:57:35.63,Default,,0000,0000,0000,,PROFESSOR TODA: No.
Dialogue: 0,1:57:35.63,1:57:44.07,Default,,0000,0000,0000,,First of all, Crarmer's rule is\Nto solve systems of equations
Dialogue: 0,1:57:44.07,1:57:47.81,Default,,0000,0000,0000,,that don't involve derivatives,\Nlike a linear system
Dialogue: 0,1:57:47.81,1:57:51.96,Default,,0000,0000,0000,,like Ax equals B.\NI'm going to have,
Dialogue: 0,1:57:51.96,1:57:56.76,Default,,0000,0000,0000,,for example, 3x1\Nplus 2x3 equals 1.
Dialogue: 0,1:57:56.76,1:58:01.00,Default,,0000,0000,0000,,5x1 plus x2 plus x3\Nequals something else.
Dialogue: 0,1:58:01.00,1:58:03.40,Default,,0000,0000,0000,,So for that I can\Nuse Cramer's rule.
Dialogue: 0,1:58:03.40,1:58:04.69,Default,,0000,0000,0000,,But look at that!
Dialogue: 0,1:58:04.69,1:58:06.06,Default,,0000,0000,0000,,This is really complicated.
Dialogue: 0,1:58:06.06,1:58:07.84,Default,,0000,0000,0000,,It's a dynamical system.
Dialogue: 0,1:58:07.84,1:58:11.58,Default,,0000,0000,0000,,At every moment of time\Nthe vectors are changing.
Dialogue: 0,1:58:11.58,1:58:13.42,Default,,0000,0000,0000,,So it's a crazy [INAUDIBLE].
Dialogue: 0,1:58:13.42,1:58:19.10,Default,,0000,0000,0000,,Like A of t times\Nsomething, so some vector
Dialogue: 0,1:58:19.10,1:58:22.86,Default,,0000,0000,0000,,that is also depending on\Ntime equals the derivative
Dialogue: 0,1:58:22.86,1:58:25.00,Default,,0000,0000,0000,,of that vector that [INAUDIBLE].
Dialogue: 0,1:58:25.00,1:58:31.56,Default,,0000,0000,0000,,So that's a OD system that\Nyou should learn in 3351.
Dialogue: 0,1:58:31.56,1:58:33.36,Default,,0000,0000,0000,,So I don't know what\Nyour degree plan is,
Dialogue: 0,1:58:33.36,1:58:35.16,Default,,0000,0000,0000,,but most of you in\Nengineering will
Dialogue: 0,1:58:35.16,1:58:43.80,Default,,0000,0000,0000,,take my class, 2316 in algebra,\NOD1 3350 where they teach you
Dialogue: 0,1:58:43.80,1:58:45.01,Default,,0000,0000,0000,,about differential equations.
Dialogue: 0,1:58:45.01,1:58:48.07,Default,,0000,0000,0000,,These are all differential\Nequations, all three of them.
Dialogue: 0,1:58:48.07,1:58:51.33,Default,,0000,0000,0000,,In 3351 you learn\Nabout this system
Dialogue: 0,1:58:51.33,1:58:54.46,Default,,0000,0000,0000,,which is a system of\Ndifferential equation.
Dialogue: 0,1:58:54.46,1:58:57.21,Default,,0000,0000,0000,,And then you\Npractically say, now I
Dialogue: 0,1:58:57.21,1:58:59.84,Default,,0000,0000,0000,,know everything I need to\Nknow in math, and you say,
Dialogue: 0,1:58:59.84,1:59:01.10,Default,,0000,0000,0000,,goodbye math.
Dialogue: 0,1:59:01.10,1:59:02.74,Default,,0000,0000,0000,,If you guys wanted\Nto learn more,
Dialogue: 0,1:59:02.74,1:59:06.22,Default,,0000,0000,0000,,of course I would be very\Nhappy to learn that, hey, I
Dialogue: 0,1:59:06.22,1:59:08.81,Default,,0000,0000,0000,,like math, I'd like\Nto be a double major.
Dialogue: 0,1:59:08.81,1:59:12.32,Default,,0000,0000,0000,,I'd like to be not just an\Nengineering, but also math
Dialogue: 0,1:59:12.32,1:59:14.55,Default,,0000,0000,0000,,major if you really like it.
Dialogue: 0,1:59:14.55,1:59:18.17,Default,,0000,0000,0000,,Many people already\Nhave a minor.
Dialogue: 0,1:59:18.17,1:59:20.24,Default,,0000,0000,0000,,Many of you have a\Nminor in your plan.
Dialogue: 0,1:59:20.24,1:59:22.84,Default,,0000,0000,0000,,Like for that minor\Nyou only need--
Dialogue: 0,1:59:22.84,1:59:24.17,Default,,0000,0000,0000,,STUDENT: One extra math course.
Dialogue: 0,1:59:24.17,1:59:25.75,Default,,0000,0000,0000,,PROFESSOR TODA: One\Nextra math course.
Dialogue: 0,1:59:25.75,1:59:30.72,Default,,0000,0000,0000,,For example, with 3350 you\Ndon't need 3351 for a minor.
Dialogue: 0,1:59:30.72,1:59:31.38,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:59:31.38,1:59:34.38,Default,,0000,0000,0000,,Because you are taking the\Nprobability in stats anyway.
Dialogue: 0,1:59:34.38,1:59:35.26,Default,,0000,0000,0000,,You have to.
Dialogue: 0,1:59:35.26,1:59:38.60,Default,,0000,0000,0000,,They force you to do that, 3342.
Dialogue: 0,1:59:38.60,1:59:44.97,Default,,0000,0000,0000,,So if you take 3351 it's on top\Nof the minor that we give you.
Dialogue: 0,1:59:44.97,1:59:46.40,Default,,0000,0000,0000,,I know because that's what I do.
Dialogue: 0,1:59:46.40,1:59:47.98,Default,,0000,0000,0000,,I look at the degree plans.
Dialogue: 0,1:59:47.98,1:59:51.96,Default,,0000,0000,0000,,And I work closely to the\Nmath adviser, with Patty.
Dialogue: 0,1:59:51.96,1:59:54.26,Default,,0000,0000,0000,,She has all the [INAUDIBLE].
Dialogue: 0,1:59:54.26,1:59:55.39,Default,,0000,0000,0000,,STUDENT: So is [INAUDIBLE]?
Dialogue: 0,1:59:55.39,1:59:59.53,Default,,0000,0000,0000,,
Dialogue: 0,1:59:59.53,2:00:00.86,Default,,0000,0000,0000,,PROFESSOR TODA: You mean double?
Dialogue: 0,2:00:00.86,2:00:01.84,Default,,0000,0000,0000,,Double degree?
Dialogue: 0,2:00:01.84,2:00:04.21,Default,,0000,0000,0000,,We have this already in place.
Dialogue: 0,2:00:04.21,2:00:05.38,Default,,0000,0000,0000,,We've had it for many years.
Dialogue: 0,2:00:05.38,2:00:07.40,Default,,0000,0000,0000,,It's an excellent plan.
Dialogue: 0,2:00:07.40,2:00:09.87,Default,,0000,0000,0000,,162 hours it is now.
Dialogue: 0,2:00:09.87,2:00:12.87,Default,,0000,0000,0000,,It used to be 159.
Dialogue: 0,2:00:12.87,2:00:17.70,Default,,0000,0000,0000,,Double major, computer\Nscience and mathematics.
Dialogue: 0,2:00:17.70,2:00:22.84,Default,,0000,0000,0000,,And I could say they were\Nsome of the most successful
Dialogue: 0,2:00:22.84,2:00:26.52,Default,,0000,0000,0000,,in terms of finding jobs.
Dialogue: 0,2:00:26.52,2:00:28.66,Default,,0000,0000,0000,,What would you take\Non top of that?
Dialogue: 0,2:00:28.66,2:00:30.94,Default,,0000,0000,0000,,Well, as a math major you\Nhave a few more courses
Dialogue: 0,2:00:30.94,2:00:32.86,Default,,0000,0000,0000,,to take one top of that.
Dialogue: 0,2:00:32.86,2:00:36.72,Default,,0000,0000,0000,,You can link your computer\Nscience with the mathematics,
Dialogue: 0,2:00:36.72,2:00:39.63,Default,,0000,0000,0000,,for example, by taking\Nnumerical analysis.
Dialogue: 0,2:00:39.63,2:00:42.30,Default,,0000,0000,0000,,If you love computers\Nand you like calculus
Dialogue: 0,2:00:42.30,2:00:46.70,Default,,0000,0000,0000,,and you want to put\Ntogether all the information
Dialogue: 0,2:00:46.70,2:00:49.14,Default,,0000,0000,0000,,you have in both, then\Nnumerical analysis
Dialogue: 0,2:00:49.14,2:00:50.43,Default,,0000,0000,0000,,would be your best bet.
Dialogue: 0,2:00:50.43,2:00:55.22,Default,,0000,0000,0000,,And they require that in\Nboth computer science degree
Dialogue: 0,2:00:55.22,2:00:58.23,Default,,0000,0000,0000,,if you are a double major,\Nand your math degree.
Dialogue: 0,2:00:58.23,2:01:03.05,Default,,0000,0000,0000,,So the good thing is that some\Nthings count for both degrees.
Dialogue: 0,2:01:03.05,2:01:06.75,Default,,0000,0000,0000,,And so with those 160\Nhours you are very happy.
Dialogue: 0,2:01:06.75,2:01:10.06,Default,,0000,0000,0000,,Oh, I'm done, I got\Na few more hours.
Dialogue: 0,2:01:10.06,2:01:12.42,Default,,0000,0000,0000,,Many math majors\Nalready have around 130.
Dialogue: 0,2:01:12.42,2:01:13.83,Default,,0000,0000,0000,,They're not supposed to.
Dialogue: 0,2:01:13.83,2:01:16.08,Default,,0000,0000,0000,,They are supposed\Nto stop at 120.
Dialogue: 0,2:01:16.08,2:01:19.69,Default,,0000,0000,0000,,So why not go the extra 20 hours\Nand get two degrees in one?
Dialogue: 0,2:01:19.69,2:01:21.13,Default,,0000,0000,0000,,STUDENT: It's a semester.
Dialogue: 0,2:01:21.13,2:01:22.10,Default,,0000,0000,0000,,PROFESSOR TODA: Yeah.
Dialogue: 0,2:01:22.10,2:01:23.43,Default,,0000,0000,0000,,Of course, it's a lot more work.
Dialogue: 0,2:01:23.43,2:01:26.43,Default,,0000,0000,0000,,But we have people\Nwho like-- really they
Dialogue: 0,2:01:26.43,2:01:30.17,Default,,0000,0000,0000,,are nerdy people who loved\Ncomputer science from when
Dialogue: 0,2:01:30.17,2:01:31.99,Default,,0000,0000,0000,,they were three or four.
Dialogue: 0,2:01:31.99,2:01:33.41,Default,,0000,0000,0000,,And they also like math.
Dialogue: 0,2:01:33.41,2:01:37.16,Default,,0000,0000,0000,,And they say, OK,\NI want to do both.
Dialogue: 0,2:01:37.16,2:01:41.64,Default,,0000,0000,0000,,OK, a little bit more\Nand I'll let you go.
Dialogue: 0,2:01:41.64,2:01:44.82,Default,,0000,0000,0000,,Now I want you to ask\Nme other questions
Dialogue: 0,2:01:44.82,2:01:48.49,Default,,0000,0000,0000,,you may have had about the\Nhomework, anything that
Dialogue: 0,2:01:48.49,2:01:59.22,Default,,0000,0000,0000,,gave you headache, anything that\Nyou feel you need a little bit
Dialogue: 0,2:01:59.22,2:02:00.71,Default,,0000,0000,0000,,more of an explanation about.
Dialogue: 0,2:02:00.71,2:02:12.47,Default,,0000,0000,0000,,
Dialogue: 0,2:02:12.47,2:02:12.97,Default,,0000,0000,0000,,Yes?
Dialogue: 0,2:02:12.97,2:02:14.01,Default,,0000,0000,0000,,STUDENT: I just have one.
Dialogue: 0,2:02:14.01,2:02:15.80,Default,,0000,0000,0000,,In WeBWork, what\Nis the easiest way
Dialogue: 0,2:02:15.80,2:02:17.65,Default,,0000,0000,0000,,to take the square\Nroot of something?
Dialogue: 0,2:02:17.65,2:02:18.61,Default,,0000,0000,0000,,STUDENT: sqrt.
Dialogue: 0,2:02:18.61,2:02:21.97,Default,,0000,0000,0000,,PROFESSOR TODA: sqrt\Nis what you type.
Dialogue: 0,2:02:21.97,2:02:24.87,Default,,0000,0000,0000,,But of course you can\Nalso go to the caret 1/2.
Dialogue: 0,2:02:24.87,2:02:27.93,Default,,0000,0000,0000,,
Dialogue: 0,2:02:27.93,2:02:29.49,Default,,0000,0000,0000,,Something non-technical?
Dialogue: 0,2:02:29.49,2:02:34.35,Default,,0000,0000,0000,,Any question, yes sir,\Nfrom the homework?
Dialogue: 0,2:02:34.35,2:02:38.69,Default,,0000,0000,0000,,Or in relation to [INAUDIBLE]?
Dialogue: 0,2:02:38.69,2:02:41.51,Default,,0000,0000,0000,,STUDENT: I don't understand\Nwhy is the tangent unit vector,
Dialogue: 0,2:02:41.51,2:02:44.16,Default,,0000,0000,0000,,it's just the slope off\Nof that line, right?
Dialogue: 0,2:02:44.16,2:02:45.14,Default,,0000,0000,0000,,The drunk bug?
Dialogue: 0,2:02:45.14,2:02:47.10,Default,,0000,0000,0000,,Whatever line the\Ndrunk bug is on?
Dialogue: 0,2:02:47.10,2:02:49.12,Default,,0000,0000,0000,,PROFESSOR TODA: So it\Nwould be the tangent
Dialogue: 0,2:02:49.12,2:02:52.04,Default,,0000,0000,0000,,to the directional\Nmotion, which is a curve.
Dialogue: 0,2:02:52.04,2:02:54.62,Default,,0000,0000,0000,,
Dialogue: 0,2:02:54.62,2:02:58.14,Default,,0000,0000,0000,,And normalized to\Nhave length one.
Dialogue: 0,2:02:58.14,2:03:01.70,Default,,0000,0000,0000,,Because otherwise our\Nprime is-- you may say,
Dialogue: 0,2:03:01.70,2:03:04.21,Default,,0000,0000,0000,,why do you need T to be unitary?
Dialogue: 0,2:03:04.21,2:03:07.15,Default,,0000,0000,0000,,
Dialogue: 0,2:03:07.15,2:03:11.36,Default,,0000,0000,0000,,OK, computations become\Nhorrible unless your speed
Dialogue: 0,2:03:11.36,2:03:13.82,Default,,0000,0000,0000,,is 1 or 5 or 9.
Dialogue: 0,2:03:13.82,2:03:18.14,Default,,0000,0000,0000,,If the speed is a constant,\Neverything else becomes easier.
Dialogue: 0,2:03:18.14,2:03:20.18,Default,,0000,0000,0000,,So that's one reason.
Dialogue: 0,2:03:20.18,2:03:22.09,Default,,0000,0000,0000,,STUDENT: And why\Nis the derivative
Dialogue: 0,2:03:22.09,2:03:24.47,Default,,0000,0000,0000,,of T then perpendicular?
Dialogue: 0,2:03:24.47,2:03:26.50,Default,,0000,0000,0000,,Why does it always turn into--
Dialogue: 0,2:03:26.50,2:03:27.96,Default,,0000,0000,0000,,PROFESSOR TODA:\NPerpendicular to T?
Dialogue: 0,2:03:27.96,2:03:30.94,Default,,0000,0000,0000,,We've done that last time,\Nbut I'm glad to do it again.
Dialogue: 0,2:03:30.94,2:03:34.43,Default,,0000,0000,0000,,And I forgot what we\Nwrote in the book,
Dialogue: 0,2:03:34.43,2:03:36.72,Default,,0000,0000,0000,,and I also saw in\Nthe book this thing
Dialogue: 0,2:03:36.72,2:03:42.80,Default,,0000,0000,0000,,that if you have R, in\Nabsolute value, constant--
Dialogue: 0,2:03:42.80,2:03:44.81,Default,,0000,0000,0000,,and I've done that\Nwith you guys--
Dialogue: 0,2:03:44.81,2:03:52.02,Default,,0000,0000,0000,,prove that R and R prime had\Nevery point perpendicular.
Dialogue: 0,2:03:52.02,2:03:54.85,Default,,0000,0000,0000,,So if you have-- we've\Ndone that before.
Dialogue: 0,2:03:54.85,2:03:57.09,Default,,0000,0000,0000,,Now, what do you do then?
Dialogue: 0,2:03:57.09,2:04:00.56,Default,,0000,0000,0000,,T [INAUDIBLE] T is 1.
Dialogue: 0,2:04:00.56,2:04:04.50,Default,,0000,0000,0000,,The scalar [INAUDIBLE]\Nthe product.
Dialogue: 0,2:04:04.50,2:04:09.54,Default,,0000,0000,0000,,T prime times T plus\NT prime T prime.
Dialogue: 0,2:04:09.54,2:04:12.09,Default,,0000,0000,0000,,So 0.
Dialogue: 0,2:04:12.09,2:04:16.54,Default,,0000,0000,0000,,And T is perpendicular\Nto T prime,
Dialogue: 0,2:04:16.54,2:04:20.61,Default,,0000,0000,0000,,because that means T\Nor T prime equals 0.
Dialogue: 0,2:04:20.61,2:04:27.86,Default,,0000,0000,0000,,
Dialogue: 0,2:04:27.86,2:04:30.36,Default,,0000,0000,0000,,When you run in a\Ncircle, you say--
Dialogue: 0,2:04:30.36,2:04:33.79,Default,,0000,0000,0000,,OK, let's run in a circle.
Dialogue: 0,2:04:33.79,2:04:40.65,Default,,0000,0000,0000,,I say, this is my T. I can feel\Nthat there is something that's
Dialogue: 0,2:04:40.65,2:04:42.53,Default,,0000,0000,0000,,trying to bend me this way.
Dialogue: 0,2:04:42.53,2:04:44.37,Default,,0000,0000,0000,,That is my acceleration.
Dialogue: 0,2:04:44.37,2:04:49.04,Default,,0000,0000,0000,,And I have to-- but I don't\Nknow-- how familiar are you
Dialogue: 0,2:04:49.04,2:04:51.38,Default,,0000,0000,0000,,with the winter sports?
Dialogue: 0,2:04:51.38,2:04:54.15,Default,,0000,0000,0000,,
Dialogue: 0,2:04:54.15,2:04:58.07,Default,,0000,0000,0000,,In many winter sports, the\NFrenet Trihedron is crucial.
Dialogue: 0,2:04:58.07,2:05:01.09,Default,,0000,0000,0000,,Imagine that you have\None of those slopes,
Dialogue: 0,2:05:01.09,2:05:04.89,Default,,0000,0000,0000,,and all of the sudden the\Ntorsion becomes too weak.
Dialogue: 0,2:05:04.89,2:05:06.68,Default,,0000,0000,0000,,That means it becomes dangerous.
Dialogue: 0,2:05:06.68,2:05:09.87,Default,,0000,0000,0000,,That means that the\Nvehicle you're in,
Dialogue: 0,2:05:09.87,2:05:15.01,Default,,0000,0000,0000,,the snow vehicle or any kind\Nof-- your skis, [INAUDIBLE],
Dialogue: 0,2:05:15.01,2:05:20.85,Default,,0000,0000,0000,,if the torsion of your body\Nmoving can become too big,
Dialogue: 0,2:05:20.85,2:05:21.88,Default,,0000,0000,0000,,that will be a problem.
Dialogue: 0,2:05:21.88,2:05:24.67,Default,,0000,0000,0000,,So you have to redesign\Nthat some more.
Dialogue: 0,2:05:24.67,2:05:26.66,Default,,0000,0000,0000,,And this is what they do.
Dialogue: 0,2:05:26.66,2:05:28.57,Default,,0000,0000,0000,,You know there have\Nbeen many accidents.
Dialogue: 0,2:05:28.57,2:05:32.36,Default,,0000,0000,0000,,And many times they say,\Neven in Formula One,
Dialogue: 0,2:05:32.36,2:05:38.17,Default,,0000,0000,0000,,the people who project\Na certain racetrack,
Dialogue: 0,2:05:38.17,2:05:41.62,Default,,0000,0000,0000,,like a track in\NIndianapolis or Montecarlo
Dialogue: 0,2:05:41.62,2:05:44.46,Default,,0000,0000,0000,,or whatever, they\Nhave to have in mind
Dialogue: 0,2:05:44.46,2:05:47.66,Default,,0000,0000,0000,,that Frenet frame every second.
Dialogue: 0,2:05:47.66,2:05:50.69,Default,,0000,0000,0000,,So there are\Nsimulators showing how
Dialogue: 0,2:05:50.69,2:05:52.87,Default,,0000,0000,0000,,the Frenet frame is changing.
Dialogue: 0,2:05:52.87,2:05:55.72,Default,,0000,0000,0000,,There are programs that\Nmeasure the curvature
Dialogue: 0,2:05:55.72,2:05:59.95,Default,,0000,0000,0000,,in a torsion for those\Nsimulators at every point.
Dialogue: 0,2:05:59.95,2:06:02.69,Default,,0000,0000,0000,,Neither the curvature\Nnor the torsion
Dialogue: 0,2:06:02.69,2:06:04.36,Default,,0000,0000,0000,,can exceed a certain value.
Dialogue: 0,2:06:04.36,2:06:06.90,Default,,0000,0000,0000,,Otherwise it becomes dangerous.
Dialogue: 0,2:06:06.90,2:06:09.54,Default,,0000,0000,0000,,You say, oh, I thought\Nonly the speed is a danger.
Dialogue: 0,2:06:09.54,2:06:10.91,Default,,0000,0000,0000,,Nope.
Dialogue: 0,2:06:10.91,2:06:14.85,Default,,0000,0000,0000,,It's also the way that the\Nmotion, if it's a skew curve,
Dialogue: 0,2:06:14.85,2:06:16.71,Default,,0000,0000,0000,,it's really complicated.
Dialogue: 0,2:06:16.71,2:06:20.03,Default,,0000,0000,0000,,Because you twist and turn\Nand bend in many ways.
Dialogue: 0,2:06:20.03,2:06:22.24,Default,,0000,0000,0000,,And it can become\Nreally dangerous.
Dialogue: 0,2:06:22.24,2:06:23.28,Default,,0000,0000,0000,,Speed is not [INAUDIBLE].
Dialogue: 0,2:06:23.28,2:06:26.26,Default,,0000,0000,0000,,
Dialogue: 0,2:06:26.26,2:06:30.25,Default,,0000,0000,0000,,STUDENT: So the torsion was\Nthe twists in the track?
Dialogue: 0,2:06:30.25,2:06:31.96,Default,,0000,0000,0000,,PROFESSOR TODA: The\Ntorsion is the twist.
Dialogue: 0,2:06:31.96,2:06:34.58,Default,,0000,0000,0000,,And by the way, keep your idea.
Dialogue: 0,2:06:34.58,2:06:37.29,Default,,0000,0000,0000,,You wanted to ask\Nsomething more?
Dialogue: 0,2:06:37.29,2:06:43.09,Default,,0000,0000,0000,,When you twist-- suppose you\Nhave something like a race car.
Dialogue: 0,2:06:43.09,2:06:47.19,Default,,0000,0000,0000,,And the race car is at\Nthe walls of the track.
Dialogue: 0,2:06:47.19,2:06:57.98,Default,,0000,0000,0000,,And here's-- when you have\Na very abrupt curvature
Dialogue: 0,2:06:57.98,2:07:03.76,Default,,0000,0000,0000,,and torsion, and you can have\Nthat in Formula One as well,
Dialogue: 0,2:07:03.76,2:07:09.92,Default,,0000,0000,0000,,why do they build one wall\Na lot higher than the other?
Dialogue: 0,2:07:09.92,2:07:13.98,Default,,0000,0000,0000,,Because the poor car-- I\Ndon't know how passionate you
Dialogue: 0,2:07:13.98,2:07:19.55,Default,,0000,0000,0000,,are about Formula\NOne or car races--
Dialogue: 0,2:07:19.55,2:07:24.69,Default,,0000,0000,0000,,the poor car is going\Nto be close to the wall.
Dialogue: 0,2:07:24.69,2:07:28.38,Default,,0000,0000,0000,,It's going to bend like that,\Nthat wall would be round.
Dialogue: 0,2:07:28.38,2:07:32.85,Default,,0000,0000,0000,,And as a builder, you have to\Nbuild the wall really high.
Dialogue: 0,2:07:32.85,2:07:35.82,Default,,0000,0000,0000,,Because that kind of high\Nspeed, high velocity,
Dialogue: 0,2:07:35.82,2:07:39.35,Default,,0000,0000,0000,,high curvature, the poor\Ncar's going szhhhhh-- then
Dialogue: 0,2:07:39.35,2:07:42.05,Default,,0000,0000,0000,,again on a normal track.
Dialogue: 0,2:07:42.05,2:07:45.14,Default,,0000,0000,0000,,Imagine what happens if the\Nwall is not high enough.
Dialogue: 0,2:07:45.14,2:07:48.49,Default,,0000,0000,0000,,The wheels of the car\Nwill go up and get over.
Dialogue: 0,2:07:48.49,2:07:50.04,Default,,0000,0000,0000,,And it's going to be a disaster.
Dialogue: 0,2:07:50.04,2:07:52.74,Default,,0000,0000,0000,,
Dialogue: 0,2:07:52.74,2:07:57.49,Default,,0000,0000,0000,,So that engineer ha to study\Nall the parametric equations
Dialogue: 0,2:07:57.49,2:08:01.24,Default,,0000,0000,0000,,and the Frenet frame and\Ndeep down make a simulator,
Dialogue: 0,2:08:01.24,2:08:04.45,Default,,0000,0000,0000,,compute how tall the walls\Nshould be in order for the car
Dialogue: 0,2:08:04.45,2:08:10.22,Default,,0000,0000,0000,,not to get over on the other\Nside or get off the track.
Dialogue: 0,2:08:10.22,2:08:12.35,Default,,0000,0000,0000,,It's really complicated stuff.
Dialogue: 0,2:08:12.35,2:08:14.89,Default,,0000,0000,0000,,It's all mathematics\Nand physics,
Dialogue: 0,2:08:14.89,2:08:18.68,Default,,0000,0000,0000,,but all the applications are\Nrun by engineers and-- yes, sir?
Dialogue: 0,2:08:18.68,2:08:22.03,Default,,0000,0000,0000,,STUDENT: What's the difference\N[INAUDIBLE] centrifugal force?
Dialogue: 0,2:08:22.03,2:08:23.95,Default,,0000,0000,0000,,PROFESSOR TODA: The\Ncentrifugal force
Dialogue: 0,2:08:23.95,2:08:26.38,Default,,0000,0000,0000,,is related to our double prime.
Dialogue: 0,2:08:26.38,2:08:32.13,Default,,0000,0000,0000,,Our double prime is related\Nto N and T at the same time.
Dialogue: 0,2:08:32.13,2:08:36.16,Default,,0000,0000,0000,,So at some point, let me ask you\None last question and I'm done.
Dialogue: 0,2:08:36.16,2:08:39.21,Default,,0000,0000,0000,,
Dialogue: 0,2:08:39.21,2:08:43.24,Default,,0000,0000,0000,,What's the relationship between\Nacceleration or double prime?
Dialogue: 0,2:08:43.24,2:08:45.98,Default,,0000,0000,0000,,And are they the same thing?
Dialogue: 0,2:08:45.98,2:08:50.30,Default,,0000,0000,0000,,And when are they\Nnot the same thing?
Dialogue: 0,2:08:50.30,2:08:52.38,Default,,0000,0000,0000,,Because you say, OK,\Npractically the centrifugal--
Dialogue: 0,2:08:52.38,2:08:54.18,Default,,0000,0000,0000,,STUDENT: They're\Nthe same on a curve.
Dialogue: 0,2:08:54.18,2:08:55.74,Default,,0000,0000,0000,,PROFESSOR TODA:\NThey are the same--
Dialogue: 0,2:08:55.74,2:08:56.82,Default,,0000,0000,0000,,STUDENT: Like on a circle.
Dialogue: 0,2:08:56.82,2:08:58.52,Default,,0000,0000,0000,,PROFESSOR TODA: On a circle!
Dialogue: 0,2:08:58.52,2:09:00.09,Default,,0000,0000,0000,,And you are getting so close.
Dialogue: 0,2:09:00.09,2:09:01.37,Default,,0000,0000,0000,,It's hot, hot, hot.
Dialogue: 0,2:09:01.37,2:09:08.10,Default,,0000,0000,0000,,On a circle and on a helix they\Nare the same up to a constant.
Dialogue: 0,2:09:08.10,2:09:11.31,Default,,0000,0000,0000,,So what do you think the\Nmagic answer will be?
Dialogue: 0,2:09:11.31,2:09:12.48,Default,,0000,0000,0000,,N was what, guys?
Dialogue: 0,2:09:12.48,2:09:15.22,Default,,0000,0000,0000,,N was-- remind me again.
Dialogue: 0,2:09:15.22,2:09:18.48,Default,,0000,0000,0000,,That was T prime over\Nabsolute value of T prime.
Dialogue: 0,2:09:18.48,2:09:22.29,Default,,0000,0000,0000,,But that doesn't mean,\Ndoes not equal, in general,
Dialogue: 0,2:09:22.29,2:09:26.35,Default,,0000,0000,0000,,does not equal to\NR double prime.
Dialogue: 0,2:09:26.35,2:09:28.10,Default,,0000,0000,0000,,When is it equal?
Dialogue: 0,2:09:28.10,2:09:29.93,Default,,0000,0000,0000,,In general it's not equal.
Dialogue: 0,2:09:29.93,2:09:31.07,Default,,0000,0000,0000,,When is it equal?
Dialogue: 0,2:09:31.07,2:09:35.44,Default,,0000,0000,0000,,If you are in aclength, you\Nsee the advantage of aclength.
Dialogue: 0,2:09:35.44,2:09:36.96,Default,,0000,0000,0000,,It's wonderful.
Dialogue: 0,2:09:36.96,2:09:40.07,Default,,0000,0000,0000,,In arclength, T is R prime of s.
Dialogue: 0,2:09:40.07,2:09:45.94,Default,,0000,0000,0000,,And in arclength that means T\Nprime is R double prime of s.
Dialogue: 0,2:09:45.94,2:09:48.59,Default,,0000,0000,0000,,And in arclength\NI just told you,
Dialogue: 0,2:09:48.59,2:09:50.33,Default,,0000,0000,0000,,T prime is the first\NFrenet formula.
Dialogue: 0,2:09:50.33,2:09:55.51,Default,,0000,0000,0000,,It'll be curvature times the N.
Dialogue: 0,2:09:55.51,2:10:02.18,Default,,0000,0000,0000,,So the acceleration\Npractically and the N
Dialogue: 0,2:10:02.18,2:10:06.56,Default,,0000,0000,0000,,will be the same in arclength,\Nup to a scalar multiplication.
Dialogue: 0,2:10:06.56,2:10:11.68,Default,,0000,0000,0000,,But what if your speed\Nis not even constant?
Dialogue: 0,2:10:11.68,2:10:12.53,Default,,0000,0000,0000,,Then God help you.
Dialogue: 0,2:10:12.53,2:10:16.85,Default,,0000,0000,0000,,Because the acceleration\NR double prime and N
Dialogue: 0,2:10:16.85,2:10:19.78,Default,,0000,0000,0000,,are not colinear.
Dialogue: 0,2:10:19.78,2:10:24.37,Default,,0000,0000,0000,,So if I were to draw-- and\Nthat's my last picture--
Dialogue: 0,2:10:24.37,2:10:26.95,Default,,0000,0000,0000,,let me give you a\Nwild motion here.
Dialogue: 0,2:10:26.95,2:10:32.47,Default,,0000,0000,0000,,You start slow and then you go\Ncrazy and fast and slow down.
Dialogue: 0,2:10:32.47,2:10:35.66,Default,,0000,0000,0000,,Just like most of the\Nphysical models from the bugs
Dialogue: 0,2:10:35.66,2:10:38.90,Default,,0000,0000,0000,,and the flies and so on.
Dialogue: 0,2:10:38.90,2:10:44.95,Default,,0000,0000,0000,,In that kind of crazy motion you\Nhave a T and N at every point.
Dialogue: 0,2:10:44.95,2:10:45.45,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,2:10:45.45,2:10:48.43,Default,,0000,0000,0000,,
Dialogue: 0,2:10:48.43,2:10:50.86,Default,,0000,0000,0000,,[? v ?] will be down.
Dialogue: 0,2:10:50.86,2:10:53.36,Default,,0000,0000,0000,,And T is here.
Dialogue: 0,2:10:53.36,2:10:56.94,Default,,0000,0000,0000,,So can you draw arc\Ndouble prime for me?
Dialogue: 0,2:10:56.94,2:10:59.43,Default,,0000,0000,0000,,It will still be\Ntowards the inside.
Dialogue: 0,2:10:59.43,2:11:04.20,Default,,0000,0000,0000,,But it's still going to\Ncoincide with N. Maybe this one.
Dialogue: 0,2:11:04.20,2:11:12.64,Default,,0000,0000,0000,,What's the magic thing is\Nthat T, N, and R double prime
Dialogue: 0,2:11:12.64,2:11:15.74,Default,,0000,0000,0000,,are in the same plane always.
Dialogue: 0,2:11:15.74,2:11:18.17,Default,,0000,0000,0000,,That's another\Nsecret other students
Dialogue: 0,2:11:18.17,2:11:19.63,Default,,0000,0000,0000,,don't know in Calculus 3.
Dialogue: 0,2:11:19.63,2:11:22.46,Default,,0000,0000,0000,,That same thing is\Ncalled osculating plane.
Dialogue: 0,2:11:22.46,2:11:25.63,Default,,0000,0000,0000,,
Dialogue: 0,2:11:25.63,2:11:31.27,Default,,0000,0000,0000,,We have a few magic\Nnames for these things.
Dialogue: 0,2:11:31.27,2:11:36.51,Default,,0000,0000,0000,,So T and N, the plane that\Nis-- how shall I say that?
Dialogue: 0,2:11:36.51,2:11:37.05,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,2:11:37.05,2:11:43.64,Default,,0000,0000,0000,,The plane given by T and N\Nis called osculating plane.
Dialogue: 0,2:11:43.64,2:11:46.71,Default,,0000,0000,0000,,
Dialogue: 0,2:11:46.71,2:11:49.08,Default,,0000,0000,0000,,The acceleration is\Nalways on that plane.
Dialogue: 0,2:11:49.08,2:11:52.46,Default,,0000,0000,0000,,So imagine T and N are\Nin the same shaded plane.
Dialogue: 0,2:11:52.46,2:11:55.50,Default,,0000,0000,0000,,R double prime is\Nin the same plane.
Dialogue: 0,2:11:55.50,2:11:56.55,Default,,0000,0000,0000,,OK?
Dialogue: 0,2:11:56.55,2:11:58.51,Default,,0000,0000,0000,,Now, can you guess\Nthe other two names?
Dialogue: 0,2:11:58.51,2:12:03.64,Default,,0000,0000,0000,,So this is T, this\Nis N. And B is up.
Dialogue: 0,2:12:03.64,2:12:04.80,Default,,0000,0000,0000,,This is my body's direction.
Dialogue: 0,2:12:04.80,2:12:06.20,Default,,0000,0000,0000,,T and N, look at me.
Dialogue: 0,2:12:06.20,2:12:10.16,Default,,0000,0000,0000,,T, N, and B. I'm the\NFrenet Trihedron.
Dialogue: 0,2:12:10.16,2:12:13.36,Default,,0000,0000,0000,,Which one is the\Nosculating plane?
Dialogue: 0,2:12:13.36,2:12:16.74,Default,,0000,0000,0000,,It's the horizontal xy plane.
Dialogue: 0,2:12:16.74,2:12:20.94,Default,,0000,0000,0000,,OK, do you know-- maybe you're\Na mechanical engineering major,
Dialogue: 0,2:12:20.94,2:12:23.65,Default,,0000,0000,0000,,and after that I\Nwill let you go.
Dialogue: 0,2:12:23.65,2:12:25.92,Default,,0000,0000,0000,,No extra credit,\Nthough for this task.
Dialogue: 0,2:12:25.92,2:12:29.13,Default,,0000,0000,0000,,Maybe I'm going to start asking\Nquestions and give you $1.
Dialogue: 0,2:12:29.13,2:12:31.33,Default,,0000,0000,0000,,I used to do that a lot\Nin differential equations,
Dialogue: 0,2:12:31.33,2:12:34.58,Default,,0000,0000,0000,,like ask a hard question,\Nwhoever gets it first,
Dialogue: 0,2:12:34.58,2:12:36.42,Default,,0000,0000,0000,,give her a dollar.
Dialogue: 0,2:12:36.42,2:12:41.60,Default,,0000,0000,0000,,Until a point when they asked\Nme to teach Honors 3350 when
Dialogue: 0,2:12:41.60,2:12:44.21,Default,,0000,0000,0000,,I started having three or four\Npeople answering the question
Dialogue: 0,2:12:44.21,2:12:44.92,Default,,0000,0000,0000,,at the same time.
Dialogue: 0,2:12:44.92,2:12:49.02,Default,,0000,0000,0000,,And that was a\Nsignificant expense,
Dialogue: 0,2:12:49.02,2:12:52.22,Default,,0000,0000,0000,,because I had to give $4\Naway at the same time.
Dialogue: 0,2:12:52.22,2:12:53.84,Default,,0000,0000,0000,,STUDENT: I feel like\Nyou should've just
Dialogue: 0,2:12:53.84,2:12:54.59,Default,,0000,0000,0000,,split it between--
Dialogue: 0,2:12:54.59,2:12:57.95,Default,,0000,0000,0000,,PROFESSOR TODA: So that's\Nnormal and binormal.
Dialogue: 0,2:12:57.95,2:13:00.80,Default,,0000,0000,0000,,This is me, the binormal,\Nand this is the normal.
Dialogue: 0,2:13:00.80,2:13:03.08,Default,,0000,0000,0000,,Does anybody know the\Nname of this plane,
Dialogue: 0,2:13:03.08,2:13:05.85,Default,,0000,0000,0000,,between normal and bionormal?
Dialogue: 0,2:13:05.85,2:13:08.17,Default,,0000,0000,0000,,This would be this plane.
Dialogue: 0,2:13:08.17,2:13:10.75,Default,,0000,0000,0000,,STUDENT: The skew [INAUDIBLE].
Dialogue: 0,2:13:10.75,2:13:12.25,Default,,0000,0000,0000,,PROFESSOR TODA:\NNormal and binormal.
Dialogue: 0,2:13:12.25,2:13:13.88,Default,,0000,0000,0000,,They call that normal plane.
Dialogue: 0,2:13:13.88,2:13:16.54,Default,,0000,0000,0000,,
Dialogue: 0,2:13:16.54,2:13:22.51,Default,,0000,0000,0000,,So it's tricky if you are not\Na mechanical engineering major.
Dialogue: 0,2:13:22.51,2:13:28.46,Default,,0000,0000,0000,,But some of you are maybe\Nand will learn that later.
Dialogue: 0,2:13:28.46,2:13:29.94,Default,,0000,0000,0000,,Any other questions for me?
Dialogue: 0,2:13:29.94,2:13:33.89,Default,,0000,0000,0000,,Now, in my office I'm\Ngoing to do review.
Dialogue: 0,2:13:33.89,2:13:37.84,Default,,0000,0000,0000,,I was wondering\Nif you have time,
Dialogue: 0,2:13:37.84,2:13:39.96,Default,,0000,0000,0000,,I don't know if you have\Ntime to come to my office,
Dialogue: 0,2:13:39.96,2:13:43.34,Default,,0000,0000,0000,,but should you have any kind\Nof homework related question,
Dialogue: 0,2:13:43.34,2:13:46.29,Default,,0000,0000,0000,,I'll be very happy\Nto answer it now.
Dialogue: 0,2:13:46.29,2:13:49.00,Default,,0000,0000,0000,,3:00 to 5:00.
Dialogue: 0,2:13:49.00,2:13:51.02,Default,,0000,0000,0000,,Now, one time I\Nhad a student who
Dialogue: 0,2:13:51.02,2:13:53.18,Default,,0000,0000,0000,,only had seven questions left.
Dialogue: 0,2:13:53.18,2:13:55.69,Default,,0000,0000,0000,,He came to my office and\Nhe left with no homework.
Dialogue: 0,2:13:55.69,2:13:57.59,Default,,0000,0000,0000,,We finished all of them.
Dialogue: 0,2:13:57.59,2:13:58.38,Default,,0000,0000,0000,,And I felt guilty.
Dialogue: 0,2:13:58.38,2:14:00.87,Default,,0000,0000,0000,,But at the same, he\Nsaid, well, no, it's
Dialogue: 0,2:14:00.87,2:14:03.17,Default,,0000,0000,0000,,better I came to you instead\Nof going to my tutor.
Dialogue: 0,2:14:03.17,2:14:05.18,Default,,0000,0000,0000,,It was fine.
Dialogue: 0,2:14:05.18,2:14:08.67,Default,,0000,0000,0000,,So we can try some\Nproblems together today
Dialogue: 0,2:14:08.67,2:14:11.85,Default,,0000,0000,0000,,if you want between 3:00 and\N5:00, if you have the time.
Dialogue: 0,2:14:11.85,2:14:13.84,Default,,0000,0000,0000,,Some of you don't have the time.
Dialogue: 0,2:14:13.84,2:14:14.83,Default,,0000,0000,0000,,All right?
Dialogue: 0,2:14:14.83,2:14:16.32,Default,,0000,0000,0000,,If you don't have\Nthe time today,
Dialogue: 0,2:14:16.32,2:14:19.30,Default,,0000,0000,0000,,and you would like to\Nbe helped [INAUDIBLE],
Dialogue: 0,2:14:19.30,2:14:21.29,Default,,0000,0000,0000,,click Email Instructor.
Dialogue: 0,2:14:21.29,2:14:24.28,Default,,0000,0000,0000,,I'm going to get the\Nquestions [INAUDIBLE].
Dialogue: 0,2:14:24.28,2:14:26.01,Default,,0000,0000,0000,,You're welcome to\Nask me anything
Dialogue: 0,2:14:26.01,2:14:27.26,Default,,0000,0000,0000,,at any time over there.
Dialogue: 0,2:14:27.26,2:14:38.19,Default,,0000,0000,0000,,
Dialogue: 0,2:14:38.19,2:14:41.17,Default,,0000,0000,0000,,[CLASSROOM CHATTER]
Dialogue: 0,2:14:41.17,2:15:12.35,Default,,0000,0000,0000,,
Dialogue: 0,2:15:12.35,2:15:14.47,Default,,0000,0000,0000,,PROFESSOR TODA: I have\Nsomebody who's taking notes.
Dialogue: 0,2:15:14.47,2:15:15.17,Default,,0000,0000,0000,,STUDENT: Yeah, I know.
Dialogue: 0,2:15:15.17,2:15:15.96,Default,,0000,0000,0000,,And that's why I was like--
Dialogue: 0,2:15:15.96,2:15:16.96,Default,,0000,0000,0000,,PROFESSOR TODA: He's\Ngoing to make a copy
Dialogue: 0,2:15:16.96,2:15:18.45,Default,,0000,0000,0000,,and I'll give you a copy.
Dialogue: 0,2:15:18.45,2:15:19.44,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:15:19.44,2:15:23.62,Default,,0000,0000,0000,,My Cal 1 teacher,\NDr. [INAUDIBLE].
Dialogue: 0,2:15:23.62,2:15:24.41,Default,,0000,0000,0000,,STUDENT: Thank you.
Dialogue: 0,2:15:24.41,2:15:25.50,Default,,0000,0000,0000,,PROFESSOR TODA: Yes, yeah.
Dialogue: 0,2:15:25.50,2:15:26.90,Default,,0000,0000,0000,,Have a nice day.
Dialogue: 0,2:15:26.90,2:15:29.03,Default,,0000,0000,0000,,STUDENT: --got really mad\Nwhen I don't take notes.
Dialogue: 0,2:15:29.03,2:15:34.68,Default,,0000,0000,0000,,Because he felt like\NI was not, I guess--
Dialogue: 0,2:15:34.68,2:15:36.50,Default,,0000,0000,0000,,