[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:02.15,Default,,0000,0000,0000,,PROFESSOR TODA: Any\Nquestions so far?
Dialogue: 0,0:00:02.15,0:00:06.45,Default,,0000,0000,0000,,I mean, conceptual,\Ntheoretical questions first,
Dialogue: 0,0:00:06.45,0:00:08.75,Default,,0000,0000,0000,,and then we will\Ndo the second part
Dialogue: 0,0:00:08.75,0:00:10.18,Default,,0000,0000,0000,,of [INAUDIBLE] applications.
Dialogue: 0,0:00:10.18,0:00:14.48,Default,,0000,0000,0000,,Then you can ask\Nfor more questions.
Dialogue: 0,0:00:14.48,0:00:15.91,Default,,0000,0000,0000,,No questions so far?
Dialogue: 0,0:00:15.91,0:00:18.68,Default,,0000,0000,0000,,I have not finished 11-4.
Dialogue: 0,0:00:18.68,0:00:25.74,Default,,0000,0000,0000,,I still owe you a long\Nexplanation about 11-4.
Dialogue: 0,0:00:25.74,0:00:28.27,Default,,0000,0000,0000,,Hopefully it's going to\Nmake more sense today
Dialogue: 0,0:00:28.27,0:00:30.85,Default,,0000,0000,0000,,than it made last time.
Dialogue: 0,0:00:30.85,0:00:34.22,Default,,0000,0000,0000,,I was just saying\Nthat I'm doing 11-4.
Dialogue: 0,0:00:34.22,0:00:36.35,Default,,0000,0000,0000,,This is a lot of chapter.
Dialogue: 0,0:00:36.35,0:00:47.96,Default,,0000,0000,0000,,So second part of 11-4 today--\Ntangent plane and applications.
Dialogue: 0,0:00:47.96,0:00:50.87,Default,,0000,0000,0000,,
Dialogue: 0,0:00:50.87,0:00:53.74,Default,,0000,0000,0000,,Now, we don't say what\Nthose applications are
Dialogue: 0,0:00:53.74,0:00:58.97,Default,,0000,0000,0000,,from the start, but these are\Nsome very important concepts
Dialogue: 0,0:00:58.97,0:01:00.84,Default,,0000,0000,0000,,called the total differential.
Dialogue: 0,0:01:00.84,0:01:07.16,Default,,0000,0000,0000,,
Dialogue: 0,0:01:07.16,0:01:13.96,Default,,0000,0000,0000,,And the linear\Napproximation number
Dialogue: 0,0:01:13.96,0:01:15.50,Default,,0000,0000,0000,,is going under the [INAUDIBLE].
Dialogue: 0,0:01:15.50,0:01:17.18,Default,,0000,0000,0000,,Thank you, sir.
Dialogue: 0,0:01:17.18,0:01:23.88,Default,,0000,0000,0000,,Linear approximation for\Nfunctions of the type z
Dialogue: 0,0:01:23.88,0:01:29.48,Default,,0000,0000,0000,,equals f of xy, which means\Ngraphs of two variables.
Dialogue: 0,0:01:29.48,0:01:33.70,Default,,0000,0000,0000,,At the end of the chapter, I'll\Ntake the notes copy from you.
Dialogue: 0,0:01:33.70,0:01:37.13,Default,,0000,0000,0000,,So don't give me\Nanything until it's over.
Dialogue: 0,0:01:37.13,0:01:38.91,Default,,0000,0000,0000,,When is that going to be over?
Dialogue: 0,0:01:38.91,0:01:41.86,Default,,0000,0000,0000,,We have four more\Nsections to go.
Dialogue: 0,0:01:41.86,0:01:47.14,Default,,0000,0000,0000,,So I guess right before\Nspring break you give me
Dialogue: 0,0:01:47.14,0:01:50.33,Default,,0000,0000,0000,,the notes for chapter 11.
Dialogue: 0,0:01:50.33,0:01:52.47,Default,,0000,0000,0000,,All right, and then\NI'm thinking of making
Dialogue: 0,0:01:52.47,0:01:54.89,Default,,0000,0000,0000,,copies of both chapters.
Dialogue: 0,0:01:54.89,0:02:00.26,Default,,0000,0000,0000,,You get the-- I'm\Ndistributing them to you.
Dialogue: 0,0:02:00.26,0:02:02.66,Default,,0000,0000,0000,,I haven't started\Nand yet go ahead.
Dialogue: 0,0:02:02.66,0:02:09.60,Default,,0000,0000,0000,,Could anybody tell\Nme what the equation
Dialogue: 0,0:02:09.60,0:02:13.52,Default,,0000,0000,0000,,that we used last time--\Nwe proved it, actually.
Dialogue: 0,0:02:13.52,0:02:16.26,Default,,0000,0000,0000,,
Dialogue: 0,0:02:16.26,0:02:20.98,Default,,0000,0000,0000,,What is the equation\Nof the tangent plane
Dialogue: 0,0:02:20.98,0:02:27.27,Default,,0000,0000,0000,,to a smooth surface or a patch\Nof a surface at the point
Dialogue: 0,0:02:27.27,0:02:33.78,Default,,0000,0000,0000,,m of coordinates x0, y0,\Nz0, where the graph is
Dialogue: 0,0:02:33.78,0:02:37.48,Default,,0000,0000,0000,,given by z equals f of x and y.
Dialogue: 0,0:02:37.48,0:02:40.86,Default,,0000,0000,0000,,I'm going to label it on\Nthe patch of a surface.
Dialogue: 0,0:02:40.86,0:02:44.03,Default,,0000,0000,0000,,OK, imagine it\Nlabeled brown there.
Dialogue: 0,0:02:44.03,0:02:52.54,Default,,0000,0000,0000,,And can somebody tell me the\Nequation of the other plane?
Dialogue: 0,0:02:52.54,0:02:54.45,Default,,0000,0000,0000,,But because you\Nhave better memory,
Dialogue: 0,0:02:54.45,0:03:00.30,Default,,0000,0000,0000,,being much younger, about 25\Nyears younger than me or so.
Dialogue: 0,0:03:00.30,0:03:05.65,Default,,0000,0000,0000,,So could you-- could anybody\Ntell me what the tangent
Dialogue: 0,0:03:05.65,0:03:08.62,Default,,0000,0000,0000,,planes equation-- I'll start.
Dialogue: 0,0:03:08.62,0:03:10.29,Default,,0000,0000,0000,,And it's going to come to you.
Dialogue: 0,0:03:10.29,0:03:14.88,Default,,0000,0000,0000,,z minus z0 equals.
Dialogue: 0,0:03:14.88,0:03:15.94,Default,,0000,0000,0000,,And now let's see.
Dialogue: 0,0:03:15.94,0:03:18.38,Default,,0000,0000,0000,,I'll pick a nice color.
Dialogue: 0,0:03:18.38,0:03:19.10,Default,,0000,0000,0000,,I'll wait.
Dialogue: 0,0:03:19.10,0:03:21.84,Default,,0000,0000,0000,,
Dialogue: 0,0:03:21.84,0:03:23.65,Default,,0000,0000,0000,,STUDENT: fx of x.
Dialogue: 0,0:03:23.65,0:03:27.22,Default,,0000,0000,0000,,PROFESSOR TODA: f sub x, the\Npartial derivative measured
Dialogue: 0,0:03:27.22,0:03:35.78,Default,,0000,0000,0000,,at f0 i0 times the\Nquantity x minus x0 plus--
Dialogue: 0,0:03:35.78,0:03:36.98,Default,,0000,0000,0000,,STUDENT: f sub y.
Dialogue: 0,0:03:36.98,0:03:38.73,Default,,0000,0000,0000,,PROFESSOR TODA: f\Nsub y, excellent.
Dialogue: 0,0:03:38.73,0:03:40.68,Default,,0000,0000,0000,,f sub y.
Dialogue: 0,0:03:40.68,0:03:41.92,Default,,0000,0000,0000,,STUDENT: x0, y0.
Dialogue: 0,0:03:41.92,0:03:44.93,Default,,0000,0000,0000,,PROFESSOR TODA: x0,\Ny0 times y minus y0.
Dialogue: 0,0:03:44.93,0:03:49.26,Default,,0000,0000,0000,,
Dialogue: 0,0:03:49.26,0:03:50.22,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:03:50.22,0:03:51.63,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:03:51.63,0:03:59.11,Default,,0000,0000,0000,,Now thinking of what those\Nquantities mean, x minus x0, y
Dialogue: 0,0:03:59.11,0:04:03.73,Default,,0000,0000,0000,,minus y0, z minus\Nz0, what are they?
Dialogue: 0,0:04:03.73,0:04:06.83,Default,,0000,0000,0000,,They are small\Ndisplacements, aren't they?
Dialogue: 0,0:04:06.83,0:04:10.38,Default,,0000,0000,0000,,I mean, what does it\Nmean small displacement?
Dialogue: 0,0:04:10.38,0:04:20.86,Default,,0000,0000,0000,,Imagine that you are near\Nthe point on both surfaces.
Dialogue: 0,0:04:20.86,0:04:23.50,Default,,0000,0000,0000,,So what is a small\Nneighborhood--
Dialogue: 0,0:04:23.50,0:04:27.94,Default,,0000,0000,0000,,what's a typical small\Nneighborhood [INAUDIBLE]?
Dialogue: 0,0:04:27.94,0:04:30.28,Default,,0000,0000,0000,,It's a disk, right?
Dialogue: 0,0:04:30.28,0:04:32.67,Default,,0000,0000,0000,,There are many kinds of\Nneighborhoods, but one of them,
Dialogue: 0,0:04:32.67,0:04:36.96,Default,,0000,0000,0000,,I'd say, would be\Nthis open disk, OK?
Dialogue: 0,0:04:36.96,0:04:38.86,Default,,0000,0000,0000,,I'll draw that.
Dialogue: 0,0:04:38.86,0:04:44.71,Default,,0000,0000,0000,,Now, if I have a\Nred point-- I don't
Dialogue: 0,0:04:44.71,0:04:53.19,Default,,0000,0000,0000,,know how to do that pink point--\Nsomewhere nearby in planes--
Dialogue: 0,0:04:53.19,0:04:54.55,Default,,0000,0000,0000,,this is the plane.
Dialogue: 0,0:04:54.55,0:04:59.36,Default,,0000,0000,0000,,In plane, I have this\Npoint that is close.
Dialogue: 0,0:04:59.36,0:05:01.03,Default,,0000,0000,0000,,And that point is xyz.
Dialogue: 0,0:05:01.03,0:05:04.35,Default,,0000,0000,0000,,
Dialogue: 0,0:05:04.35,0:05:08.75,Default,,0000,0000,0000,,And you think, OK, can\NI visualize that better?
Dialogue: 0,0:05:08.75,0:05:11.79,Default,,0000,0000,0000,,Well, guys, it's hard to\Nvisualize that better.
Dialogue: 0,0:05:11.79,0:05:14.69,Default,,0000,0000,0000,,But I'll draw a triangle\N[? doing ?] a better job.
Dialogue: 0,0:05:14.69,0:05:17.44,Default,,0000,0000,0000,,
Dialogue: 0,0:05:17.44,0:05:18.14,Default,,0000,0000,0000,,That's the frame.
Dialogue: 0,0:05:18.14,0:05:22.40,Default,,0000,0000,0000,,
Dialogue: 0,0:05:22.40,0:05:24.87,Default,,0000,0000,0000,,This is a surface.
Dialogue: 0,0:05:24.87,0:05:27.45,Default,,0000,0000,0000,,Imagine it's a surface, OK?
Dialogue: 0,0:05:27.45,0:05:32.06,Default,,0000,0000,0000,,That's the point of x0, y0.
Dialogue: 0,0:05:32.06,0:05:34.72,Default,,0000,0000,0000,,[? It's ?] the 0 and that.
Dialogue: 0,0:05:34.72,0:05:36.69,Default,,0000,0000,0000,,Where is the point xyz again?
Dialogue: 0,0:05:36.69,0:05:40.64,Default,,0000,0000,0000,,The point xyz is not\Non the pink stuff.
Dialogue: 0,0:05:40.64,0:05:41.79,Default,,0000,0000,0000,,This is a pink surface.
Dialogue: 0,0:05:41.79,0:05:45.33,Default,,0000,0000,0000,,It looks like Pepto\NBismol or something.
Dialogue: 0,0:05:45.33,0:05:46.31,Default,,0000,0000,0000,,You shaded it.
Dialogue: 0,0:05:46.31,0:05:47.06,Default,,0000,0000,0000,,No.
Dialogue: 0,0:05:47.06,0:05:48.23,Default,,0000,0000,0000,,That's not what I want.
Dialogue: 0,0:05:48.23,0:05:55.56,Default,,0000,0000,0000,,I want the close enough\Npoint on the blue plane.
Dialogue: 0,0:05:55.56,0:06:01.18,Default,,0000,0000,0000,,It's actually in the blue plane\Npie and this guy would be xyz.
Dialogue: 0,0:06:01.18,0:06:05.19,Default,,0000,0000,0000,,So now say, OK, how\Nfar I x be from x0?
Dialogue: 0,0:06:05.19,0:06:06.09,Default,,0000,0000,0000,,Well, I don't know.
Dialogue: 0,0:06:06.09,0:06:13.51,Default,,0000,0000,0000,,We would have to check\Nthe points, the set 0,
Dialogue: 0,0:06:13.51,0:06:15.95,Default,,0000,0000,0000,,check the blue point.
Dialogue: 0,0:06:15.95,0:06:18.47,Default,,0000,0000,0000,,This is x.
Dialogue: 0,0:06:18.47,0:06:23.94,Default,,0000,0000,0000,,So between x and x0, I\Nhave this difference,
Dialogue: 0,0:06:23.94,0:06:34.07,Default,,0000,0000,0000,,which is delta x displacement,\Ndisplacement along the x-axis,
Dialogue: 0,0:06:34.07,0:06:38.92,Default,,0000,0000,0000,,away from the\Npoint, fixed point.
Dialogue: 0,0:06:38.92,0:06:41.82,Default,,0000,0000,0000,,
Dialogue: 0,0:06:41.82,0:06:44.74,Default,,0000,0000,0000,,This is the fixed\Npoint, this point.
Dialogue: 0,0:06:44.74,0:06:47.16,Default,,0000,0000,0000,,This point is p.
Dialogue: 0,0:06:47.16,0:06:48.15,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:06:48.15,0:06:51.23,Default,,0000,0000,0000,,y minus y0, let's call\Nthat delta y, which
Dialogue: 0,0:06:51.23,0:06:53.43,Default,,0000,0000,0000,,is the displacement\Nalong the y-axis.
Dialogue: 0,0:06:53.43,0:06:56.42,Default,,0000,0000,0000,,
Dialogue: 0,0:06:56.42,0:07:02.02,Default,,0000,0000,0000,,And then the z minus z0 can be.
Dialogue: 0,0:07:02.02,0:07:05.70,Default,,0000,0000,0000,,Just because I'm a mathematician\Nand I don't like writing down
Dialogue: 0,0:07:05.70,0:07:11.28,Default,,0000,0000,0000,,a lot, I would use\Ns batch as I can,
Dialogue: 0,0:07:11.28,0:07:16.78,Default,,0000,0000,0000,,compact symbols, to\Nspeed up my computation.
Dialogue: 0,0:07:16.78,0:07:19.20,Default,,0000,0000,0000,,So I can rewrite\Nthis whole thing
Dialogue: 0,0:07:19.20,0:07:27.56,Default,,0000,0000,0000,,as a delta z equals f sub\Nx, x0 y0, which is a number.
Dialogue: 0,0:07:27.56,0:07:28.52,Default,,0000,0000,0000,,It's a slope.
Dialogue: 0,0:07:28.52,0:07:31.57,Default,,0000,0000,0000,,We discussed about\Nthat last time.
Dialogue: 0,0:07:31.57,0:07:33.88,Default,,0000,0000,0000,,We even went skiing\Nlast time, when
Dialogue: 0,0:07:33.88,0:07:38.29,Default,,0000,0000,0000,,we said that's like the slope\Nin-- what's the x direction?
Dialogue: 0,0:07:38.29,0:07:41.97,Default,,0000,0000,0000,,Slope in the x direction\Nand slope in the y direction
Dialogue: 0,0:07:41.97,0:07:49.00,Default,,0000,0000,0000,,on the graph that was the\Nwhite covered with snow hill.
Dialogue: 0,0:07:49.00,0:07:50.89,Default,,0000,0000,0000,,That was what we had last time.
Dialogue: 0,0:07:50.89,0:07:54.55,Default,,0000,0000,0000,,Delta x plus f sub\N0, another slope
Dialogue: 0,0:07:54.55,0:07:56.74,Default,,0000,0000,0000,,in the y direction, delta y.
Dialogue: 0,0:07:56.74,0:08:01.62,Default,,0000,0000,0000,,
Dialogue: 0,0:08:01.62,0:08:07.89,Default,,0000,0000,0000,,And fortunately-- OK, the book\Nis a very good book, obviously,
Dialogue: 0,0:08:07.89,0:08:09.25,Default,,0000,0000,0000,,right?
Dialogue: 0,0:08:09.25,0:08:15.63,Default,,0000,0000,0000,,But I wish we could've done\Ncertain things better in terms
Dialogue: 0,0:08:15.63,0:08:21.80,Default,,0000,0000,0000,,of comparisons between\Nthis notion in Calc III
Dialogue: 0,0:08:21.80,0:08:27.26,Default,,0000,0000,0000,,and some corresponding\Nnotion in Calc I.
Dialogue: 0,0:08:27.26,0:08:29.61,Default,,0000,0000,0000,,So you're probably\Nthinking, what the heck
Dialogue: 0,0:08:29.61,0:08:31.24,Default,,0000,0000,0000,,is this witch thinking about?
Dialogue: 0,0:08:31.24,0:08:34.59,Default,,0000,0000,0000,,Well, I'm thinking\Nof something that you
Dialogue: 0,0:08:34.59,0:08:39.73,Default,,0000,0000,0000,,may want to remember\Nfrom Calc I.
Dialogue: 0,0:08:39.73,0:08:42.88,Default,,0000,0000,0000,,And that's going to come\Ninto place beautifully
Dialogue: 0,0:08:42.88,0:08:47.81,Default,,0000,0000,0000,,right now because you have the\NCalc I, Calc III comparison.
Dialogue: 0,0:08:47.81,0:08:52.80,Default,,0000,0000,0000,,And that's why it would be\Ngreat-- the books don't even
Dialogue: 0,0:08:52.80,0:08:55.27,Default,,0000,0000,0000,,talk about this comparison.
Dialogue: 0,0:08:55.27,0:08:59.81,Default,,0000,0000,0000,,In Calc I, I reminded\Nyou about Mr. Leibniz.
Dialogue: 0,0:08:59.81,0:09:01.11,Default,,0000,0000,0000,,He was a very nice guy.
Dialogue: 0,0:09:01.11,0:09:02.57,Default,,0000,0000,0000,,I have no idea, right?
Dialogue: 0,0:09:02.57,0:09:04.13,Default,,0000,0000,0000,,Never met him.
Dialogue: 0,0:09:04.13,0:09:06.66,Default,,0000,0000,0000,,One of the fathers of calculus.
Dialogue: 0,0:09:06.66,0:09:10.63,Default,,0000,0000,0000,,And he introduced the\Nso-called Leibniz notation.
Dialogue: 0,0:09:10.63,0:09:15.72,Default,,0000,0000,0000,,And one of you in office\Nhours last Wednesday
Dialogue: 0,0:09:15.72,0:09:19.28,Default,,0000,0000,0000,,told me, so the\NLeibnitz notation
Dialogue: 0,0:09:19.28,0:09:23.46,Default,,0000,0000,0000,,for a function g of\Nx-- I'm intentionally
Dialogue: 0,0:09:23.46,0:09:26.27,Default,,0000,0000,0000,,changing notation-- is what?
Dialogue: 0,0:09:26.27,0:09:31.63,Default,,0000,0000,0000,,Well, this is just\Nthe derivative
Dialogue: 0,0:09:31.63,0:09:34.00,Default,,0000,0000,0000,,which is the limit of\Nthe different quotients
Dialogue: 0,0:09:34.00,0:09:38.48,Default,,0000,0000,0000,,of your delta g over\Ndelta x-- as done by some
Dialogue: 0,0:09:38.48,0:09:43.18,Default,,0000,0000,0000,,blutches-- 0, right, which\Nwould be the same as lim
Dialogue: 0,0:09:43.18,0:09:50.84,Default,,0000,0000,0000,,of g of x minus g of x0 over\Nx minus x0 as x approaches x0,
Dialogue: 0,0:09:50.84,0:09:52.07,Default,,0000,0000,0000,,right?
Dialogue: 0,0:09:52.07,0:09:52.57,Default,,0000,0000,0000,,Right.
Dialogue: 0,0:09:52.57,0:09:57.72,Default,,0000,0000,0000,,So we've done that in Calc I.\NBut it was a long time ago.
Dialogue: 0,0:09:57.72,0:10:00.63,Default,,0000,0000,0000,,My mission is to teach\Nyou all Calc III,
Dialogue: 0,0:10:00.63,0:10:03.84,Default,,0000,0000,0000,,but I feel that\Nmy mission is also
Dialogue: 0,0:10:03.84,0:10:08.64,Default,,0000,0000,0000,,to teach you what you may not\Nremember very well from Calc I,
Dialogue: 0,0:10:08.64,0:10:11.64,Default,,0000,0000,0000,,because everything is related.
Dialogue: 0,0:10:11.64,0:10:17.69,Default,,0000,0000,0000,,So what was the way we\Ncould have written this,
Dialogue: 0,0:10:17.69,0:10:21.26,Default,,0000,0000,0000,,not delta g over delta\Nx equals g prime.
Dialogue: 0,0:10:21.26,0:10:22.49,Default,,0000,0000,0000,,No.
Dialogue: 0,0:10:22.49,0:10:29.04,Default,,0000,0000,0000,,But it's an approximation of\Ng prime around a very small
Dialogue: 0,0:10:29.04,0:10:33.74,Default,,0000,0000,0000,,[INAUDIBLE], very close to x0.
Dialogue: 0,0:10:33.74,0:10:36.55,Default,,0000,0000,0000,,
Dialogue: 0,0:10:36.55,0:10:39.68,Default,,0000,0000,0000,,So if you wanted to\Nrewrite this approximation,
Dialogue: 0,0:10:39.68,0:10:42.09,Default,,0000,0000,0000,,how would you have rewritten it?
Dialogue: 0,0:10:42.09,0:10:47.41,Default,,0000,0000,0000,,
Dialogue: 0,0:10:47.41,0:10:48.14,Default,,0000,0000,0000,,Delta g--
Dialogue: 0,0:10:48.14,0:10:54.87,Default,,0000,0000,0000,,
Dialogue: 0,0:10:54.87,0:10:57.31,Default,,0000,0000,0000,,STUDENT: g prime sub x.
Dialogue: 0,0:10:57.31,0:11:02.49,Default,,0000,0000,0000,,PROFESSOR TODA: g prime\Nof x0 times delta x.
Dialogue: 0,0:11:02.49,0:11:03.93,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:11:03.93,0:11:08.28,Default,,0000,0000,0000,,Now, why this approximation?
Dialogue: 0,0:11:08.28,0:11:11.71,Default,,0000,0000,0000,,What if I had put equal?
Dialogue: 0,0:11:11.71,0:11:14.12,Default,,0000,0000,0000,,If I had put equal, it\Nwould be all nonsense.
Dialogue: 0,0:11:14.12,0:11:15.40,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:11:15.40,0:11:19.21,Default,,0000,0000,0000,,Well, say, Magdalena, if you\Nput equal, it's another object.
Dialogue: 0,0:11:19.21,0:11:19.77,Default,,0000,0000,0000,,What object?
Dialogue: 0,0:11:19.77,0:11:20.33,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:11:20.33,0:11:22.12,Default,,0000,0000,0000,,Let's look at the objects.
Dialogue: 0,0:11:22.12,0:11:22.100,Default,,0000,0000,0000,,Let's draw a picture.
Dialogue: 0,0:11:22.100,0:11:25.73,Default,,0000,0000,0000,,
Dialogue: 0,0:11:25.73,0:11:27.18,Default,,0000,0000,0000,,This is g.
Dialogue: 0,0:11:27.18,0:11:28.62,Default,,0000,0000,0000,,This is x0.
Dialogue: 0,0:11:28.62,0:11:30.56,Default,,0000,0000,0000,,This is g of x.
Dialogue: 0,0:11:30.56,0:11:32.44,Default,,0000,0000,0000,,What's g prime?
Dialogue: 0,0:11:32.44,0:11:39.42,Default,,0000,0000,0000,,g prime-- thank god-- is the\Nslope of g prime x0 over here.
Dialogue: 0,0:11:39.42,0:11:46.61,Default,,0000,0000,0000,,So if I want to write the\Nline, the line is exactly this.
Dialogue: 0,0:11:46.61,0:11:50.17,Default,,0000,0000,0000,,The red object is the line.
Dialogue: 0,0:11:50.17,0:11:52.77,Default,,0000,0000,0000,,So what is the red object again?
Dialogue: 0,0:11:52.77,0:11:58.35,Default,,0000,0000,0000,,It's y minus y over x\Nminus x0 equals m, which
Dialogue: 0,0:11:58.35,0:12:00.29,Default,,0000,0000,0000,,is g prime number 0.
Dialogue: 0,0:12:00.29,0:12:01.75,Default,,0000,0000,0000,,m is the slope.
Dialogue: 0,0:12:01.75,0:12:05.21,Default,,0000,0000,0000,,That's the point slope\Nformula, thank you very much.
Dialogue: 0,0:12:05.21,0:12:06.77,Default,,0000,0000,0000,,So the red object is this.
Dialogue: 0,0:12:06.77,0:12:08.99,Default,,0000,0000,0000,,This is the line.
Dialogue: 0,0:12:08.99,0:12:10.77,Default,,0000,0000,0000,,Attention is not the same.
Dialogue: 0,0:12:10.77,0:12:15.62,Default,,0000,0000,0000,,The blue thing is my\Ncurve, more precisely
Dialogue: 0,0:12:15.62,0:12:17.60,Default,,0000,0000,0000,,a tiny portion of my curve.
Dialogue: 0,0:12:17.60,0:12:21.61,Default,,0000,0000,0000,,This neighborhood around the\Npoint is what I have here.
Dialogue: 0,0:12:21.61,0:12:22.80,Default,,0000,0000,0000,,What I'm actually-- what?
Dialogue: 0,0:12:22.80,0:12:26.01,Default,,0000,0000,0000,,
Dialogue: 0,0:12:26.01,0:12:29.51,Default,,0000,0000,0000,,I'm trying to\Napproximate my curve
Dialogue: 0,0:12:29.51,0:12:32.31,Default,,0000,0000,0000,,function with a little line.
Dialogue: 0,0:12:32.31,0:12:36.42,Default,,0000,0000,0000,,And I say, I would rather\Napproximate with a red line
Dialogue: 0,0:12:36.42,0:12:38.58,Default,,0000,0000,0000,,because this is the\Nbest approximation
Dialogue: 0,0:12:38.58,0:12:44.20,Default,,0000,0000,0000,,to the blue arc of a curve\Nwhich is on the curve, right?
Dialogue: 0,0:12:44.20,0:12:46.98,Default,,0000,0000,0000,,So this is what it is\Nis just an approximation
Dialogue: 0,0:12:46.98,0:12:54.62,Default,,0000,0000,0000,,of a curve, approximation of\Na curve of an arc of a curve.
Dialogue: 0,0:12:54.62,0:12:57.59,Default,,0000,0000,0000,,But Magdalena's lazy\Ntoday-- approximation
Dialogue: 0,0:12:57.59,0:13:03.55,Default,,0000,0000,0000,,of an arc of a curve\Nwith a segment of a line,
Dialogue: 0,0:13:03.55,0:13:07.10,Default,,0000,0000,0000,,with a segment of\Nthe tangent line
Dialogue: 0,0:13:07.10,0:13:10.74,Default,,0000,0000,0000,,of the tangent [INAUDIBLE].
Dialogue: 0,0:13:10.74,0:13:13.36,Default,,0000,0000,0000,,How do we call\Nsuch a phenomenon?
Dialogue: 0,0:13:13.36,0:13:17.65,Default,,0000,0000,0000,,An approximation of\Nan arc of a circle
Dialogue: 0,0:13:17.65,0:13:23.12,Default,,0000,0000,0000,,with a little segment\Nof a tangent line
Dialogue: 0,0:13:23.12,0:13:26.04,Default,,0000,0000,0000,,is like a discretization, right?
Dialogue: 0,0:13:26.04,0:13:29.42,Default,,0000,0000,0000,,But we call it\Nlinear approximation.
Dialogue: 0,0:13:29.42,0:13:32.46,Default,,0000,0000,0000,,It's called a linear\Napproximation.
Dialogue: 0,0:13:32.46,0:13:36.59,Default,,0000,0000,0000,,
Dialogue: 0,0:13:36.59,0:13:40.22,Default,,0000,0000,0000,,A-P-P, approx.
Dialogue: 0,0:13:40.22,0:13:42.46,Default,,0000,0000,0000,,Have you ever seen a\Nlinear approximation
Dialogue: 0,0:13:42.46,0:13:46.88,Default,,0000,0000,0000,,before coming from Calc II?
Dialogue: 0,0:13:46.88,0:13:49.70,Default,,0000,0000,0000,,Well, in Calc II you've\Nseen the Taylor's formula.
Dialogue: 0,0:13:49.70,0:13:51.51,Default,,0000,0000,0000,,What is the Taylor's formula?
Dialogue: 0,0:13:51.51,0:13:55.25,Default,,0000,0000,0000,,It's a beautiful\Nthing that said what?
Dialogue: 0,0:13:55.25,0:13:55.99,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:13:55.99,0:13:56.99,Default,,0000,0000,0000,,Let's remember together.
Dialogue: 0,0:13:56.99,0:14:00.21,Default,,0000,0000,0000,,So relationship\Nwith Calc II, I'm
Dialogue: 0,0:14:00.21,0:14:04.67,Default,,0000,0000,0000,,going to go and make an arrow--\Nrelationship with Calc II,
Dialogue: 0,0:14:04.67,0:14:08.16,Default,,0000,0000,0000,,because everything\Nis actually related.
Dialogue: 0,0:14:08.16,0:14:13.75,Default,,0000,0000,0000,,In Calc II-- how did we\Nintroduce Taylor's formula?
Dialogue: 0,0:14:13.75,0:14:16.93,Default,,0000,0000,0000,,Well, instead of little a that\Nyou're so used to in Calc II,
Dialogue: 0,0:14:16.93,0:14:21.17,Default,,0000,0000,0000,,we are going to put x0\Nis the same thing, right?
Dialogue: 0,0:14:21.17,0:14:23.55,Default,,0000,0000,0000,,So what was Taylor's\Nformula saying?
Dialogue: 0,0:14:23.55,0:14:28.15,Default,,0000,0000,0000,,You have this kind of\Nsmooth, beautiful curve.
Dialogue: 0,0:14:28.15,0:14:30.86,Default,,0000,0000,0000,,But being smooth is not enough.
Dialogue: 0,0:14:30.86,0:14:33.86,Default,,0000,0000,0000,,You have that real analytic.
Dialogue: 0,0:14:33.86,0:14:36.07,Default,,0000,0000,0000,,Real analytic means\Nthat the function can be
Dialogue: 0,0:14:36.07,0:14:41.10,Default,,0000,0000,0000,,expanded in Taylor's formula.
Dialogue: 0,0:14:41.10,0:14:42.25,Default,,0000,0000,0000,,So what does it mean?
Dialogue: 0,0:14:42.25,0:14:53.00,Default,,0000,0000,0000,,It means that we have f of x\Nprime is f of x0 equals-- or g.
Dialogue: 0,0:14:53.00,0:14:54.92,Default,,0000,0000,0000,,You want-- it doesn't matter.
Dialogue: 0,0:14:54.92,0:15:01.16,Default,,0000,0000,0000,,f prime of x0 times\Nx minus x0 plus
Dialogue: 0,0:15:01.16,0:15:06.01,Default,,0000,0000,0000,,dot, dot, dot, dot something\Nthat I'm going to put.
Dialogue: 0,0:15:06.01,0:15:09.30,Default,,0000,0000,0000,,This is [? O. ?] It's a small\Nquantity that's maybe not
Dialogue: 0,0:15:09.30,0:15:12.90,Default,,0000,0000,0000,,so small, but I declare\Nit to be negligible.
Dialogue: 0,0:15:12.90,0:15:14.69,Default,,0000,0000,0000,,And so they're going\Nto be negligible.
Dialogue: 0,0:15:14.69,0:15:18.92,Default,,0000,0000,0000,,I have to make a face,\Na smiley face and eyes,
Dialogue: 0,0:15:18.92,0:15:23.53,Default,,0000,0000,0000,,meaning that it's OK to\Nneglect the second order
Dialogue: 0,0:15:23.53,0:15:25.42,Default,,0000,0000,0000,,term, the third order term.
Dialogue: 0,0:15:25.42,0:15:28.37,Default,,0000,0000,0000,,So what happens, that\Nlittle h, when I square it,
Dialogue: 0,0:15:28.37,0:15:29.34,Default,,0000,0000,0000,,say the heck with it.
Dialogue: 0,0:15:29.34,0:15:30.80,Default,,0000,0000,0000,,It's going to be very small.
Dialogue: 0,0:15:30.80,0:15:36.70,Default,,0000,0000,0000,,Like if h is 0.1 and then\Nh squared will be 0.0001.
Dialogue: 0,0:15:36.70,0:15:40.44,Default,,0000,0000,0000,,And I have a certain range\Nof error that I allow,
Dialogue: 0,0:15:40.44,0:15:41.54,Default,,0000,0000,0000,,a threshold.
Dialogue: 0,0:15:41.54,0:15:43.47,Default,,0000,0000,0000,,I say that's negligible.
Dialogue: 0,0:15:43.47,0:15:47.43,Default,,0000,0000,0000,,If h squared and h cubed and h\Nto the fourth are negligible,
Dialogue: 0,0:15:47.43,0:15:49.93,Default,,0000,0000,0000,,then I'm fine.
Dialogue: 0,0:15:49.93,0:15:53.44,Default,,0000,0000,0000,,If I take all the\Nother spot, that's
Dialogue: 0,0:15:53.44,0:15:55.96,Default,,0000,0000,0000,,the linear approximation.
Dialogue: 0,0:15:55.96,0:15:59.73,Default,,0000,0000,0000,,And that's exactly\Nwhat I wrote here
Dialogue: 0,0:15:59.73,0:16:02.14,Default,,0000,0000,0000,,with little g instead of f.
Dialogue: 0,0:16:02.14,0:16:05.12,Default,,0000,0000,0000,,The only difference is this is\Nlittle f and this is little g.
Dialogue: 0,0:16:05.12,0:16:09.34,Default,,0000,0000,0000,,But it's the same exact\Nformula, linear approximation.
Dialogue: 0,0:16:09.34,0:16:14.60,Default,,0000,0000,0000,,Do you guys remember then next\Nterms of the Taylor's formula?
Dialogue: 0,0:16:14.60,0:16:15.31,Default,,0000,0000,0000,,STUDENT: fw--
Dialogue: 0,0:16:15.31,0:16:16.44,Default,,0000,0000,0000,,PROFESSOR TODA: fw--
Dialogue: 0,0:16:16.44,0:16:19.92,Default,,0000,0000,0000,,STUDENT: w over--
Dialogue: 0,0:16:19.92,0:16:23.43,Default,,0000,0000,0000,,PROFESSOR TODA: So\Nfw prime at x0 over--
Dialogue: 0,0:16:23.43,0:16:24.38,Default,,0000,0000,0000,,STUDENT: 1 factorial.
Dialogue: 0,0:16:24.38,0:16:25.55,Default,,0000,0000,0000,,PROFESSOR TODA: 2 factorial.
Dialogue: 0,0:16:25.55,0:16:26.62,Default,,0000,0000,0000,,This was 1 factorial.
Dialogue: 0,0:16:26.62,0:16:28.95,Default,,0000,0000,0000,,This was over 1 factorial.
Dialogue: 0,0:16:28.95,0:16:30.57,Default,,0000,0000,0000,,But I don't write\Nit because it's one.
Dialogue: 0,0:16:30.57,0:16:31.20,Default,,0000,0000,0000,,STUDENT: Right.
Dialogue: 0,0:16:31.20,0:16:35.82,Default,,0000,0000,0000,,PROFESSOR TODA: Here I would\Nhave f double prime of blah,
Dialogue: 0,0:16:35.82,0:16:41.10,Default,,0000,0000,0000,,blah, blah over-- what did\Nyou say-- 2 factorial times x
Dialogue: 0,0:16:41.10,0:16:44.38,Default,,0000,0000,0000,,minus x0 squared plus, plus,\Nplus, the cubic [INAUDIBLE]
Dialogue: 0,0:16:44.38,0:16:49.73,Default,,0000,0000,0000,,of the-- this is the quadratic\Nterm that I neglect, right?
Dialogue: 0,0:16:49.73,0:16:51.18,Default,,0000,0000,0000,,So that was Taylor's formula.
Dialogue: 0,0:16:51.18,0:16:54.79,Default,,0000,0000,0000,,Do I mention anything\Nabout it now?
Dialogue: 0,0:16:54.79,0:16:55.90,Default,,0000,0000,0000,,We should.
Dialogue: 0,0:16:55.90,0:16:58.25,Default,,0000,0000,0000,,But practically, the\Nauthors of the book
Dialogue: 0,0:16:58.25,0:17:00.40,Default,,0000,0000,0000,,thought, well, everything\Nis in the book.
Dialogue: 0,0:17:00.40,0:17:02.12,Default,,0000,0000,0000,,You can go back and forth.
Dialogue: 0,0:17:02.12,0:17:05.30,Default,,0000,0000,0000,,It's not like that unless\Nsomebody opens your eyes.
Dialogue: 0,0:17:05.30,0:17:09.93,Default,,0000,0000,0000,,For example, I didn't\Nsee that when I was 21.
Dialogue: 0,0:17:09.93,0:17:13.04,Default,,0000,0000,0000,,I couldn't make any connection\Nbetween these Calc I,
Dialogue: 0,0:17:13.04,0:17:14.92,Default,,0000,0000,0000,,Calc II, Calc III notions.
Dialogue: 0,0:17:14.92,0:17:17.89,Default,,0000,0000,0000,,Because nobody told me, hey,\NMagdalena, open your eyes
Dialogue: 0,0:17:17.89,0:17:20.12,Default,,0000,0000,0000,,and look at that in\Nperspective and make
Dialogue: 0,0:17:20.12,0:17:24.72,Default,,0000,0000,0000,,a comparison between what you\Nlearned in different chapters.
Dialogue: 0,0:17:24.72,0:17:26.22,Default,,0000,0000,0000,,I had to grow.
Dialogue: 0,0:17:26.22,0:17:29.03,Default,,0000,0000,0000,,After 20 years, I\Nsaid, oh, I finally
Dialogue: 0,0:17:29.03,0:17:33.68,Default,,0000,0000,0000,,see the picture of linearization\Nof a function of, let's say,
Dialogue: 0,0:17:33.68,0:17:35.39,Default,,0000,0000,0000,,n variables.
Dialogue: 0,0:17:35.39,0:17:38.48,Default,,0000,0000,0000,,So all these total\Ndifferentials will come in place
Dialogue: 0,0:17:38.48,0:17:41.05,Default,,0000,0000,0000,,when time comes.
Dialogue: 0,0:17:41.05,0:17:46.41,Default,,0000,0000,0000,,You have a so-called\Ndifferential in Calc I.
Dialogue: 0,0:17:46.41,0:17:47.92,Default,,0000,0000,0000,,And that's not delta g.
Dialogue: 0,0:17:47.92,0:17:49.89,Default,,0000,0000,0000,,Some people say, OK,\Nno, that's delta g.
Dialogue: 0,0:17:49.89,0:17:52.00,Default,,0000,0000,0000,,No, no, no, no.
Dialogue: 0,0:17:52.00,0:17:53.61,Default,,0000,0000,0000,,The delta x is a displacement.
Dialogue: 0,0:17:53.61,0:17:57.30,Default,,0000,0000,0000,,The delta g is the\Ninduced displacement.
Dialogue: 0,0:17:57.30,0:17:59.98,Default,,0000,0000,0000,,If you want this to be\Ncome a differential,
Dialogue: 0,0:17:59.98,0:18:02.84,Default,,0000,0000,0000,,then you shrink\Nthat displacement
Dialogue: 0,0:18:02.84,0:18:05.64,Default,,0000,0000,0000,,to infinitesimally small.
Dialogue: 0,0:18:05.64,0:18:06.23,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:18:06.23,0:18:09.68,Default,,0000,0000,0000,,So it's like going from\Na molecule to an atom
Dialogue: 0,0:18:09.68,0:18:13.99,Default,,0000,0000,0000,,to an electron to subatomic\Nparticles but even more,
Dialogue: 0,0:18:13.99,0:18:16.06,Default,,0000,0000,0000,,something infinitesimally small.
Dialogue: 0,0:18:16.06,0:18:17.07,Default,,0000,0000,0000,,So what do we do?
Dialogue: 0,0:18:17.07,0:18:22.81,Default,,0000,0000,0000,,We shrink delta x into dx\Nwhich is infinitesimally small.
Dialogue: 0,0:18:22.81,0:18:26.39,Default,,0000,0000,0000,,
Dialogue: 0,0:18:26.39,0:18:28.93,Default,,0000,0000,0000,,It's like the notion of\NGod but microscopically
Dialogue: 0,0:18:28.93,0:18:33.92,Default,,0000,0000,0000,,or like microbiology\Ncompared to the universe, OK?
Dialogue: 0,0:18:33.92,0:18:42.21,Default,,0000,0000,0000,,So dx is multiplied\Nby g prime of x0.
Dialogue: 0,0:18:42.21,0:18:46.43,Default,,0000,0000,0000,,And instead of delta g, I'm\Ngoing to have a so-called dg,
Dialogue: 0,0:18:46.43,0:18:49.06,Default,,0000,0000,0000,,and that's a form.
Dialogue: 0,0:18:49.06,0:18:53.26,Default,,0000,0000,0000,,In mathematics, this is\Ncalled a form or a one form.
Dialogue: 0,0:18:53.26,0:18:58.52,Default,,0000,0000,0000,,And it's a special\Nkind of object, OK?
Dialogue: 0,0:18:58.52,0:19:01.55,Default,,0000,0000,0000,,So Mr. Leibniz was very smart.
Dialogue: 0,0:19:01.55,0:19:09.72,Default,,0000,0000,0000,,He said, but I can rewrite this\Nform like dg dx equals g prime.
Dialogue: 0,0:19:09.72,0:19:13.45,Default,,0000,0000,0000,,So if you ever forget\Nabout this form which
Dialogue: 0,0:19:13.45,0:19:18.17,Default,,0000,0000,0000,,is called differential,\Ndifferential form,
Dialogue: 0,0:19:18.17,0:19:20.78,Default,,0000,0000,0000,,you remember Mr.\NLeibniz, he taught you
Dialogue: 0,0:19:20.78,0:19:25.32,Default,,0000,0000,0000,,how to write the derivative in\Ntwo different ways, dg dx or g
Dialogue: 0,0:19:25.32,0:19:26.63,Default,,0000,0000,0000,,prime.
Dialogue: 0,0:19:26.63,0:19:30.22,Default,,0000,0000,0000,,What you do is just formally\Nmultiply g prime by dx
Dialogue: 0,0:19:30.22,0:19:31.67,Default,,0000,0000,0000,,and you get dg.
Dialogue: 0,0:19:31.67,0:19:34.70,Default,,0000,0000,0000,,Say it again, Magdalena--\Nmultiply g prime by dx
Dialogue: 0,0:19:34.70,0:19:35.88,Default,,0000,0000,0000,,and you get dg.
Dialogue: 0,0:19:35.88,0:19:38.89,Default,,0000,0000,0000,,And that's your\Nso-called differential.
Dialogue: 0,0:19:38.89,0:19:42.50,Default,,0000,0000,0000,,Now, why do you say total\Ndifferential-- total
Dialogue: 0,0:19:42.50,0:19:46.87,Default,,0000,0000,0000,,differential, my god, like\Ncomplete differentiation?
Dialogue: 0,0:19:46.87,0:19:52.28,Default,,0000,0000,0000,,In 11.4, we deal with\Nfunctions of two variables.
Dialogue: 0,0:19:52.28,0:19:54.75,Default,,0000,0000,0000,,So can we say differentials?
Dialogue: 0,0:19:54.75,0:19:57.29,Default,,0000,0000,0000,,Mmm, it's a little bit\Nlike a differential
Dialogue: 0,0:19:57.29,0:20:00.03,Default,,0000,0000,0000,,with respect to what variable?
Dialogue: 0,0:20:00.03,0:20:02.59,Default,,0000,0000,0000,,If you say with respect\Nto all the variables,
Dialogue: 0,0:20:02.59,0:20:08.96,Default,,0000,0000,0000,,then you have to be thinking\Nto be smart and event,
Dialogue: 0,0:20:08.96,0:20:11.69,Default,,0000,0000,0000,,create this new object.
Dialogue: 0,0:20:11.69,0:20:17.31,Default,,0000,0000,0000,,If one would write\NTaylor's formula,
Dialogue: 0,0:20:17.31,0:20:22.72,Default,,0000,0000,0000,,there is a Taylor's\Nformula that we don't give.
Dialogue: 0,0:20:22.72,0:20:23.26,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:20:23.26,0:20:26.21,Default,,0000,0000,0000,,Now, you guys are looking\Nat me with excitement.
Dialogue: 0,0:20:26.21,0:20:30.74,Default,,0000,0000,0000,,For one point extra\Ncredit, on the internet,
Dialogue: 0,0:20:30.74,0:20:35.31,Default,,0000,0000,0000,,find Taylor's formula for\Nn variables, functions
Dialogue: 0,0:20:35.31,0:20:38.59,Default,,0000,0000,0000,,of n variables or at\Nleast two variables,
Dialogue: 0,0:20:38.59,0:20:43.72,Default,,0000,0000,0000,,which was going to look\Nlike z minus z0 equals
Dialogue: 0,0:20:43.72,0:20:49.14,Default,,0000,0000,0000,,f sub x at the point x0\Nat 0 times x minus x0 plus
Dialogue: 0,0:20:49.14,0:21:00.20,Default,,0000,0000,0000,,f sub y at x0 y0 times x minus\Nx0 plus second order terms
Dialogue: 0,0:21:00.20,0:21:04.01,Default,,0000,0000,0000,,plus third order terms\Nplus fourth order terms.
Dialogue: 0,0:21:04.01,0:21:06.72,Default,,0000,0000,0000,,And the video cannot see me.
Dialogue: 0,0:21:06.72,0:21:08.85,Default,,0000,0000,0000,,So what do we do?
Dialogue: 0,0:21:08.85,0:21:13.83,Default,,0000,0000,0000,,We just truncate this\Npart of Taylor's I say,
Dialogue: 0,0:21:13.83,0:21:18.17,Default,,0000,0000,0000,,I already take the Taylor\Npolynomial of degree one.
Dialogue: 0,0:21:18.17,0:21:21.47,Default,,0000,0000,0000,,And the quadratic terms and\Neverything else, the heck
Dialogue: 0,0:21:21.47,0:21:22.85,Default,,0000,0000,0000,,with that.
Dialogue: 0,0:21:22.85,0:21:25.02,Default,,0000,0000,0000,,And I call that a\Nlinear approximation,
Dialogue: 0,0:21:25.02,0:21:28.33,Default,,0000,0000,0000,,but it's actually Taylor's\Nformula being discussed.
Dialogue: 0,0:21:28.33,0:21:30.68,Default,,0000,0000,0000,,We don't tell you in\Nthe book because we
Dialogue: 0,0:21:30.68,0:21:31.74,Default,,0000,0000,0000,,don't want to scare you.
Dialogue: 0,0:21:31.74,0:21:34.86,Default,,0000,0000,0000,,I think we would better\Ntell you at some point,
Dialogue: 0,0:21:34.86,0:21:38.01,Default,,0000,0000,0000,,so I decided to tell you now.
Dialogue: 0,0:21:38.01,0:21:38.85,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:21:38.85,0:21:42.44,Default,,0000,0000,0000,,So this is Taylor's formula\Nfor functions of two variables.
Dialogue: 0,0:21:42.44,0:21:45.63,Default,,0000,0000,0000,,We have to create\Nnot out of nothing
Dialogue: 0,0:21:45.63,0:21:49.81,Default,,0000,0000,0000,,but out of this the\Ntotal differential.
Dialogue: 0,0:21:49.81,0:21:51.19,Default,,0000,0000,0000,,Who tells me?
Dialogue: 0,0:21:51.19,0:21:54.03,Default,,0000,0000,0000,,Shrink the\Ndisplacement, Magdalena.
Dialogue: 0,0:21:54.03,0:21:58.14,Default,,0000,0000,0000,,The delta x shrunk to\Nan infinitesimally small
Dialogue: 0,0:21:58.14,0:21:58.64,Default,,0000,0000,0000,,will be dx.
Dialogue: 0,0:21:58.64,0:22:01.11,Default,,0000,0000,0000,,Delta y will become dy.
Dialogue: 0,0:22:01.11,0:22:06.39,Default,,0000,0000,0000,,The line is a smiley from the\Nskies, just looking at us.
Dialogue: 0,0:22:06.39,0:22:08.04,Default,,0000,0000,0000,,He loves our notations.
Dialogue: 0,0:22:08.04,0:22:10.90,Default,,0000,0000,0000,,And this is dz.
Dialogue: 0,0:22:10.90,0:22:18.97,Default,,0000,0000,0000,,So I'm going to write dz or df's\Nthe same thing equals f sub x.
Dialogue: 0,0:22:18.97,0:22:22.42,Default,,0000,0000,0000,,At the point, you\Ncould be at any point
Dialogue: 0,0:22:22.42,0:22:29.78,Default,,0000,0000,0000,,you are taking in particular,\Ndx plus f sub y xy dy.
Dialogue: 0,0:22:29.78,0:22:34.01,Default,,0000,0000,0000,,So this is at any point\Nat the arbitrary point xy
Dialogue: 0,0:22:34.01,0:22:39.31,Default,,0000,0000,0000,,in the domain where your\Nfunction e is at least c1.
Dialogue: 0,0:22:39.31,0:22:40.73,Default,,0000,0000,0000,,What does it mean, c1?
Dialogue: 0,0:22:40.73,0:22:43.28,Default,,0000,0000,0000,,It means the function\Nis differentiable
Dialogue: 0,0:22:43.28,0:22:47.40,Default,,0000,0000,0000,,and the partial\Nderivatives are continuous.
Dialogue: 0,0:22:47.40,0:22:50.85,Default,,0000,0000,0000,,I said several times, I\Nwant even more than that.
Dialogue: 0,0:22:50.85,0:22:56.79,Default,,0000,0000,0000,,I want it maybe second\Norder derivatives
Dialogue: 0,0:22:56.79,0:23:02.87,Default,,0000,0000,0000,,to exist and be continuous\Nand so on and so forth.
Dialogue: 0,0:23:02.87,0:23:08.46,Default,,0000,0000,0000,,And I will assume\Nthat the function can
Dialogue: 0,0:23:08.46,0:23:11.58,Default,,0000,0000,0000,,be expanded [INAUDIBLE] series.
Dialogue: 0,0:23:11.58,0:23:14.44,Default,,0000,0000,0000,,
Dialogue: 0,0:23:14.44,0:23:17.44,Default,,0000,0000,0000,,All right, now example\Nof a final problem
Dialogue: 0,0:23:17.44,0:23:22.26,Default,,0000,0000,0000,,that was my first problem\Non the final many times
Dialogue: 0,0:23:22.26,0:23:26.31,Default,,0000,0000,0000,,and also on the common\Nfinal departmental final.
Dialogue: 0,0:23:26.31,0:23:28.32,Default,,0000,0000,0000,,And many students\Nscrewed up, and I
Dialogue: 0,0:23:28.32,0:23:32.38,Default,,0000,0000,0000,,don't want you to ever\Nmake such a mistake.
Dialogue: 0,0:23:32.38,0:23:37.32,Default,,0000,0000,0000,,So this is a mistake not\Nto make, OK, mistake not
Dialogue: 0,0:23:37.32,0:23:43.73,Default,,0000,0000,0000,,to make because after 20\Nsomething years of teaching,
Dialogue: 0,0:23:43.73,0:23:46.01,Default,,0000,0000,0000,,I'm quite familiar with\Nthe mistakes students
Dialogue: 0,0:23:46.01,0:23:49.23,Default,,0000,0000,0000,,make in general and I don't\Nwant you to make them.
Dialogue: 0,0:23:49.23,0:23:50.66,Default,,0000,0000,0000,,You are too good to do this.
Dialogue: 0,0:23:50.66,0:23:52.10,Default,,0000,0000,0000,,So problem 1.
Dialogue: 0,0:23:52.10,0:23:56.60,Default,,0000,0000,0000,,On the final, I said-- we\Nsaid-- the only difference was
Dialogue: 0,0:23:56.60,0:24:00.90,Default,,0000,0000,0000,,on some departmental finals,\Nwe gave a more sophisticated
Dialogue: 0,0:24:00.90,0:24:02.47,Default,,0000,0000,0000,,function.
Dialogue: 0,0:24:02.47,0:24:06.58,Default,,0000,0000,0000,,I'm going to give only\Nsome simple function
Dialogue: 0,0:24:06.58,0:24:07.82,Default,,0000,0000,0000,,for this polynomial.
Dialogue: 0,0:24:07.82,0:24:09.77,Default,,0000,0000,0000,,That's beautiful.
Dialogue: 0,0:24:09.77,0:24:18.93,Default,,0000,0000,0000,,And then I said we said\Nwrite the differential
Dialogue: 0,0:24:18.93,0:24:28.09,Default,,0000,0000,0000,,of this function at an\Narbitrary point x, y.
Dialogue: 0,0:24:28.09,0:24:28.61,Default,,0000,0000,0000,,And done.
Dialogue: 0,0:24:28.61,0:24:31.08,Default,,0000,0000,0000,,And [INAUDIBLE].
Dialogue: 0,0:24:31.08,0:24:34.64,Default,,0000,0000,0000,,Well, let me tell you what\Nsome of my students-- some
Dialogue: 0,0:24:34.64,0:24:36.35,Default,,0000,0000,0000,,of my studentss-- don't do that.
Dialogue: 0,0:24:36.35,0:24:38.30,Default,,0000,0000,0000,,I'm going to cross it with red.
Dialogue: 0,0:24:38.30,0:24:41.77,Default,,0000,0000,0000,,And some of my students\Nwrote me very beautifully df
Dialogue: 0,0:24:41.77,0:24:44.39,Default,,0000,0000,0000,,equals 2x plus 2y.
Dialogue: 0,0:24:44.39,0:24:47.55,Default,,0000,0000,0000,,And that can send\Nme to the hospital.
Dialogue: 0,0:24:47.55,0:24:53.32,Default,,0000,0000,0000,,If you want to go to the ER\Nsoon, do this on the exam
Dialogue: 0,0:24:53.32,0:24:55.96,Default,,0000,0000,0000,,because this is nonsense.
Dialogue: 0,0:24:55.96,0:24:57.48,Default,,0000,0000,0000,,Why is this nonsense?
Dialogue: 0,0:24:57.48,0:24:58.36,Default,,0000,0000,0000,,This is not--
Dialogue: 0,0:24:58.36,0:24:59.84,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] dx or dy.
Dialogue: 0,0:24:59.84,0:25:00.84,Default,,0000,0000,0000,,PROFESSOR TODA: Exactly.
Dialogue: 0,0:25:00.84,0:25:06.98,Default,,0000,0000,0000,,So the most important thing\Nis that the df is like-- OK,
Dialogue: 0,0:25:06.98,0:25:09.06,Default,,0000,0000,0000,,let me come back to driving.
Dialogue: 0,0:25:09.06,0:25:14.48,Default,,0000,0000,0000,,I'm driving to Amarillo-- and I\Ngive this example to my calc 1
Dialogue: 0,0:25:14.48,0:25:18.20,Default,,0000,0000,0000,,students all the time because\Nit's a linear motion in terms
Dialogue: 0,0:25:18.20,0:25:18.70,Default,,0000,0000,0000,,of time.
Dialogue: 0,0:25:18.70,0:25:21.09,Default,,0000,0000,0000,,And let's say I'm on\Ncruise control or not.
Dialogue: 0,0:25:21.09,0:25:22.78,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:25:22.78,0:25:30.19,Default,,0000,0000,0000,,When we drive and I'm looking at\Nthe speedometer and I see 60--
Dialogue: 0,0:25:30.19,0:25:37.00,Default,,0000,0000,0000,,I didn't want to say more, but\Nlet's say 80, 80 miles an hour.
Dialogue: 0,0:25:37.00,0:25:38.62,Default,,0000,0000,0000,,That is a miles an hour.
Dialogue: 0,0:25:38.62,0:25:43.34,Default,,0000,0000,0000,,That means the hour is a huge\Nchunk delta h or delta t.
Dialogue: 0,0:25:43.34,0:25:45.01,Default,,0000,0000,0000,,Let's call it delta\Nt because it's time.
Dialogue: 0,0:25:45.01,0:25:45.64,Default,,0000,0000,0000,,I'm silly.
Dialogue: 0,0:25:45.64,0:25:47.66,Default,,0000,0000,0000,,Delta t is 1.
Dialogue: 0,0:25:47.66,0:25:51.31,Default,,0000,0000,0000,,Delta s, the space,\Nthe space, is going
Dialogue: 0,0:25:51.31,0:25:54.97,Default,,0000,0000,0000,,to be the chunk of 60 miles.
Dialogue: 0,0:25:54.97,0:26:00.36,Default,,0000,0000,0000,,But then that is the\Naverage speed that I had.
Dialogue: 0,0:26:00.36,0:26:02.13,Default,,0000,0000,0000,,So that's why I said 60.
Dialogue: 0,0:26:02.13,0:26:04.81,Default,,0000,0000,0000,,That's the average\Nspeed I had in my trip,
Dialogue: 0,0:26:04.81,0:26:05.93,Default,,0000,0000,0000,,during my trip [INAUDIBLE].
Dialogue: 0,0:26:05.93,0:26:10.60,Default,,0000,0000,0000,,There were moments when my\Nspeed was 0 or close to 0.
Dialogue: 0,0:26:10.60,0:26:12.39,Default,,0000,0000,0000,,Let's assume it was never 0.
Dialogue: 0,0:26:12.39,0:26:14.93,Default,,0000,0000,0000,,But that means there were many\Nmoments when my speed could've
Dialogue: 0,0:26:14.93,0:26:18.99,Default,,0000,0000,0000,,been 100, and nobody knows\Nbecause they didn't catch me.
Dialogue: 0,0:26:18.99,0:26:21.45,Default,,0000,0000,0000,,So I was just lucky.
Dialogue: 0,0:26:21.45,0:26:26.30,Default,,0000,0000,0000,,So in average, if somebody is\Nasking you what is the average,
Dialogue: 0,0:26:26.30,0:26:30.44,Default,,0000,0000,0000,,that doesn't tell them anything.
Dialogue: 0,0:26:30.44,0:26:34.09,Default,,0000,0000,0000,,That reminds me of that\Njoke-- overall I'm good,
Dialogue: 0,0:26:34.09,0:26:38.19,Default,,0000,0000,0000,,the statistician joke\Nwho was, are you cold?
Dialogue: 0,0:26:38.19,0:26:39.00,Default,,0000,0000,0000,,Are you warm?
Dialogue: 0,0:26:39.00,0:26:44.14,Default,,0000,0000,0000,,And he was actually sitting\Non with one half of him
Dialogue: 0,0:26:44.14,0:26:47.09,Default,,0000,0000,0000,,on a block of ice and the\Nother half on the stove,
Dialogue: 0,0:26:47.09,0:26:49.17,Default,,0000,0000,0000,,and he says, in\Naverage, I'm fine.
Dialogue: 0,0:26:49.17,0:26:52.40,Default,,0000,0000,0000,,But he was dying.
Dialogue: 0,0:26:52.40,0:26:53.91,Default,,0000,0000,0000,,This is the same kind of thing.
Dialogue: 0,0:26:53.91,0:26:58.36,Default,,0000,0000,0000,,My average was 60 miles\Nan hour, but I almost
Dialogue: 0,0:26:58.36,0:27:02.11,Default,,0000,0000,0000,,got caught when I was\Ndriving almost 100.
Dialogue: 0,0:27:02.11,0:27:06.25,Default,,0000,0000,0000,,But nobody knows because I'm\Nnot giving you that information.
Dialogue: 0,0:27:06.25,0:27:12.44,Default,,0000,0000,0000,,That's the infinitesimally small\Ninformation that I have not
Dialogue: 0,0:27:12.44,0:27:16.61,Default,,0000,0000,0000,,put correctly here\Nmeans that what is
Dialogue: 0,0:27:16.61,0:27:18.99,Default,,0000,0000,0000,,what I see on the speedometer?
Dialogue: 0,0:27:18.99,0:27:21.06,Default,,0000,0000,0000,,It's the instantaneous\Nrate of change
Dialogue: 0,0:27:21.06,0:27:23.88,Default,,0000,0000,0000,,that I see that\Nfraction of second.
Dialogue: 0,0:27:23.88,0:27:30.94,Default,,0000,0000,0000,,So that means maybe a few feet\Nper a fraction of a second.
Dialogue: 0,0:27:30.94,0:27:33.92,Default,,0000,0000,0000,,It means how many\Nfeet did I travel
Dialogue: 0,0:27:33.92,0:27:36.47,Default,,0000,0000,0000,,in that fraction of a second?
Dialogue: 0,0:27:36.47,0:27:41.24,Default,,0000,0000,0000,,And if that fraction of a second\Nis very tiny that I cannot even
Dialogue: 0,0:27:41.24,0:27:44.00,Default,,0000,0000,0000,,express it properly, that's\Nwhat I'm going to have--
Dialogue: 0,0:27:44.00,0:27:46.61,Default,,0000,0000,0000,,df equals f prime dx.
Dialogue: 0,0:27:46.61,0:27:52.01,Default,,0000,0000,0000,,So df and dx have to be small\Nbecause their ratio will be
Dialogue: 0,0:27:52.01,0:27:56.18,Default,,0000,0000,0000,,a good number, like 60, like\N80, but [? them in ?] themselves
Dialogue: 0,0:27:56.18,0:27:58.64,Default,,0000,0000,0000,,delta m delta [? srv, ?]\Nvery tiny things.
Dialogue: 0,0:27:58.64,0:28:03.42,Default,,0000,0000,0000,,It's the ratio that matters\Nin the end to be 60, or 80,
Dialogue: 0,0:28:03.42,0:28:04.47,Default,,0000,0000,0000,,or whatever.
Dialogue: 0,0:28:04.47,0:28:08.52,Default,,0000,0000,0000,,So I have 2x dx plus 2y dy.
Dialogue: 0,0:28:08.52,0:28:10.92,Default,,0000,0000,0000,,Never say that the\Ndifferential, which
Dialogue: 0,0:28:10.92,0:28:13.16,Default,,0000,0000,0000,,is something\Ninfinitesimally small,
Dialogue: 0,0:28:13.16,0:28:17.38,Default,,0000,0000,0000,,is equal to this scalar\Nfunction that it doesn't even
Dialogue: 0,0:28:17.38,0:28:18.16,Default,,0000,0000,0000,,make any sense.
Dialogue: 0,0:28:18.16,0:28:20.06,Default,,0000,0000,0000,,Don't do that because\Nyou get 0 points
Dialogue: 0,0:28:20.06,0:28:21.90,Default,,0000,0000,0000,,and then we argue,\Nand I don't want
Dialogue: 0,0:28:21.90,0:28:25.45,Default,,0000,0000,0000,,you to get 0 points on\Nthis problem, right.
Dialogue: 0,0:28:25.45,0:28:27.25,Default,,0000,0000,0000,,So it's a very simple problem.
Dialogue: 0,0:28:27.25,0:28:31.08,Default,,0000,0000,0000,,All I want to test you on\Nwould be this definition.
Dialogue: 0,0:28:31.08,0:28:36.00,Default,,0000,0000,0000,,Remember, you're going to\Nsee that again on the midterm
Dialogue: 0,0:28:36.00,0:28:39.02,Default,,0000,0000,0000,,and on the final, or\Njust on the final.
Dialogue: 0,0:28:39.02,0:28:41.65,Default,,0000,0000,0000,,Any questions about that?
Dialogue: 0,0:28:41.65,0:28:42.25,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:28:42.25,0:28:53.98,Default,,0000,0000,0000,,So I want to give you the\Nfollowing homework out
Dialogue: 0,0:28:53.98,0:29:00.68,Default,,0000,0000,0000,,of section 11.4 on\Ntop of the web work.
Dialogue: 0,0:29:00.68,0:29:07.25,Default,,0000,0000,0000,,
Dialogue: 0,0:29:07.25,0:29:16.64,Default,,0000,0000,0000,,Read all the solved\Nexamples of the section.
Dialogue: 0,0:29:16.64,0:29:23.53,Default,,0000,0000,0000,,
Dialogue: 0,0:29:23.53,0:29:24.03,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:29:24.03,0:29:30.47,Default,,0000,0000,0000,,So for example,\Nsomebody tells you
Dialogue: 0,0:29:30.47,0:29:40.11,Default,,0000,0000,0000,,I have to apply this\Nknowing that I have
Dialogue: 0,0:29:40.11,0:29:44.61,Default,,0000,0000,0000,,an error of measurement of\Nsome sort in the s direction
Dialogue: 0,0:29:44.61,0:29:48.21,Default,,0000,0000,0000,,and an error of measurement of\Nsome sort in the y direction.
Dialogue: 0,0:29:48.21,0:29:51.01,Default,,0000,0000,0000,,There are two or three\Nexamples like that.
Dialogue: 0,0:29:51.01,0:29:54.91,Default,,0000,0000,0000,,They will give you all this\Ndata, including the error
Dialogue: 0,0:29:54.91,0:29:55.64,Default,,0000,0000,0000,,measurement.
Dialogue: 0,0:29:55.64,0:29:58.49,Default,,0000,0000,0000,,For delta, it should be 0.1.
Dialogue: 0,0:29:58.49,0:30:04.24,Default,,0000,0000,0000,,Don't confuse the 0.1 with\Ndx. dx is not a quantity.
Dialogue: 0,0:30:04.24,0:30:08.61,Default,,0000,0000,0000,,dx is something like\Nmicro cosmic thing.
Dialogue: 0,0:30:08.61,0:30:14.13,Default,,0000,0000,0000,,It's like infinitely\N[? small ?].
Dialogue: 0,0:30:14.13,0:30:15.05,Default,,0000,0000,0000,,Infinitesimally small.
Dialogue: 0,0:30:15.05,0:30:19.56,Default,,0000,0000,0000,,So saying that dx should be\N0.1 doesn't make any sense,
Dialogue: 0,0:30:19.56,0:30:22.88,Default,,0000,0000,0000,,but delta x being\N0.1 make sense.
Dialogue: 0,0:30:22.88,0:30:26.35,Default,,0000,0000,0000,,Delta y being 0.3 makes sense.
Dialogue: 0,0:30:26.35,0:30:29.56,Default,,0000,0000,0000,,And they ask you to\Nplug it in and find
Dialogue: 0,0:30:29.56,0:30:32.13,Default,,0000,0000,0000,,the general difference.
Dialogue: 0,0:30:32.13,0:30:33.73,Default,,0000,0000,0000,,For example, where\Ncould that happen?
Dialogue: 0,0:30:33.73,0:30:35.76,Default,,0000,0000,0000,,And you see examples\Nin the book.
Dialogue: 0,0:30:35.76,0:30:40.91,Default,,0000,0000,0000,,Somebody measures something--\Nan area of a rectangle
Dialogue: 0,0:30:40.91,0:30:42.97,Default,,0000,0000,0000,,or a volume of a cube.
Dialogue: 0,0:30:42.97,0:30:46.11,Default,,0000,0000,0000,,But when you measure,\Nyou make mistakes.
Dialogue: 0,0:30:46.11,0:30:48.27,Default,,0000,0000,0000,,You have measurement errors.
Dialogue: 0,0:30:48.27,0:30:53.25,Default,,0000,0000,0000,,In the delta x, you have\Nan error of plus minus 0.1.
Dialogue: 0,0:30:53.25,0:31:00.87,Default,,0000,0000,0000,,In the y direction, you have\Ndisplacement error 0.2 or 0.3,
Dialogue: 0,0:31:00.87,0:31:02.22,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:31:02.22,0:31:05.09,Default,,0000,0000,0000,,What is the overall\Nerror you are
Dialogue: 0,0:31:05.09,0:31:08.10,Default,,0000,0000,0000,,going to make when you measure\Nthat function of two variables?
Dialogue: 0,0:31:08.10,0:31:09.73,Default,,0000,0000,0000,,That's what you have.
Dialogue: 0,0:31:09.73,0:31:12.14,Default,,0000,0000,0000,,So you plug in all\Nthose displacements
Dialogue: 0,0:31:12.14,0:31:14.79,Default,,0000,0000,0000,,and you come up with the\Ncomputational problem.
Dialogue: 0,0:31:14.79,0:31:20.20,Default,,0000,0000,0000,,Several of you Wednesday we\Ndiscussed in my office already
Dialogue: 0,0:31:20.20,0:31:24.70,Default,,0000,0000,0000,,solved those problems through\Nweb work and came to me,
Dialogue: 0,0:31:24.70,0:31:27.51,Default,,0000,0000,0000,,and I said, how did you know\Nto plug in those [? numbers ?]?
Dialogue: 0,0:31:27.51,0:31:28.90,Default,,0000,0000,0000,,Well, it's not so hard.
Dialogue: 0,0:31:28.90,0:31:30.12,Default,,0000,0000,0000,,It's sort of common sense.
Dialogue: 0,0:31:30.12,0:31:32.99,Default,,0000,0000,0000,,Plus, I looked in the book\Nand that gave me the idea
Dialogue: 0,0:31:32.99,0:31:34.52,Default,,0000,0000,0000,,to remind you to\Nlook in the book
Dialogue: 0,0:31:34.52,0:31:37.25,Default,,0000,0000,0000,,for those numerical examples.
Dialogue: 0,0:31:37.25,0:31:40.37,Default,,0000,0000,0000,,You will have to\Nuse your calculator.
Dialogue: 0,0:31:40.37,0:31:42.99,Default,,0000,0000,0000,,So you don't have it with\Nyou, you generally, we
Dialogue: 0,0:31:42.99,0:31:45.00,Default,,0000,0000,0000,,don't use in the classroom,\Nbut it's very easy.
Dialogue: 0,0:31:45.00,0:31:48.39,Default,,0000,0000,0000,,All you have to do is use the\Ncalculator and [INAUDIBLE]
Dialogue: 0,0:31:48.39,0:31:51.31,Default,,0000,0000,0000,,examples and see how it goes.
Dialogue: 0,0:31:51.31,0:31:57.43,Default,,0000,0000,0000,,I wanted to show you\Nsomething more interesting
Dialogue: 0,0:31:57.43,0:32:09.41,Default,,0000,0000,0000,,even, more beautiful\Nregarding something
Dialogue: 0,0:32:09.41,0:32:12.93,Default,,0000,0000,0000,,we don't show in the\Nbook until later on,
Dialogue: 0,0:32:12.93,0:32:18.24,Default,,0000,0000,0000,,and I'm uncomfortable with the\Nidea of not showing this to you
Dialogue: 0,0:32:18.24,0:32:19.61,Default,,0000,0000,0000,,now.
Dialogue: 0,0:32:19.61,0:32:26.56,Default,,0000,0000,0000,,An alternate way, or\Nmore advanced way,
Dialogue: 0,0:32:26.56,0:32:38.39,Default,,0000,0000,0000,,more advanced way, to\Ndefine the tangent plane--
Dialogue: 0,0:32:38.39,0:32:49.19,Default,,0000,0000,0000,,the tangent plane-- to a\Nsurface S at the point p.
Dialogue: 0,0:32:49.19,0:32:51.69,Default,,0000,0000,0000,,And I'll draw again.
Dialogue: 0,0:32:51.69,0:32:56.47,Default,,0000,0000,0000,,Half of my job is drawing\Nin this class, which I like.
Dialogue: 0,0:32:56.47,0:32:59.91,Default,,0000,0000,0000,,I mean, I was having an argument\Nwith one of my colleagues who
Dialogue: 0,0:32:59.91,0:33:03.48,Default,,0000,0000,0000,,said, I hate when they are\Ngiving me to teach calculus 3
Dialogue: 0,0:33:03.48,0:33:07.66,Default,,0000,0000,0000,,because I cannot draw.
Dialogue: 0,0:33:07.66,0:33:09.91,Default,,0000,0000,0000,,I think that the\Nmost beautiful part
Dialogue: 0,0:33:09.91,0:33:15.45,Default,,0000,0000,0000,,is that we can represent\Nthings visually,
Dialogue: 0,0:33:15.45,0:33:20.26,Default,,0000,0000,0000,,and this is just pi, the\Ntangent plane I'm after,
Dialogue: 0,0:33:20.26,0:33:24.88,Default,,0000,0000,0000,,and p will be a\Ncoordinate 0 by 0, z0.
Dialogue: 0,0:33:24.88,0:33:26.90,Default,,0000,0000,0000,,And what was the label?
Dialogue: 0,0:33:26.90,0:33:27.79,Default,,0000,0000,0000,,Oh, the label.
Dialogue: 0,0:33:27.79,0:33:28.36,Default,,0000,0000,0000,,The label.
Dialogue: 0,0:33:28.36,0:33:34.33,Default,,0000,0000,0000,,The label was internal\Nwhere z equals f of xy.
Dialogue: 0,0:33:34.33,0:33:40.16,Default,,0000,0000,0000,,But more generally, I'll say\Nthis time plus more generally,
Dialogue: 0,0:33:40.16,0:33:58.97,Default,,0000,0000,0000,,what if you have f of xyz\Nequals c for that surface.
Dialogue: 0,0:33:58.97,0:34:00.56,Default,,0000,0000,0000,,Let's call it [INAUDIBLE].
Dialogue: 0,0:34:00.56,0:34:04.80,Default,,0000,0000,0000,,F of xy is [INAUDIBLE].
Dialogue: 0,0:34:04.80,0:34:08.21,Default,,0000,0000,0000,,And somebody even said, can\Nyou have a parametrization?
Dialogue: 0,0:34:08.21,0:34:10.44,Default,,0000,0000,0000,,And this is where\NI wanted to go.
Dialogue: 0,0:34:10.44,0:34:14.47,Default,,0000,0000,0000,,
Dialogue: 0,0:34:14.47,0:34:16.23,Default,,0000,0000,0000,,Ryan was the first\None who asked me,
Dialogue: 0,0:34:16.23,0:34:18.87,Default,,0000,0000,0000,,but then there were\Nthree more of you
Dialogue: 0,0:34:18.87,0:34:21.16,Default,,0000,0000,0000,,who have restless\Nminds plus you--
Dialogue: 0,0:34:21.16,0:34:25.67,Default,,0000,0000,0000,,because that's the essence\Nof being active here.
Dialogue: 0,0:34:25.67,0:34:29.84,Default,,0000,0000,0000,,We don't lose our connections.
Dialogue: 0,0:34:29.84,0:34:34.30,Default,,0000,0000,0000,,We lose neurons anyway, but\Nwe don't lose our connections
Dialogue: 0,0:34:34.30,0:34:37.95,Default,,0000,0000,0000,,if we think, and\Nanticipate things,
Dialogue: 0,0:34:37.95,0:34:40.08,Default,,0000,0000,0000,,and try to relate concepts.
Dialogue: 0,0:34:40.08,0:34:42.59,Default,,0000,0000,0000,,So if you don't want to\Nget Alzheimer's, just
Dialogue: 0,0:34:42.59,0:34:45.73,Default,,0000,0000,0000,,think about the parametrization.
Dialogue: 0,0:34:45.73,0:34:49.70,Default,,0000,0000,0000,,So can I have a\Nparametrization for a surface?
Dialogue: 0,0:34:49.70,0:34:52.18,Default,,0000,0000,0000,,All righty, what do you mean?
Dialogue: 0,0:34:52.18,0:34:58.24,Default,,0000,0000,0000,,What if somebody says for a\Ncurve, we have r of t, right,
Dialogue: 0,0:34:58.24,0:34:59.08,Default,,0000,0000,0000,,which was what?
Dialogue: 0,0:34:59.08,0:35:06.50,Default,,0000,0000,0000,,It was x of ti plus y of tj plus\Nz of tk, and we were so happy
Dialogue: 0,0:35:06.50,0:35:09.82,Default,,0000,0000,0000,,and we were happy\Nbecause we were traveling
Dialogue: 0,0:35:09.82,0:35:12.32,Default,,0000,0000,0000,,in time with respect\Nto the origin,
Dialogue: 0,0:35:12.32,0:35:15.64,Default,,0000,0000,0000,,and this was r of t at time t.
Dialogue: 0,0:35:15.64,0:35:18.33,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:35:18.33,0:35:20.21,Default,,0000,0000,0000,,But somebody asked\Nme, [INAUDIBLE],
Dialogue: 0,0:35:20.21,0:35:27.01,Default,,0000,0000,0000,,can you have such a position\Nvector moving on a surface?
Dialogue: 0,0:35:27.01,0:35:30.24,Default,,0000,0000,0000,,Like look, it's a rigid motion.
Dialogue: 0,0:35:30.24,0:35:32.77,Default,,0000,0000,0000,,If you went to the\Nrobotics science
Dialogue: 0,0:35:32.77,0:35:36.34,Default,,0000,0000,0000,,fair, Texas Tech, or something\Nlike that, you know about that.
Dialogue: 0,0:35:36.34,0:35:37.18,Default,,0000,0000,0000,,Yeah, cities.
Dialogue: 0,0:35:37.18,0:35:39.98,Default,,0000,0000,0000,,So how do we introduce\Nsuch a parametrization?
Dialogue: 0,0:35:39.98,0:35:44.47,Default,,0000,0000,0000,,We have an origin of course.
Dialogue: 0,0:35:44.47,0:35:46.39,Default,,0000,0000,0000,,An origin is always important.
Dialogue: 0,0:35:46.39,0:35:48.33,Default,,0000,0000,0000,,Everybody has an origin.
Dialogue: 0,0:35:48.33,0:35:53.17,Default,,0000,0000,0000,,
Dialogue: 0,0:35:53.17,0:35:57.61,Default,,0000,0000,0000,,And I take that position\Nvector, and where does it start?
Dialogue: 0,0:35:57.61,0:36:02.12,Default,,0000,0000,0000,,It starts at the origin, and\Nthe tip of it is on the surface,
Dialogue: 0,0:36:02.12,0:36:05.38,Default,,0000,0000,0000,,And it's gliding on the\Nsurface, the tip of it.
Dialogue: 0,0:36:05.38,0:36:10.50,Default,,0000,0000,0000,,And that's going to be r, but\Nit's not going to be r of t.
Dialogue: 0,0:36:10.50,0:36:12.93,Default,,0000,0000,0000,,It's going to be r of\Nlongitude and latitude.
Dialogue: 0,0:36:12.93,0:36:16.11,Default,,0000,0000,0000,,Like imagine, that would\Nbe the radius coming
Dialogue: 0,0:36:16.11,0:36:18.36,Default,,0000,0000,0000,,from the center of the earth.
Dialogue: 0,0:36:18.36,0:36:20.98,Default,,0000,0000,0000,,And it depends on\Ntwo parameters.
Dialogue: 0,0:36:20.98,0:36:24.78,Default,,0000,0000,0000,,One of them would be latitude.
Dialogue: 0,0:36:24.78,0:36:26.14,Default,,0000,0000,0000,,Am I drawing this right?
Dialogue: 0,0:36:26.14,0:36:26.64,Default,,0000,0000,0000,,Latitude--
Dialogue: 0,0:36:26.64,0:36:28.73,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] longitude.
Dialogue: 0,0:36:28.73,0:36:30.87,Default,,0000,0000,0000,,PROFESSOR TODA:\N--from a latitude 0.
Dialogue: 0,0:36:30.87,0:36:32.01,Default,,0000,0000,0000,,I'm at the equator.
Dialogue: 0,0:36:32.01,0:36:33.76,Default,,0000,0000,0000,,Then latitude 90 degrees.
Dialogue: 0,0:36:33.76,0:36:35.97,Default,,0000,0000,0000,,I'm at the North Pole.
Dialogue: 0,0:36:35.97,0:36:37.76,Default,,0000,0000,0000,,In mathematics, we are funny.
Dialogue: 0,0:36:37.76,0:36:40.88,Default,,0000,0000,0000,,We say latitude 0,\Nlatitude 90 North Pole,
Dialogue: 0,0:36:40.88,0:36:45.16,Default,,0000,0000,0000,,latitude negative 90,\Nwhich is South Pole.
Dialogue: 0,0:36:45.16,0:36:49.29,Default,,0000,0000,0000,,And longitude from 0 to 2 pi.
Dialogue: 0,0:36:49.29,0:36:53.74,Default,,0000,0000,0000,,Meridian 0 to all around.
Dialogue: 0,0:36:53.74,0:36:58.20,Default,,0000,0000,0000,,So r will be not a function of\Nt but a function of u and b,
Dialogue: 0,0:36:58.20,0:37:02.24,Default,,0000,0000,0000,,thank god, because u and b\Nare the latitude and longitude
Dialogue: 0,0:37:02.24,0:37:03.32,Default,,0000,0000,0000,,sort of.
Dialogue: 0,0:37:03.32,0:37:12.32,Default,,0000,0000,0000,,So we have x of uv i plus\Ny of uv j plus z of uv k.
Dialogue: 0,0:37:12.32,0:37:20.62,Default,,0000,0000,0000,,
Dialogue: 0,0:37:20.62,0:37:23.03,Default,,0000,0000,0000,,You can do that.
Dialogue: 0,0:37:23.03,0:37:26.01,Default,,0000,0000,0000,,And you say, but can you give\Nus an example, because this
Dialogue: 0,0:37:26.01,0:37:28.21,Default,,0000,0000,0000,,looks so abstract for god sake.
Dialogue: 0,0:37:28.21,0:37:31.83,Default,,0000,0000,0000,,If you give me the graph\Nthe way you gave it to me
Dialogue: 0,0:37:31.83,0:37:37.31,Default,,0000,0000,0000,,before z equals f of xy,\Nplease parametrize this for me.
Dialogue: 0,0:37:37.31,0:37:41.88,Default,,0000,0000,0000,,
Dialogue: 0,0:37:41.88,0:37:44.64,Default,,0000,0000,0000,,Parametrize it for\Nme because I'm lost.
Dialogue: 0,0:37:44.64,0:37:45.61,Default,,0000,0000,0000,,You are not lost.
Dialogue: 0,0:37:45.61,0:37:47.53,Default,,0000,0000,0000,,We can do this together.
Dialogue: 0,0:37:47.53,0:37:51.48,Default,,0000,0000,0000,,Now what's the simplest\Nway to parametrize
Dialogue: 0,0:37:51.48,0:37:57.26,Default,,0000,0000,0000,,a graph of the type\Nz equals f of xy?
Dialogue: 0,0:37:57.26,0:38:01.97,Default,,0000,0000,0000,,Take the xy to be\Nu and v. Take x
Dialogue: 0,0:38:01.97,0:38:05.36,Default,,0000,0000,0000,,and y to be your\Nindependent variables
Dialogue: 0,0:38:05.36,0:38:07.85,Default,,0000,0000,0000,,and take z to be the\Ndependent variable.
Dialogue: 0,0:38:07.85,0:38:12.70,Default,,0000,0000,0000,,
Dialogue: 0,0:38:12.70,0:38:16.93,Default,,0000,0000,0000,,I'm again expressing these\Nthings in terms of variables
Dialogue: 0,0:38:16.93,0:38:18.34,Default,,0000,0000,0000,,like I did last time.
Dialogue: 0,0:38:18.34,0:38:23.37,Default,,0000,0000,0000,,Then I say, let's take this kind\Nof parametrization. [INAUDIBLE]
Dialogue: 0,0:38:23.37,0:38:24.38,Default,,0000,0000,0000,,vu, right.
Dialogue: 0,0:38:24.38,0:38:33.08,Default,,0000,0000,0000,,y would be v. Then I'm\Ngoing to write r of x and y
Dialogue: 0,0:38:33.08,0:38:36.71,Default,,0000,0000,0000,,just like that guy will\Nbe [INAUDIBLE] of xn.
Dialogue: 0,0:38:36.71,0:38:38.77,Default,,0000,0000,0000,,[? y ?] will say, wait a minute.
Dialogue: 0,0:38:38.77,0:38:42.88,Default,,0000,0000,0000,,I will have to re-denote\Neverybody with capitals.
Dialogue: 0,0:38:42.88,0:38:46.30,Default,,0000,0000,0000,,Then my life will become\Nbetter because you
Dialogue: 0,0:38:46.30,0:38:47.30,Default,,0000,0000,0000,,don't have to erase.
Dialogue: 0,0:38:47.30,0:38:50.67,Default,,0000,0000,0000,,You just make little\Nx big, little y bigs,
Dialogue: 0,0:38:50.67,0:38:53.89,Default,,0000,0000,0000,,bigs, big, capitalized XYZ.
Dialogue: 0,0:38:53.89,0:39:02.15,Default,,0000,0000,0000,,And then I'll say OK, XYZ\Nwill be my setting here in 3D.
Dialogue: 0,0:39:02.15,0:39:07.02,Default,,0000,0000,0000,,
Dialogue: 0,0:39:07.02,0:39:07.56,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:39:07.56,0:39:10.29,Default,,0000,0000,0000,,So how am I going\Nto re-parametrize
Dialogue: 0,0:39:10.29,0:39:12.58,Default,,0000,0000,0000,,the whole surface?
Dialogue: 0,0:39:12.58,0:39:22.22,Default,,0000,0000,0000,,Whole surface will be r of\Nxy equals in this case, well,
Dialogue: 0,0:39:22.22,0:39:23.28,Default,,0000,0000,0000,,let's think about it.
Dialogue: 0,0:39:23.28,0:39:29.02,Default,,0000,0000,0000,,In this case, I'm\Ngoing to have xy.
Dialogue: 0,0:39:29.02,0:39:31.35,Default,,0000,0000,0000,,And where's the little f?
Dialogue: 0,0:39:31.35,0:39:32.55,Default,,0000,0000,0000,,I just erased it.
Dialogue: 0,0:39:32.55,0:39:35.08,Default,,0000,0000,0000,,I was smart, right,\Nthat I erased f of xy.
Dialogue: 0,0:39:35.08,0:39:37.83,Default,,0000,0000,0000,,
Dialogue: 0,0:39:37.83,0:39:46.01,Default,,0000,0000,0000,,So I have x, y, and\Nz, which is f of xy.
Dialogue: 0,0:39:46.01,0:39:53.24,Default,,0000,0000,0000,,
Dialogue: 0,0:39:53.24,0:40:01.43,Default,,0000,0000,0000,,And this is the generic point\Np of coordinates xy f of xy.
Dialogue: 0,0:40:01.43,0:40:04.98,Default,,0000,0000,0000,,
Dialogue: 0,0:40:04.98,0:40:07.58,Default,,0000,0000,0000,,So I say, OK, what does it mean?
Dialogue: 0,0:40:07.58,0:40:10.10,Default,,0000,0000,0000,,I will project this point.
Dialogue: 0,0:40:10.10,0:40:13.18,Default,,0000,0000,0000,,And this is the point\Nwhen big x becomes little
Dialogue: 0,0:40:13.18,0:40:17.86,Default,,0000,0000,0000,,x, when big y becomes--\Nwhere is my y-axis?
Dialogue: 0,0:40:17.86,0:40:20.09,Default,,0000,0000,0000,,Somebody ate my y axis.
Dialogue: 0,0:40:20.09,0:40:22.19,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:40:22.19,0:40:28.40,Default,,0000,0000,0000,,So when big Y becomes\Nlittle y, little y
Dialogue: 0,0:40:28.40,0:40:33.83,Default,,0000,0000,0000,,is just an instance of big Y.\NAnd big Z will take what value?
Dialogue: 0,0:40:33.83,0:40:35.63,Default,,0000,0000,0000,,Well, I need to project that.
Dialogue: 0,0:40:35.63,0:40:39.12,Default,,0000,0000,0000,,How do you project from\Na point to the z-axis?
Dialogue: 0,0:40:39.12,0:40:42.68,Default,,0000,0000,0000,,You have to take the\Nparallel from the point
Dialogue: 0,0:40:42.68,0:40:47.63,Default,,0000,0000,0000,,to the horizontal\Nplane until you
Dialogue: 0,0:40:47.63,0:40:52.94,Default,,0000,0000,0000,,hit the-- [INAUDIBLE] the whole\Nplane parallel to the floor
Dialogue: 0,0:40:52.94,0:40:54.21,Default,,0000,0000,0000,,through the point p.
Dialogue: 0,0:40:54.21,0:40:55.45,Default,,0000,0000,0000,,And what do I get here?
Dialogue: 0,0:40:55.45,0:40:56.41,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:40:56.41,0:40:58.67,Default,,0000,0000,0000,,PROFESSOR TODA: Not\Nz0, but it's little z
Dialogue: 0,0:40:58.67,0:41:03.12,Default,,0000,0000,0000,,equals f of xy, which is an\Ninstance of the variable xz.
Dialogue: 0,0:41:03.12,0:41:06.46,Default,,0000,0000,0000,,For you programmers, you know\Nthat big z will be a variable
Dialogue: 0,0:41:06.46,0:41:11.64,Default,,0000,0000,0000,,and little z will be\N[INAUDIBLE] a variable.
Dialogue: 0,0:41:11.64,0:41:12.14,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:41:12.14,0:41:16.61,Default,,0000,0000,0000,,So I parametrized my graph\Nin a more general way,
Dialogue: 0,0:41:16.61,0:41:18.58,Default,,0000,0000,0000,,general parametrization\Nfor a graph.
Dialogue: 0,0:41:18.58,0:41:25.96,Default,,0000,0000,0000,,
Dialogue: 0,0:41:25.96,0:41:33.42,Default,,0000,0000,0000,,And now, what are-- what's the\Nmeaning of r sub x and r sub y?
Dialogue: 0,0:41:33.42,0:41:34.49,Default,,0000,0000,0000,,What are they?
Dialogue: 0,0:41:34.49,0:41:35.36,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:41:35.36,0:41:38.18,Default,,0000,0000,0000,,
Dialogue: 0,0:41:38.18,0:41:41.66,Default,,0000,0000,0000,,PROFESSOR TODA: Now, we\Ndon't say that in the book.
Dialogue: 0,0:41:41.66,0:41:42.99,Default,,0000,0000,0000,,Shame on us.
Dialogue: 0,0:41:42.99,0:41:43.63,Default,,0000,0000,0000,,Shame on us.
Dialogue: 0,0:41:43.63,0:41:47.48,Default,,0000,0000,0000,,We should have because I was\Nbrowsing through the projects
Dialogue: 0,0:41:47.48,0:41:49.90,Default,,0000,0000,0000,,about a year and a half ago.
Dialogue: 0,0:41:49.90,0:41:52.97,Default,,0000,0000,0000,,The senior projects of\Na few of my students
Dialogue: 0,0:41:52.97,0:41:56.34,Default,,0000,0000,0000,,who are-- two of them were\Nin mechanical engineering.
Dialogue: 0,0:41:56.34,0:42:00.66,Default,,0000,0000,0000,,One of them was in\Npetroleum engineering.
Dialogue: 0,0:42:00.66,0:42:03.96,Default,,0000,0000,0000,,And he actually showed me\Nthat they were doing this.
Dialogue: 0,0:42:03.96,0:42:07.83,Default,,0000,0000,0000,,They were taking vectors\Nthat depend on parameters--
Dialogue: 0,0:42:07.83,0:42:11.25,Default,,0000,0000,0000,,this is a vector [INAUDIBLE]--\Nand differentiated them with
Dialogue: 0,0:42:11.25,0:42:13.72,Default,,0000,0000,0000,,respect to those parameters.
Dialogue: 0,0:42:13.72,0:42:17.22,Default,,0000,0000,0000,,And I was thinking OK, did we\Ndo the partial derivatives r sub
Dialogue: 0,0:42:17.22,0:42:17.96,Default,,0000,0000,0000,,x, r sub y?
Dialogue: 0,0:42:17.96,0:42:19.34,Default,,0000,0000,0000,,Not so much.
Dialogue: 0,0:42:19.34,0:42:22.38,Default,,0000,0000,0000,,But now I want to do it\Nbecause I think that prepares
Dialogue: 0,0:42:22.38,0:42:24.64,Default,,0000,0000,0000,,you better as engineers.
Dialogue: 0,0:42:24.64,0:42:29.07,Default,,0000,0000,0000,,So what is r sub x\Nand what is r sub y?
Dialogue: 0,0:42:29.07,0:42:31.25,Default,,0000,0000,0000,,And you say, well,\NOK. [INAUDIBLE],
Dialogue: 0,0:42:31.25,0:42:34.86,Default,,0000,0000,0000,,I think I know how to do\Nthat in my sleep, right.
Dialogue: 0,0:42:34.86,0:42:36.78,Default,,0000,0000,0000,,If you want me to do\Nthat theoretically
Dialogue: 0,0:42:36.78,0:42:39.72,Default,,0000,0000,0000,,from this formula,\Nbut on the picture,
Dialogue: 0,0:42:39.72,0:42:42.45,Default,,0000,0000,0000,,I really don't know what it is.
Dialogue: 0,0:42:42.45,0:42:45.59,Default,,0000,0000,0000,,So I'm asking you what\NI'm going to have in terms
Dialogue: 0,0:42:45.59,0:42:47.24,Default,,0000,0000,0000,,of r sub x and r sub y.
Dialogue: 0,0:42:47.24,0:42:48.95,Default,,0000,0000,0000,,They will be vectors.
Dialogue: 0,0:42:48.95,0:42:51.88,Default,,0000,0000,0000,,This should be a\Nvector as well, right.
Dialogue: 0,0:42:51.88,0:42:56.62,Default,,0000,0000,0000,,And for me, vector triple\Nmeans the identification
Dialogue: 0,0:42:56.62,0:42:59.93,Default,,0000,0000,0000,,between the three coordinates\Nand the physical vector.
Dialogue: 0,0:42:59.93,0:43:01.96,Default,,0000,0000,0000,,So this is the physical vector.
Dialogue: 0,0:43:01.96,0:43:06.03,Default,,0000,0000,0000,,Go ahead and write x prime\Nwith respect to x is 1.
Dialogue: 0,0:43:06.03,0:43:08.68,Default,,0000,0000,0000,,
Dialogue: 0,0:43:08.68,0:43:13.78,Default,,0000,0000,0000,,y prime with respect to x is 0.
Dialogue: 0,0:43:13.78,0:43:15.97,Default,,0000,0000,0000,,The third [INAUDIBLE]\Nprime with respect
Dialogue: 0,0:43:15.97,0:43:20.19,Default,,0000,0000,0000,,to x is just whatever\Nthis little f is,
Dialogue: 0,0:43:20.19,0:43:21.98,Default,,0000,0000,0000,,it's not any of my business.
Dialogue: 0,0:43:21.98,0:43:24.79,Default,,0000,0000,0000,,It's a [INAUDIBLE]\Nfunction f sub x.
Dialogue: 0,0:43:24.79,0:43:28.29,Default,,0000,0000,0000,,
Dialogue: 0,0:43:28.29,0:43:30.59,Default,,0000,0000,0000,,Well, what is the second vector?
Dialogue: 0,0:43:30.59,0:43:32.28,Default,,0000,0000,0000,,STUDENT: 0, 1, f sub y.
Dialogue: 0,0:43:32.28,0:43:34.81,Default,,0000,0000,0000,,PROFESSOR TODA: 0, 1, f sub y.
Dialogue: 0,0:43:34.81,0:43:36.60,Default,,0000,0000,0000,,Now, are they slopes?
Dialogue: 0,0:43:36.60,0:43:37.10,Default,,0000,0000,0000,,No.
Dialogue: 0,0:43:37.10,0:43:38.01,Default,,0000,0000,0000,,These are slopes.
Dialogue: 0,0:43:38.01,0:43:40.77,Default,,0000,0000,0000,,That's a slope and\Nthat's a slope.
Dialogue: 0,0:43:40.77,0:43:44.95,Default,,0000,0000,0000,,And we learned\Nabout those in 11.3,
Dialogue: 0,0:43:44.95,0:43:49.53,Default,,0000,0000,0000,,and we understood that those\Nare ski slopes, they were.
Dialogue: 0,0:43:49.53,0:43:52.31,Default,,0000,0000,0000,,In the direction of x\Nand the direction of y,
Dialogue: 0,0:43:52.31,0:44:00.03,Default,,0000,0000,0000,,the slopes of the tangents\Nto the coordinate lines.
Dialogue: 0,0:44:00.03,0:44:04.98,Default,,0000,0000,0000,,But this looks like I have\Na direction of a line,
Dialogue: 0,0:44:04.98,0:44:08.61,Default,,0000,0000,0000,,and this would be the lope, and\Nthat's the direction of a line,
Dialogue: 0,0:44:08.61,0:44:10.31,Default,,0000,0000,0000,,and that would be the slope.
Dialogue: 0,0:44:10.31,0:44:12.70,Default,,0000,0000,0000,,What are those lines?
Dialogue: 0,0:44:12.70,0:44:16.31,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] to\Nthe function [INAUDIBLE].
Dialogue: 0,0:44:16.31,0:44:17.48,Default,,0000,0000,0000,,PROFESSOR TODA: Let me draw.
Dialogue: 0,0:44:17.48,0:44:19.44,Default,,0000,0000,0000,,Then shall I erase\Nthe whole thing?
Dialogue: 0,0:44:19.44,0:44:20.16,Default,,0000,0000,0000,,No.
Dialogue: 0,0:44:20.16,0:44:23.95,Default,,0000,0000,0000,,I'm just going to keep--\NI'll erase the tangent.
Dialogue: 0,0:44:23.95,0:44:27.47,Default,,0000,0000,0000,,Don't erase anything\Non your notebooks.
Dialogue: 0,0:44:27.47,0:44:28.92,Default,,0000,0000,0000,,So this is the point p.
Dialogue: 0,0:44:28.92,0:44:29.63,Default,,0000,0000,0000,,It's still there.
Dialogue: 0,0:44:29.63,0:44:30.57,Default,,0000,0000,0000,,This is the surface.
Dialogue: 0,0:44:30.57,0:44:33.06,Default,,0000,0000,0000,,It's still there.
Dialogue: 0,0:44:33.06,0:44:38.20,Default,,0000,0000,0000,,So my surface will be x,\Nslices of x, [? S ?] constant
Dialogue: 0,0:44:38.20,0:44:39.59,Default,,0000,0000,0000,,are coming towards you.
Dialogue: 0,0:44:39.59,0:44:45.80,Default,,0000,0000,0000,,They are these [? walls ?]\Nlike that, like this, yes.
Dialogue: 0,0:44:45.80,0:44:47.61,Default,,0000,0000,0000,,It's like the CT scan.
Dialogue: 0,0:44:47.61,0:44:52.19,Default,,0000,0000,0000,,I think that when they\Nslice up your body,
Dialogue: 0,0:44:52.19,0:44:54.26,Default,,0000,0000,0000,,tch tch tch tch tch\Ntch, take pictures
Dialogue: 0,0:44:54.26,0:44:57.59,Default,,0000,0000,0000,,of the slices of your body,\Nthat's the same kind of thing.
Dialogue: 0,0:44:57.59,0:44:59.51,Default,,0000,0000,0000,,So x0, x0, x0, x0.
Dialogue: 0,0:44:59.51,0:45:05.41,Default,,0000,0000,0000,,I'm going to [INAUDIBLE]\Nplanes and I had x equals x0.
Dialogue: 0,0:45:05.41,0:45:12.40,Default,,0000,0000,0000,,And in the other direction, I\Ncut and I get, what do I get?
Dialogue: 0,0:45:12.40,0:45:18.40,Default,,0000,0000,0000,,
Dialogue: 0,0:45:18.40,0:45:20.23,Default,,0000,0000,0000,,Well, I started bad.
Dialogue: 0,0:45:20.23,0:45:23.65,Default,,0000,0000,0000,,
Dialogue: 0,0:45:23.65,0:45:25.20,Default,,0000,0000,0000,,Great, Magdalena, this is--
Dialogue: 0,0:45:25.20,0:45:27.23,Default,,0000,0000,0000,,What is this pink?
Dialogue: 0,0:45:27.23,0:45:32.35,Default,,0000,0000,0000,,It's not Valentine's Day\Nanymore. y equals [INAUDIBLE].
Dialogue: 0,0:45:32.35,0:45:34.81,Default,,0000,0000,0000,,And this is the point.
Dialogue: 0,0:45:34.81,0:45:39.32,Default,,0000,0000,0000,,So, as Alex was\Ntrying to tell you,
Dialogue: 0,0:45:39.32,0:45:44.98,Default,,0000,0000,0000,,our sub x would represent the\Nvector, the physical vector
Dialogue: 0,0:45:44.98,0:45:52.26,Default,,0000,0000,0000,,in 3D, that is originating\Nat p and tangent to which
Dialogue: 0,0:45:52.26,0:45:55.76,Default,,0000,0000,0000,,of the two, to the purple\None or to the red one?
Dialogue: 0,0:45:55.76,0:45:57.18,Default,,0000,0000,0000,,STUDENT: Red.
Dialogue: 0,0:45:57.18,0:45:58.14,Default,,0000,0000,0000,,Uh, purple.
Dialogue: 0,0:45:58.14,0:45:59.56,Default,,0000,0000,0000,,PROFESSOR TODA:\NMake up your mind.
Dialogue: 0,0:45:59.56,0:46:01.49,Default,,0000,0000,0000,,STUDENT: The purple one.
Dialogue: 0,0:46:01.49,0:46:03.66,Default,,0000,0000,0000,,PROFESSOR TODA: [INAUDIBLE]\Nconstant and [INAUDIBLE]
Dialogue: 0,0:46:03.66,0:46:06.77,Default,,0000,0000,0000,,constant in the red\None, y equals y0, right?
Dialogue: 0,0:46:06.77,0:46:08.92,Default,,0000,0000,0000,,So, this depends on x.
Dialogue: 0,0:46:08.92,0:46:11.01,Default,,0000,0000,0000,,So this has r sub x.
Dialogue: 0,0:46:11.01,0:46:14.80,Default,,0000,0000,0000,,
Dialogue: 0,0:46:14.80,0:46:18.83,Default,,0000,0000,0000,,This is the velocity with\Nrespect to the variable x.
Dialogue: 0,0:46:18.83,0:46:23.20,Default,,0000,0000,0000,,And the other one, the\Nblue one, x equals x0,
Dialogue: 0,0:46:23.20,0:46:27.64,Default,,0000,0000,0000,,means x0 is held fixed\Nand y is the variable.
Dialogue: 0,0:46:27.64,0:46:30.50,Default,,0000,0000,0000,,So I have to do r sub y,\Nand what am I gonna get?
Dialogue: 0,0:46:30.50,0:46:32.70,Default,,0000,0000,0000,,I'm gonna get the blue vector.
Dialogue: 0,0:46:32.70,0:46:34.88,Default,,0000,0000,0000,,What's the property\Nof the blue vector?
Dialogue: 0,0:46:34.88,0:46:37.83,Default,,0000,0000,0000,,It's tangent to the purple line.
Dialogue: 0,0:46:37.83,0:46:44.16,Default,,0000,0000,0000,,So r sub y has to be\Ntangent to the curve.
Dialogue: 0,0:46:44.16,0:46:47.44,Default,,0000,0000,0000,,
Dialogue: 0,0:46:47.44,0:46:55.31,Default,,0000,0000,0000,,x0, y, f of x0 and\Ny is the curve.
Dialogue: 0,0:46:55.31,0:46:59.77,Default,,0000,0000,0000,,And r sub x is tangent\Nto which curve?
Dialogue: 0,0:46:59.77,0:47:02.40,Default,,0000,0000,0000,,Who is telling me which curve?
Dialogue: 0,0:47:02.40,0:47:12.02,Default,,0000,0000,0000,,x, y0 sub constant,\Nf of x and y0.
Dialogue: 0,0:47:12.02,0:47:14.49,Default,,0000,0000,0000,,So that's a curve that\Ndepends only on y,
Dialogue: 0,0:47:14.49,0:47:16.85,Default,,0000,0000,0000,,y is the time in this case.
Dialogue: 0,0:47:16.85,0:47:19.00,Default,,0000,0000,0000,,And that's the curve\Nthat depends only on x.
Dialogue: 0,0:47:19.00,0:47:21.21,Default,,0000,0000,0000,,x is the time in this case.
Dialogue: 0,0:47:21.21,0:47:24.58,Default,,0000,0000,0000,,r sub x and r sub y are\Nthe tangent vectors.
Dialogue: 0,0:47:24.58,0:47:26.83,Default,,0000,0000,0000,,What's magical about them?
Dialogue: 0,0:47:26.83,0:47:30.54,Default,,0000,0000,0000,,If I shape this\Ntriangle between them,
Dialogue: 0,0:47:30.54,0:47:32.17,Default,,0000,0000,0000,,that will be the tangent plane.
Dialogue: 0,0:47:32.17,0:47:35.95,Default,,0000,0000,0000,,
Dialogue: 0,0:47:35.95,0:47:39.17,Default,,0000,0000,0000,,And I make a smile because I\Ndiscovered the tangent plane
Dialogue: 0,0:47:39.17,0:47:43.23,Default,,0000,0000,0000,,in a different way than\Nwe did it last time.
Dialogue: 0,0:47:43.23,0:47:51.00,Default,,0000,0000,0000,,So the tangent plane represents\Nthe plane of the vector r sub
Dialogue: 0,0:47:51.00,0:47:54.53,Default,,0000,0000,0000,,x and r sub y.
Dialogue: 0,0:47:54.53,0:48:02.29,Default,,0000,0000,0000,,The tangent plane\Nrepresents the plane
Dialogue: 0,0:48:02.29,0:48:13.08,Default,,0000,0000,0000,,given by vectors r sub x and\Nr sub y with what conditions?
Dialogue: 0,0:48:13.08,0:48:14.02,Default,,0000,0000,0000,,It's a conditional.
Dialogue: 0,0:48:14.02,0:48:17.01,Default,,0000,0000,0000,,
Dialogue: 0,0:48:17.01,0:48:20.63,Default,,0000,0000,0000,,r sub x and r sub\Ny shouldn't be 0.
Dialogue: 0,0:48:20.63,0:48:24.85,Default,,0000,0000,0000,,r sub x different from 0,\Nr sub y different from 0,
Dialogue: 0,0:48:24.85,0:48:27.46,Default,,0000,0000,0000,,and r sub x and r sub\Ny are not collinear.
Dialogue: 0,0:48:27.46,0:48:32.16,Default,,0000,0000,0000,,
Dialogue: 0,0:48:32.16,0:48:35.05,Default,,0000,0000,0000,,What's gonna happen\Nif they are collinear?
Dialogue: 0,0:48:35.05,0:48:36.88,Default,,0000,0000,0000,,Well, they're gonna\Ncollapse; they are not
Dialogue: 0,0:48:36.88,0:48:38.19,Default,,0000,0000,0000,,gonna determine a plane.
Dialogue: 0,0:48:38.19,0:48:40.77,Default,,0000,0000,0000,,So there will be\Nno tangent planes.
Dialogue: 0,0:48:40.77,0:48:43.72,Default,,0000,0000,0000,,So they have to be\Nlinearly independent.
Dialogue: 0,0:48:43.72,0:48:47.94,Default,,0000,0000,0000,,For the people who are taking\Nnow linear algebra, I'm saying.
Dialogue: 0,0:48:47.94,0:48:50.94,Default,,0000,0000,0000,,So we have no other\Nchoice, we have
Dialogue: 0,0:48:50.94,0:48:54.82,Default,,0000,0000,0000,,to assume that these vectors,\Ncalled partial velocities,
Dialogue: 0,0:48:54.82,0:49:04.12,Default,,0000,0000,0000,,by the way, for the\Nmotion across the surface.
Dialogue: 0,0:49:04.12,0:49:04.62,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:49:04.62,0:49:06.97,Default,,0000,0000,0000,,These are the partial\Nvelocities, or partial velocity
Dialogue: 0,0:49:06.97,0:49:08.63,Default,,0000,0000,0000,,vectors.
Dialogue: 0,0:49:08.63,0:49:12.86,Default,,0000,0000,0000,,Partial velocity vectors\Nhave to determine a plane,
Dialogue: 0,0:49:12.86,0:49:16.56,Default,,0000,0000,0000,,so I have to assume\Nthat they are non-zero,
Dialogue: 0,0:49:16.56,0:49:20.12,Default,,0000,0000,0000,,they never become 0, and\Nthey are not collinear.
Dialogue: 0,0:49:20.12,0:49:23.27,Default,,0000,0000,0000,,If they are collinear,\Nlife is over for you.
Dialogue: 0,0:49:23.27,0:49:24.14,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:49:24.14,0:49:29.39,Default,,0000,0000,0000,,So I have to assume that I\Nthrow away all the points where
Dialogue: 0,0:49:29.39,0:49:35.10,Default,,0000,0000,0000,,the velocities become 0, and\Nall the points where--those are
Dialogue: 0,0:49:35.10,0:49:39.71,Default,,0000,0000,0000,,singularity points--where\Nmy velocity vectors are 0.
Dialogue: 0,0:49:39.71,0:49:43.71,Default,,0000,0000,0000,,
Dialogue: 0,0:49:43.71,0:49:45.82,Default,,0000,0000,0000,,Have you ever studied design?
Dialogue: 0,0:49:45.82,0:49:47.35,Default,,0000,0000,0000,,Any kind of experimental design.
Dialogue: 0,0:49:47.35,0:49:52.31,Default,,0000,0000,0000,,Like, how do you design a car,\Nthe coordinate lines on a car?
Dialogue: 0,0:49:52.31,0:49:53.28,Default,,0000,0000,0000,,I'm just dreaming.
Dialogue: 0,0:49:53.28,0:50:00.20,Default,,0000,0000,0000,,You have a car, a beautiful\Ncar, and then you have-- Well,
Dialogue: 0,0:50:00.20,0:50:04.79,Default,,0000,0000,0000,,I cannot draw really\Nwell, but anyway.
Dialogue: 0,0:50:04.79,0:50:08.73,Default,,0000,0000,0000,,I have these coordinate\Nlines on this car.
Dialogue: 0,0:50:08.73,0:50:12.06,Default,,0000,0000,0000,,It's a mesh what I have there.
Dialogue: 0,0:50:12.06,0:50:15.51,Default,,0000,0000,0000,,Actually, we do that in\Nanimation all the time.
Dialogue: 0,0:50:15.51,0:50:21.03,Default,,0000,0000,0000,,We have meshes over the\Nmodels we have in animation.
Dialogue: 0,0:50:21.03,0:50:22.66,Default,,0000,0000,0000,,Think Avatar.
Dialogue: 0,0:50:22.66,0:50:27.21,Default,,0000,0000,0000,,Now, those are all\Ncoordinate lines.
Dialogue: 0,0:50:27.21,0:50:33.65,Default,,0000,0000,0000,,Those coordinate lines would be,\Neven your singularities, where?
Dialogue: 0,0:50:33.65,0:50:38.51,Default,,0000,0000,0000,,For example, if you take a body\Nin a mesh like that, in a net,
Dialogue: 0,0:50:38.51,0:50:43.19,Default,,0000,0000,0000,,in, like, a fishnet, then\Nyou pull from the fishnet,
Dialogue: 0,0:50:43.19,0:50:52.98,Default,,0000,0000,0000,,all the coordinate lines\Nwill come together,
Dialogue: 0,0:50:52.98,0:50:55.31,Default,,0000,0000,0000,,and this would be a singularity.
Dialogue: 0,0:50:55.31,0:50:57.89,Default,,0000,0000,0000,,We avoid this kind\Nof singularity.
Dialogue: 0,0:50:57.89,0:51:00.43,Default,,0000,0000,0000,,So these are points where\Nsomething bad happened.
Dialogue: 0,0:51:00.43,0:51:05.38,Default,,0000,0000,0000,,Either the velocity\Nvectors become collinear.
Dialogue: 0,0:51:05.38,0:51:07.43,Default,,0000,0000,0000,,You see what I'm talking about?
Dialogue: 0,0:51:07.43,0:51:11.26,Default,,0000,0000,0000,,Or the velocity\Nvectors shrank to 0.
Dialogue: 0,0:51:11.26,0:51:14.19,Default,,0000,0000,0000,,So that's a bad point;\Nthat's a singularity point.
Dialogue: 0,0:51:14.19,0:51:16.87,Default,,0000,0000,0000,,They have this\Nproblem when meshing.
Dialogue: 0,0:51:16.87,0:51:20.67,Default,,0000,0000,0000,,So when they make\Nthese models that
Dialogue: 0,0:51:20.67,0:51:26.85,Default,,0000,0000,0000,,involve two-dimensional meshing\Nand three-dimensional ambient
Dialogue: 0,0:51:26.85,0:51:31.49,Default,,0000,0000,0000,,space, like it is in\Nanimation, the mesh
Dialogue: 0,0:51:31.49,0:51:34.63,Default,,0000,0000,0000,,is called regular\Nif we don't have
Dialogue: 0,0:51:34.63,0:51:39.77,Default,,0000,0000,0000,,this kind of singularity, where\Nthe velocity vectors become 0,
Dialogue: 0,0:51:39.77,0:51:42.00,Default,,0000,0000,0000,,or collinear.
Dialogue: 0,0:51:42.00,0:51:45.80,Default,,0000,0000,0000,,It's very important for a\Nperson who programs in animation
Dialogue: 0,0:51:45.80,0:51:47.22,Default,,0000,0000,0000,,to know mathematics.
Dialogue: 0,0:51:47.22,0:51:50.10,Default,,0000,0000,0000,,If they don't understand\Nthese things, it's over.
Dialogue: 0,0:51:50.10,0:51:55.91,Default,,0000,0000,0000,,Because you write the matrix,\Nand you will know the vectors
Dialogue: 0,0:51:55.91,0:51:59.95,Default,,0000,0000,0000,,will become collinear when the\Ntwo vectors--let's say two rows
Dialogue: 0,0:51:59.95,0:52:00.50,Default,,0000,0000,0000,,of a matrix--
Dialogue: 0,0:52:00.50,0:52:00.81,Default,,0000,0000,0000,,STUDENT: Parallel.
Dialogue: 0,0:52:00.81,0:52:01.88,Default,,0000,0000,0000,,PROFESSOR TODA:\NAre proportional.
Dialogue: 0,0:52:01.88,0:52:02.51,Default,,0000,0000,0000,,Or parallel.
Dialogue: 0,0:52:02.51,0:52:03.87,Default,,0000,0000,0000,,Or proportional.
Dialogue: 0,0:52:03.87,0:52:07.55,Default,,0000,0000,0000,,So, everything is numerical\Nin terms of those matrices,
Dialogue: 0,0:52:07.55,0:52:12.89,Default,,0000,0000,0000,,but it's just a discretization\Nof a continuous phenomenon,
Dialogue: 0,0:52:12.89,0:52:14.01,Default,,0000,0000,0000,,which is this one.
Dialogue: 0,0:52:14.01,0:52:17.69,Default,,0000,0000,0000,,
Dialogue: 0,0:52:17.69,0:52:19.97,Default,,0000,0000,0000,,Do you remember Toy Story?
Dialogue: 0,0:52:19.97,0:52:20.82,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:52:20.82,0:52:24.30,Default,,0000,0000,0000,,The Toy Story people,\Nthe renderers,
Dialogue: 0,0:52:24.30,0:52:27.01,Default,,0000,0000,0000,,the ones who did the rendering\Ntechniques for Toy Story,
Dialogue: 0,0:52:27.01,0:52:30.41,Default,,0000,0000,0000,,both have their\Nmaster's in mathematics.
Dialogue: 0,0:52:30.41,0:52:33.97,Default,,0000,0000,0000,,And you realize why\Nnow to do that you
Dialogue: 0,0:52:33.97,0:52:38.86,Default,,0000,0000,0000,,have to know calc I, calc\NII, calc III, linear algebra,
Dialogue: 0,0:52:38.86,0:52:41.12,Default,,0000,0000,0000,,be able to deal with matrices.
Dialogue: 0,0:52:41.12,0:52:45.61,Default,,0000,0000,0000,,Have a programming course\Nor two; that's essential.
Dialogue: 0,0:52:45.61,0:52:50.04,Default,,0000,0000,0000,,They took advanced calculus\Nbecause some people
Dialogue: 0,0:52:50.04,0:52:55.42,Default,,0000,0000,0000,,don't cover thi-- I was about to\Nskip it right now in calc III.
Dialogue: 0,0:52:55.42,0:53:00.11,Default,,0000,0000,0000,,But they teach that in\Nadvanced calculus 4350, 4351.
Dialogue: 0,0:53:00.11,0:53:02.67,Default,,0000,0000,0000,,So that's about as\Nfar as you can get,
Dialogue: 0,0:53:02.67,0:53:05.87,Default,,0000,0000,0000,,and differential equation's\Nalso very important.
Dialogue: 0,0:53:05.87,0:53:09.51,Default,,0000,0000,0000,,So, if you master those and\Nyou go into something else,
Dialogue: 0,0:53:09.51,0:53:12.32,Default,,0000,0000,0000,,like programming,\Nelectrical engineering,
Dialogue: 0,0:53:12.32,0:53:14.25,Default,,0000,0000,0000,,you're ready for animation.
Dialogue: 0,0:53:14.25,0:53:16.97,Default,,0000,0000,0000,,[INAUDIBLE] If you went\NI want to be a rendering
Dialogue: 0,0:53:16.97,0:53:20.14,Default,,0000,0000,0000,,guy for the next movie,\Nthen they'll say no,
Dialogue: 0,0:53:20.14,0:53:21.60,Default,,0000,0000,0000,,we won't take you.
Dialogue: 0,0:53:21.60,0:53:23.92,Default,,0000,0000,0000,,I have a friend who\Nworks for Disney.
Dialogue: 0,0:53:23.92,0:53:26.78,Default,,0000,0000,0000,,She wanted to get a PhD.
Dialogue: 0,0:53:26.78,0:53:29.38,Default,,0000,0000,0000,,At some point, she\Nchanged her mind
Dialogue: 0,0:53:29.38,0:53:31.97,Default,,0000,0000,0000,,and ended up just with a\Nmaster's in mathematics
Dialogue: 0,0:53:31.97,0:53:33.80,Default,,0000,0000,0000,,while I was in Kansas,\NUniversity of Kansas,
Dialogue: 0,0:53:33.80,0:53:36.56,Default,,0000,0000,0000,,and she said, "You know what?
Dialogue: 0,0:53:36.56,0:53:41.62,Default,,0000,0000,0000,,Disney's just giving me\N$65,000 as an intern."
Dialogue: 0,0:53:41.62,0:53:45.58,Default,,0000,0000,0000,,And I was like OK and probably\Nasked [INAUDIBLE] $40,000 as
Dialogue: 0,0:53:45.58,0:53:46.68,Default,,0000,0000,0000,,a postdoc.
Dialogue: 0,0:53:46.68,0:53:48.05,Default,,0000,0000,0000,,And she said,\N"Good luck to you."
Dialogue: 0,0:53:48.05,0:53:49.42,Default,,0000,0000,0000,,Good luck to you, too.
Dialogue: 0,0:53:49.42,0:53:52.52,Default,,0000,0000,0000,,But we stayed in touch,\Nand right now she's
Dialogue: 0,0:53:52.52,0:53:57.46,Default,,0000,0000,0000,,making twice as much as\NI'm making, for Disney.
Dialogue: 0,0:53:57.46,0:53:58.68,Default,,0000,0000,0000,,Is she happy?
Dialogue: 0,0:53:58.68,0:53:59.57,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:53:59.57,0:54:00.46,Default,,0000,0000,0000,,Would I be happy?
Dialogue: 0,0:54:00.46,0:54:01.41,Default,,0000,0000,0000,,No.
Dialogue: 0,0:54:01.41,0:54:05.76,Default,,0000,0000,0000,,Because she works\Nfor 11 hours a day.
Dialogue: 0,0:54:05.76,0:54:08.12,Default,,0000,0000,0000,,11 hours a day, on a chair.
Dialogue: 0,0:54:08.12,0:54:09.09,Default,,0000,0000,0000,,That would kill me.
Dialogue: 0,0:54:09.09,0:54:15.07,Default,,0000,0000,0000,,I mean, I spend about six hours\Nsitting on a chair every day
Dialogue: 0,0:54:15.07,0:54:19.16,Default,,0000,0000,0000,,of the week, but\Nit's still too much.
Dialogue: 0,0:54:19.16,0:54:20.80,Default,,0000,0000,0000,,She's a hard worker, though.
Dialogue: 0,0:54:20.80,0:54:22.82,Default,,0000,0000,0000,,She loves what she's doing.
Dialogue: 0,0:54:22.82,0:54:24.06,Default,,0000,0000,0000,,The problem is your eyes.
Dialogue: 0,0:54:24.06,0:54:27.42,Default,,0000,0000,0000,,After a while, your\Neyes are going bad.
Dialogue: 0,0:54:27.42,0:54:33.60,Default,,0000,0000,0000,,So, what is the normal for\Nthe plane in this case?
Dialogue: 0,0:54:33.60,0:54:37.30,Default,,0000,0000,0000,,I'll try my best\Nability to draw normal.
Dialogue: 0,0:54:37.30,0:54:38.71,Default,,0000,0000,0000,,The normal has to\Nbe perpendicular
Dialogue: 0,0:54:38.71,0:54:41.95,Default,,0000,0000,0000,,to the tangent space, right?
Dialogue: 0,0:54:41.95,0:54:43.70,Default,,0000,0000,0000,,Tangent plane.
Dialogue: 0,0:54:43.70,0:54:46.23,Default,,0000,0000,0000,,So, n has to be\Nperpendicular to our sub
Dialogue: 0,0:54:46.23,0:54:49.79,Default,,0000,0000,0000,,x and has to be\Nperpendicular to our sub y.
Dialogue: 0,0:54:49.79,0:54:53.04,Default,,0000,0000,0000,,
Dialogue: 0,0:54:53.04,0:54:56.24,Default,,0000,0000,0000,,So, can you have any\Nguess how in the world
Dialogue: 0,0:54:56.24,0:54:59.47,Default,,0000,0000,0000,,I'm gonna get n vector?
Dialogue: 0,0:54:59.47,0:55:01.45,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:55:01.45,0:55:02.87,Default,,0000,0000,0000,,PROFESSOR TODA:\N[INAUDIBLE] That's
Dialogue: 0,0:55:02.87,0:55:05.07,Default,,0000,0000,0000,,why you need to\Nknow linear algebra
Dialogue: 0,0:55:05.07,0:55:09.04,Default,,0000,0000,0000,,sort of at the same time, but\Nyou guys are making it fine.
Dialogue: 0,0:55:09.04,0:55:10.46,Default,,0000,0000,0000,,It's not a big deal.
Dialogue: 0,0:55:10.46,0:55:16.45,Default,,0000,0000,0000,,You have a matrix, i, j, k\Nin the front row vectors,
Dialogue: 0,0:55:16.45,0:55:21.57,Default,,0000,0000,0000,,and then you have r sub x that\Nyou gave me, and I erased it.
Dialogue: 0,0:55:21.57,0:55:23.60,Default,,0000,0000,0000,,1, 0, f sub x.
Dialogue: 0,0:55:23.60,0:55:26.59,Default,,0000,0000,0000,,
Dialogue: 0,0:55:26.59,0:55:29.15,Default,,0000,0000,0000,,0, 1, f sub y.
Dialogue: 0,0:55:29.15,0:55:40.60,Default,,0000,0000,0000,,And you have exactly 18\Nseconds to compute this vector.
Dialogue: 0,0:55:40.60,0:55:47.63,Default,,0000,0000,0000,,
Dialogue: 0,0:55:47.63,0:55:48.46,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:55:48.46,0:55:52.89,Default,,0000,0000,0000,,
Dialogue: 0,0:55:52.89,0:55:55.69,Default,,0000,0000,0000,,PROFESSOR TODA: You want k, but\NI want to leave k at the end
Dialogue: 0,0:55:55.69,0:55:58.54,Default,,0000,0000,0000,,because I always\Norder my vectors.
Dialogue: 0,0:55:58.54,0:56:02.14,Default,,0000,0000,0000,,Something i plus something\Nj plus something k.
Dialogue: 0,0:56:02.14,0:56:02.97,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:56:02.97,0:56:05.28,Default,,0000,0000,0000,,
Dialogue: 0,0:56:05.28,0:56:06.40,Default,,0000,0000,0000,,PROFESSOR TODA: Am I right?
Dialogue: 0,0:56:06.40,0:56:07.02,Default,,0000,0000,0000,,Minus f sub x--
Dialogue: 0,0:56:07.02,0:56:09.91,Default,,0000,0000,0000,,STUDENT: Minus f of x plus k.
Dialogue: 0,0:56:09.91,0:56:11.89,Default,,0000,0000,0000,,PROFESSOR TODA: --times i.
Dialogue: 0,0:56:11.89,0:56:14.26,Default,,0000,0000,0000,,For j, do I have to change sign?
Dialogue: 0,0:56:14.26,0:56:18.37,Default,,0000,0000,0000,,Yeah, because 1 plus 2 is odd.
Dialogue: 0,0:56:18.37,0:56:21.27,Default,,0000,0000,0000,,So I go minus 1.
Dialogue: 0,0:56:21.27,0:56:22.60,Default,,0000,0000,0000,,And do it slowly.
Dialogue: 0,0:56:22.60,0:56:25.74,Default,,0000,0000,0000,,You're not gonna make fun of\Nme; I gotta make fun of you, OK?
Dialogue: 0,0:56:25.74,0:56:28.10,Default,,0000,0000,0000,,And minus 1 times--
Dialogue: 0,0:56:28.10,0:56:29.44,Default,,0000,0000,0000,,STUDENT: Did you forget f y?
Dialogue: 0,0:56:29.44,0:56:37.15,Default,,0000,0000,0000,,PROFESSOR TODA: --f sub y--I go\Nlike that--sub y times j plus
Dialogue: 0,0:56:37.15,0:56:39.23,Default,,0000,0000,0000,,k.
Dialogue: 0,0:56:39.23,0:56:42.12,Default,,0000,0000,0000,,As you said very well\Nin the most elegant way
Dialogue: 0,0:56:42.12,0:56:45.75,Default,,0000,0000,0000,,without being like yours,\Nbut I say it like this.
Dialogue: 0,0:56:45.75,0:56:49.87,Default,,0000,0000,0000,,So you have minus f\Nsub x, minus f sub y,
Dialogue: 0,0:56:49.87,0:56:54.58,Default,,0000,0000,0000,,and 1 as a triple with angular\Nbrackets--You love that.
Dialogue: 0,0:56:54.58,0:57:00.25,Default,,0000,0000,0000,,I don't; I like it parentheses\N[INAUDIBLE]--equals n.
Dialogue: 0,0:57:00.25,0:57:03.48,Default,,0000,0000,0000,,But n is non-unitary,\Nbut I don't care.
Dialogue: 0,0:57:03.48,0:57:04.73,Default,,0000,0000,0000,,Why don't I care?
Dialogue: 0,0:57:04.73,0:57:08.27,Default,,0000,0000,0000,,I can write the\Ntangent plane very well
Dialogue: 0,0:57:08.27,0:57:13.22,Default,,0000,0000,0000,,without that n being\Nunitary, right?
Dialogue: 0,0:57:13.22,0:57:14.54,Default,,0000,0000,0000,,It doesn't matter in the end.
Dialogue: 0,0:57:14.54,0:57:17.68,Default,,0000,0000,0000,,These would be my a, b, c.
Dialogue: 0,0:57:17.68,0:57:18.86,Default,,0000,0000,0000,,Now I know my ABC.
Dialogue: 0,0:57:18.86,0:57:20.40,Default,,0000,0000,0000,,I know my ABC.
Dialogue: 0,0:57:20.40,0:57:26.32,Default,,0000,0000,0000,,So, the tangent plane\Nis your next guess.
Dialogue: 0,0:57:26.32,0:57:30.14,Default,,0000,0000,0000,,The tangent plane would\Nbe perpendicular to n.
Dialogue: 0,0:57:30.14,0:57:32.15,Default,,0000,0000,0000,,So this is n.
Dialogue: 0,0:57:32.15,0:57:35.52,Default,,0000,0000,0000,,The tangent plane passes\Nthrough the point p
Dialogue: 0,0:57:35.52,0:57:37.35,Default,,0000,0000,0000,,and is perpendicular to n.
Dialogue: 0,0:57:37.35,0:57:43.15,Default,,0000,0000,0000,,So, what is the equation\Nof the tangent plane?
Dialogue: 0,0:57:43.15,0:57:44.73,Default,,0000,0000,0000,,STUDENT: Do you want\Nscalar equations?
Dialogue: 0,0:57:44.73,0:57:49.16,Default,,0000,0000,0000,,PROFESSOR TODA: A by x minus 0.
Dialogue: 0,0:57:49.16,0:57:50.22,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:57:50.22,0:57:56.33,Default,,0000,0000,0000,,That's exactly what I\Nwanted you to write.
Dialogue: 0,0:57:56.33,0:58:01.39,Default,,0000,0000,0000,,All right, so, does\Nit look familiar?
Dialogue: 0,0:58:01.39,0:58:01.92,Default,,0000,0000,0000,,Not yet.
Dialogue: 0,0:58:01.92,0:58:02.39,Default,,0000,0000,0000,,[STUDENT SNEEZES]
Dialogue: 0,0:58:02.39,0:58:02.85,Default,,0000,0000,0000,,STUDENT: Bless you.
Dialogue: 0,0:58:02.85,0:58:03.76,Default,,0000,0000,0000,,STUDENT: Bless you.
Dialogue: 0,0:58:03.76,0:58:04.84,Default,,0000,0000,0000,,PROFESSOR TODA: Bless you.
Dialogue: 0,0:58:04.84,0:58:05.99,Default,,0000,0000,0000,,Who sneezed?
Dialogue: 0,0:58:05.99,0:58:08.80,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:58:08.80,0:58:10.37,Default,,0000,0000,0000,,Am I almost done?
Dialogue: 0,0:58:10.37,0:58:11.70,Default,,0000,0000,0000,,Well, I am almost done.
Dialogue: 0,0:58:11.70,0:58:14.93,Default,,0000,0000,0000,,I have to go backwards,\Nand whatever I get
Dialogue: 0,0:58:14.93,0:58:17.76,Default,,0000,0000,0000,,I'll put it big here in\Na big formula on top.
Dialogue: 0,0:58:17.76,0:58:22.37,Default,,0000,0000,0000,,I'm gonna say oh, my God.
Dialogue: 0,0:58:22.37,0:58:24.02,Default,,0000,0000,0000,,No, that's not\Nwhat I'm gonna say.
Dialogue: 0,0:58:24.02,0:58:33.33,Default,,0000,0000,0000,,I'm gonna say minus f sub x at\Nmy point p--that is a, right?
Dialogue: 0,0:58:33.33,0:58:37.09,Default,,0000,0000,0000,,Times x minus x0.
Dialogue: 0,0:58:37.09,0:58:45.96,Default,,0000,0000,0000,,Plus minus f sub y at\Nthe point p; that's b.
Dialogue: 0,0:58:45.96,0:58:54.66,Default,,0000,0000,0000,,y minus y0 plus--c is 1, right?
Dialogue: 0,0:58:54.66,0:58:55.36,Default,,0000,0000,0000,,c is 1.
Dialogue: 0,0:58:55.36,0:58:58.02,Default,,0000,0000,0000,,I'm not gonna write\Nit because if I write
Dialogue: 0,0:58:58.02,0:59:03.99,Default,,0000,0000,0000,,it you'll want to make fun\Nof me. z minus z0 equals 0.
Dialogue: 0,0:59:03.99,0:59:08.56,Default,,0000,0000,0000,,Now it starts looking like\Nsomething familiar, finally.
Dialogue: 0,0:59:08.56,0:59:14.96,Default,,0000,0000,0000,,Now we discovered\Nthat the tangent plane
Dialogue: 0,0:59:14.96,0:59:20.63,Default,,0000,0000,0000,,can be written as z minus z0.
Dialogue: 0,0:59:20.63,0:59:24.63,Default,,0000,0000,0000,,I'm keeping the guys z minus\Nz0 on the left-hand side.
Dialogue: 0,0:59:24.63,0:59:28.63,Default,,0000,0000,0000,,And these guys are gonna\Nmove to the right-hand side.
Dialogue: 0,0:59:28.63,0:59:33.57,Default,,0000,0000,0000,,So, I'm gonna have\Nagain, my friend,
Dialogue: 0,0:59:33.57,0:59:45.46,Default,,0000,0000,0000,,the equation of the tangent\Nplane for the graph z equals f
Dialogue: 0,0:59:45.46,0:59:46.18,Default,,0000,0000,0000,,of x,y.
Dialogue: 0,0:59:46.18,0:59:51.94,Default,,0000,0000,0000,,
Dialogue: 0,0:59:51.94,0:59:54.87,Default,,0000,0000,0000,,But you will say\NOK, I think by now
Dialogue: 0,0:59:54.87,0:59:57.34,Default,,0000,0000,0000,,we've learned these\Nby heart, we know
Dialogue: 0,0:59:57.34,1:00:00.48,Default,,0000,0000,0000,,the equation of the tangent\Nplane, and now we're asleep.
Dialogue: 0,1:00:00.48,1:00:06.16,Default,,0000,0000,0000,,But what if your surface\Nwould be implicit the way
Dialogue: 0,1:00:06.16,1:00:08.76,Default,,0000,0000,0000,,you gave it to us at first.
Dialogue: 0,1:00:08.76,1:00:11.84,Default,,0000,0000,0000,,Maybe you remember the sphere\Nthat was an implicit equation,
Dialogue: 0,1:00:11.84,1:00:14.72,Default,,0000,0000,0000,,x squared plus x squared\Nplus x squared equals--
Dialogue: 0,1:00:14.72,1:00:16.03,Default,,0000,0000,0000,,What do you want it to be?
Dialogue: 0,1:00:16.03,1:00:16.78,Default,,0000,0000,0000,,STUDENT: 16.
Dialogue: 0,1:00:16.78,1:00:17.61,Default,,0000,0000,0000,,PROFESSOR TODA: Huh?
Dialogue: 0,1:00:17.61,1:00:18.80,Default,,0000,0000,0000,,STUDENT: 16.
Dialogue: 0,1:00:18.80,1:00:20.92,Default,,0000,0000,0000,,PROFESSOR TODA: 16.
Dialogue: 0,1:00:20.92,1:00:22.38,Default,,0000,0000,0000,,So, radius should be 4.
Dialogue: 0,1:00:22.38,1:00:26.80,Default,,0000,0000,0000,,
Dialogue: 0,1:00:26.80,1:00:31.06,Default,,0000,0000,0000,,And in such a case, the equation\Nis of the type f of x, y, z
Dialogue: 0,1:00:31.06,1:00:33.19,Default,,0000,0000,0000,,equals constant.
Dialogue: 0,1:00:33.19,1:00:35.74,Default,,0000,0000,0000,,Can we write again the\Nequation [INAUDIBLE]?
Dialogue: 0,1:00:35.74,1:00:39.77,Default,,0000,0000,0000,,
Dialogue: 0,1:00:39.77,1:00:42.24,Default,,0000,0000,0000,,Well, you say well,\Nyou just taught
Dialogue: 0,1:00:42.24,1:00:51.24,Default,,0000,0000,0000,,us some theory that says I have\Nto think of u and v, but not x
Dialogue: 0,1:00:51.24,1:00:51.85,Default,,0000,0000,0000,,and y.
Dialogue: 0,1:00:51.85,1:00:55.19,Default,,0000,0000,0000,,Because if I think of x\Nand y, what would they be?
Dialogue: 0,1:00:55.19,1:00:57.96,Default,,0000,0000,0000,,I think the sphere\Nas being an apple.
Dialogue: 0,1:00:57.96,1:01:01.88,Default,,0000,0000,0000,,Not an apple, something\Nyou can cut easily.
Dialogue: 0,1:01:01.88,1:01:05.48,Default,,0000,0000,0000,,Well, an apple, an\Norange, something.
Dialogue: 0,1:01:05.48,1:01:07.08,Default,,0000,0000,0000,,A round piece of soft cheese.
Dialogue: 0,1:01:07.08,1:01:09.51,Default,,0000,0000,0000,,I started being hungry,\Nand I'm dreaming.
Dialogue: 0,1:01:09.51,1:01:14.19,Default,,0000,0000,0000,,So, this is a huge something\Nyou're gonna slice up.
Dialogue: 0,1:01:14.19,1:01:19.10,Default,,0000,0000,0000,,If you are gonna\Ndo it with x and y,
Dialogue: 0,1:01:19.10,1:01:21.58,Default,,0000,0000,0000,,the slices would be like this.
Dialogue: 0,1:01:21.58,1:01:24.63,Default,,0000,0000,0000,,Like that and like this, right?
Dialogue: 0,1:01:24.63,1:01:27.12,Default,,0000,0000,0000,,And in that case,\Nyour coordinate curves
Dialogue: 0,1:01:27.12,1:01:30.54,Default,,0000,0000,0000,,are sort of weird.
Dialogue: 0,1:01:30.54,1:01:33.61,Default,,0000,0000,0000,,If you want to do it in\Ndifferent coordinates,
Dialogue: 0,1:01:33.61,1:01:35.08,Default,,0000,0000,0000,,so we want to\Nchange coordinates,
Dialogue: 0,1:01:35.08,1:01:39.81,Default,,0000,0000,0000,,and those coordinates should\Nbe plotted to the longitude,
Dialogue: 0,1:01:39.81,1:01:43.63,Default,,0000,0000,0000,,then we cannot use x and y.
Dialogue: 0,1:01:43.63,1:01:44.99,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,1:01:44.99,1:01:46.59,Default,,0000,0000,0000,,We cannot use x and y.
Dialogue: 0,1:01:46.59,1:01:50.63,Default,,0000,0000,0000,,So those u and v will be\Ndifferent coordinates,
Dialogue: 0,1:01:50.63,1:01:55.16,Default,,0000,0000,0000,,and then we can do it\Nlike that, latitude.
Dialogue: 0,1:01:55.16,1:01:57.79,Default,,0000,0000,0000,,
Dialogue: 0,1:01:57.79,1:02:00.01,Default,,0000,0000,0000,,[INAUDIBLE] minus [INAUDIBLE].
Dialogue: 0,1:02:00.01,1:02:00.80,Default,,0000,0000,0000,,And longitude.
Dialogue: 0,1:02:00.80,1:02:03.08,Default,,0000,0000,0000,,We are gonna talk about\Nspherical coordinates
Dialogue: 0,1:02:03.08,1:02:05.20,Default,,0000,0000,0000,,later, not today.
Dialogue: 0,1:02:05.20,1:02:06.16,Default,,0000,0000,0000,,Latitude and longitude.
Dialogue: 0,1:02:06.16,1:02:10.34,Default,,0000,0000,0000,,
Dialogue: 0,1:02:10.34,1:02:12.89,Default,,0000,0000,0000,,1 point extra credit,\Nbecause eventually we
Dialogue: 0,1:02:12.89,1:02:16.88,Default,,0000,0000,0000,,are gonna get\Nthere, chapter 12.7.
Dialogue: 0,1:02:16.88,1:02:20.65,Default,,0000,0000,0000,,12.7 comes way\Nafter spring break.
Dialogue: 0,1:02:20.65,1:02:27.39,Default,,0000,0000,0000,,But before we get there, who\Nis in mechanical engineering
Dialogue: 0,1:02:27.39,1:02:28.83,Default,,0000,0000,0000,,again?
Dialogue: 0,1:02:28.83,1:02:32.71,Default,,0000,0000,0000,,You know about Euler's\Nangles, and stuff like that.
Dialogue: 0,1:02:32.71,1:02:33.55,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:02:33.55,1:02:40.33,Default,,0000,0000,0000,,Can you write me\Nthe equations of x
Dialogue: 0,1:02:40.33,1:02:47.85,Default,,0000,0000,0000,,and y and z of the sphere\Nwith respect to u and v,
Dialogue: 0,1:02:47.85,1:02:51.20,Default,,0000,0000,0000,,u being latitude and\Nv being longitude?
Dialogue: 0,1:02:51.20,1:02:53.98,Default,,0000,0000,0000,,
Dialogue: 0,1:02:53.98,1:02:58.64,Default,,0000,0000,0000,,These have to be\Ntrigonometric functions.
Dialogue: 0,1:02:58.64,1:03:03.86,Default,,0000,0000,0000,,
Dialogue: 0,1:03:03.86,1:03:10.77,Default,,0000,0000,0000,,In terms of u and v, when u is\Nlatitude and v is longitude.
Dialogue: 0,1:03:10.77,1:03:15.31,Default,,0000,0000,0000,,1 point extra credit\Nuntil a week from today.
Dialogue: 0,1:03:15.31,1:03:16.28,Default,,0000,0000,0000,,How about that?
Dialogue: 0,1:03:16.28,1:03:20.65,Default,,0000,0000,0000,,
Dialogue: 0,1:03:20.65,1:03:23.85,Default,,0000,0000,0000,,U and v are latitude\Nand longitude.
Dialogue: 0,1:03:23.85,1:03:33.80,Default,,0000,0000,0000,,And express the xyz point in\Nthe ambient space on the sphere.
Dialogue: 0,1:03:33.80,1:03:36.46,Default,,0000,0000,0000,,x squared plus x squared\Nplus x squared would be 16.
Dialogue: 0,1:03:36.46,1:03:40.02,Default,,0000,0000,0000,,So you'll have lots of\Ncosines and sines [INAUDIBLE]
Dialogue: 0,1:03:40.02,1:03:46.02,Default,,0000,0000,0000,,of those angles, the latitude\Nangle and the longitude angle.
Dialogue: 0,1:03:46.02,1:03:49.80,Default,,0000,0000,0000,,And I would suggest to you that\Nyou take--for the extra credit
Dialogue: 0,1:03:49.80,1:03:54.91,Default,,0000,0000,0000,,thing--you take the longitude\Nangle to be from 0 to 2pi,
Dialogue: 0,1:03:54.91,1:04:00.15,Default,,0000,0000,0000,,from the Greenwich 0 meridian\Ngoing back to himself,
Dialogue: 0,1:04:00.15,1:04:07.72,Default,,0000,0000,0000,,and--well, there are two ways\Nwe do this in mathematics
Dialogue: 0,1:04:07.72,1:04:09.81,Default,,0000,0000,0000,,because mathematicians\Nare so diverse.
Dialogue: 0,1:04:09.81,1:04:14.85,Default,,0000,0000,0000,,Some of us, say, for me,\NI measure the latitude
Dialogue: 0,1:04:14.85,1:04:17.10,Default,,0000,0000,0000,,starting from the North Pole.
Dialogue: 0,1:04:17.10,1:04:20.27,Default,,0000,0000,0000,,I think that's because we all\Nbelieve in Santa or something.
Dialogue: 0,1:04:20.27,1:04:23.44,Default,,0000,0000,0000,,So, we start measuring\Nalways from the North Pole
Dialogue: 0,1:04:23.44,1:04:27.03,Default,,0000,0000,0000,,because that's the most\Nimportant place on Earth.
Dialogue: 0,1:04:27.03,1:04:35.63,Default,,0000,0000,0000,,They go 0, pi over 2, and then--\Nwhat is our lat--shame on me.
Dialogue: 0,1:04:35.63,1:04:36.48,Default,,0000,0000,0000,,STUDENT: It's 33.
Dialogue: 0,1:04:36.48,1:04:37.27,Default,,0000,0000,0000,,PROFESSOR TODA: 33?
Dialogue: 0,1:04:37.27,1:04:39.22,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:04:39.22,1:04:44.06,Default,,0000,0000,0000,,Then pi would be the\Nequator, and then pi
Dialogue: 0,1:04:44.06,1:04:45.83,Default,,0000,0000,0000,,would be the South Pole.
Dialogue: 0,1:04:45.83,1:04:50.62,Default,,0000,0000,0000,,But some other mathematicians,\Nespecially biologists
Dialogue: 0,1:04:50.62,1:04:54.53,Default,,0000,0000,0000,,and differential geometry\Npeople, I'm one of them,
Dialogue: 0,1:04:54.53,1:04:56.09,Default,,0000,0000,0000,,we go like that.
Dialogue: 0,1:04:56.09,1:05:01.62,Default,,0000,0000,0000,,Minus pi over 2, South Pole\N0, pi over 2 North Pole.
Dialogue: 0,1:05:01.62,1:05:06.82,Default,,0000,0000,0000,,So we shift that\Nkind of interval.
Dialogue: 0,1:05:06.82,1:05:10.28,Default,,0000,0000,0000,,Then for us, the trigonometric\Nfunctions of these angles
Dialogue: 0,1:05:10.28,1:05:12.02,Default,,0000,0000,0000,,would be a little\Nbit different when we
Dialogue: 0,1:05:12.02,1:05:14.40,Default,,0000,0000,0000,,do the spherical coordinates.
Dialogue: 0,1:05:14.40,1:05:16.34,Default,,0000,0000,0000,,OK, that's just extra credit.
Dialogue: 0,1:05:16.34,1:05:19.07,Default,,0000,0000,0000,,It has nothing to do with\Nwhat I'm gonna do right now.
Dialogue: 0,1:05:19.07,1:05:22.96,Default,,0000,0000,0000,,What I'm gonna do right now\Nis to pick a point on Earth.
Dialogue: 0,1:05:22.96,1:05:26.00,Default,,0000,0000,0000,,We have to find Lubbock.
Dialogue: 0,1:05:26.00,1:05:27.21,Default,,0000,0000,0000,,STUDENT: It's on the left.
Dialogue: 0,1:05:27.21,1:05:28.74,Default,,0000,0000,0000,,PROFESSOR TODA: Here?
Dialogue: 0,1:05:28.74,1:05:29.87,Default,,0000,0000,0000,,Is that a good point?
Dialogue: 0,1:05:29.87,1:05:32.40,Default,,0000,0000,0000,,
Dialogue: 0,1:05:32.40,1:05:34.49,Default,,0000,0000,0000,,This is LBB.
Dialogue: 0,1:05:34.49,1:05:38.43,Default,,0000,0000,0000,,That's Lubbock\NInternational Airport.
Dialogue: 0,1:05:38.43,1:05:47.53,Default,,0000,0000,0000,,So, for Lubbock--let's call it\Np as well--draw the r sub u,
Dialogue: 0,1:05:47.53,1:05:52.55,Default,,0000,0000,0000,,r sub v. So, u was latitude.
Dialogue: 0,1:05:52.55,1:05:55.75,Default,,0000,0000,0000,,So if I fix the latitude,\Nthat means I fix
Dialogue: 0,1:05:55.75,1:05:58.65,Default,,0000,0000,0000,,the 33 point whatever you said.
Dialogue: 0,1:05:58.65,1:06:00.06,Default,,0000,0000,0000,,u equals u0.
Dialogue: 0,1:06:00.06,1:06:09.63,Default,,0000,0000,0000,,It is fixed, so I have u\Nfixed, and v equals v0 is that.
Dialogue: 0,1:06:09.63,1:06:14.34,Default,,0000,0000,0000,,I fixed the meridian\Nwhere we are.
Dialogue: 0,1:06:14.34,1:06:15.99,Default,,0000,0000,0000,,What is this tangent vector?
Dialogue: 0,1:06:15.99,1:06:20.52,Default,,0000,0000,0000,,
Dialogue: 0,1:06:20.52,1:06:22.95,Default,,0000,0000,0000,,To the pink parallel,\Nthe tangent vector
Dialogue: 0,1:06:22.95,1:06:25.66,Default,,0000,0000,0000,,would be r sub what?
Dialogue: 0,1:06:25.66,1:06:26.16,Default,,0000,0000,0000,,STUDENT: v.
Dialogue: 0,1:06:26.16,1:06:27.78,Default,,0000,0000,0000,,PROFESSOR TODA: r\Nsub v. You are right.
Dialogue: 0,1:06:27.78,1:06:28.92,Default,,0000,0000,0000,,You've got the idea.
Dialogue: 0,1:06:28.92,1:06:33.37,Default,,0000,0000,0000,,And the blue vector would\Nbe the partial velocity.
Dialogue: 0,1:06:33.37,1:06:39.47,Default,,0000,0000,0000,,That's the tangent vector\Nto the blue meridian,
Dialogue: 0,1:06:39.47,1:06:43.92,Default,,0000,0000,0000,,which is r sub u.
Dialogue: 0,1:06:43.92,1:06:48.68,Default,,0000,0000,0000,,And what is n gonna be? n's\Ngonna be r sub u [INAUDIBLE].
Dialogue: 0,1:06:48.68,1:06:53.37,Default,,0000,0000,0000,,But is there any other way\Nto do it in a simpler way
Dialogue: 0,1:06:53.37,1:06:55.52,Default,,0000,0000,0000,,without you guys going oh, man.
Dialogue: 0,1:06:55.52,1:06:58.06,Default,,0000,0000,0000,,Suppose some of you don't\Nwanna do the extra credit
Dialogue: 0,1:06:58.06,1:07:00.33,Default,,0000,0000,0000,,and then say the\Nheck with it; I don't
Dialogue: 0,1:07:00.33,1:07:03.61,Default,,0000,0000,0000,,care about her stinking extra\Ncredit until chapter 12,
Dialogue: 0,1:07:03.61,1:07:07.70,Default,,0000,0000,0000,,when I have to study the\Nspherical coordinates,
Dialogue: 0,1:07:07.70,1:07:11.17,Default,,0000,0000,0000,,and is there another\Nway to get n.
Dialogue: 0,1:07:11.17,1:07:13.41,Default,,0000,0000,0000,,I told you another way to get n.
Dialogue: 0,1:07:13.41,1:07:15.38,Default,,0000,0000,0000,,Well, we are getting there.
Dialogue: 0,1:07:15.38,1:07:21.75,Default,,0000,0000,0000,,n was the gradient of f\Nover the length of that.
Dialogue: 0,1:07:21.75,1:07:26.49,Default,,0000,0000,0000,,And if we want it unitary,\Nthe length of f was what?
Dialogue: 0,1:07:26.49,1:07:31.72,Default,,0000,0000,0000,,f sub x, f sub y, f\Nsub z vector, where
Dialogue: 0,1:07:31.72,1:07:36.53,Default,,0000,0000,0000,,the implicit equation of\Nthe surface was f of x, y, z
Dialogue: 0,1:07:36.53,1:07:38.40,Default,,0000,0000,0000,,equals c.
Dialogue: 0,1:07:38.40,1:07:40.24,Default,,0000,0000,0000,,So now we've done this before.
Dialogue: 0,1:07:40.24,1:07:42.47,Default,,0000,0000,0000,,You say Magdalena, you're\Nrepeating yourself.
Dialogue: 0,1:07:42.47,1:07:47.21,Default,,0000,0000,0000,,I know I'm repeating myself, but\NI want you to learn this twice
Dialogue: 0,1:07:47.21,1:07:49.26,Default,,0000,0000,0000,,so you can remember it.
Dialogue: 0,1:07:49.26,1:07:52.41,Default,,0000,0000,0000,,What is f of x, y, z?
Dialogue: 0,1:07:52.41,1:07:56.70,Default,,0000,0000,0000,,In my case, it's x squared\Nplus y squared plus z squared
Dialogue: 0,1:07:56.70,1:07:59.93,Default,,0000,0000,0000,,minus 16, or even nothing.
Dialogue: 0,1:07:59.93,1:08:01.85,Default,,0000,0000,0000,,Because the constant\Ndoesn't matter anyway
Dialogue: 0,1:08:01.85,1:08:04.43,Default,,0000,0000,0000,,when I do the gradient.
Dialogue: 0,1:08:04.43,1:08:05.60,Default,,0000,0000,0000,,You guys are doing homework.
Dialogue: 0,1:08:05.60,1:08:08.21,Default,,0000,0000,0000,,You saw how the gradient goes.
Dialogue: 0,1:08:08.21,1:08:13.73,Default,,0000,0000,0000,,So gradient of f would\Nbe 2x times-- and that's
Dialogue: 0,1:08:13.73,1:08:19.38,Default,,0000,0000,0000,,the partial derivative times i\Nplus 2y times j plus 2z times
Dialogue: 0,1:08:19.38,1:08:22.96,Default,,0000,0000,0000,,k-- that's very important.
Dialogue: 0,1:08:22.96,1:08:28.27,Default,,0000,0000,0000,,[? Lovett ?] has some\Ncoordinates we plug in.
Dialogue: 0,1:08:28.27,1:08:33.50,Default,,0000,0000,0000,,Now, can we write-- two things.
Dialogue: 0,1:08:33.50,1:08:35.62,Default,,0000,0000,0000,,I want two things from you.
Dialogue: 0,1:08:35.62,1:08:41.34,Default,,0000,0000,0000,,Write me a total\Ndifferential b tangent plane
Dialogue: 0,1:08:41.34,1:08:46.14,Default,,0000,0000,0000,,at the point-- so, a, write\Nthe total differential.
Dialogue: 0,1:08:46.14,1:08:50.97,Default,,0000,0000,0000,,
Dialogue: 0,1:08:50.97,1:08:53.67,Default,,0000,0000,0000,,I'm not going to ask you you\Nto do a linear approximation.
Dialogue: 0,1:08:53.67,1:08:55.81,Default,,0000,0000,0000,,I could.
Dialogue: 0,1:08:55.81,1:09:23.66,Default,,0000,0000,0000,,B, write the tangent plane\Nto the sphere at the point
Dialogue: 0,1:09:23.66,1:09:25.19,Default,,0000,0000,0000,,that-- I don't know.
Dialogue: 0,1:09:25.19,1:09:26.87,Default,,0000,0000,0000,,I don't want one that's trivial.
Dialogue: 0,1:09:26.87,1:09:30.04,Default,,0000,0000,0000,,
Dialogue: 0,1:09:30.04,1:09:37.77,Default,,0000,0000,0000,,Let's take this 0, square root\Nof 8, and square root of 8.
Dialogue: 0,1:09:37.77,1:09:39.64,Default,,0000,0000,0000,,I just have to make\Nsure that I don't
Dialogue: 0,1:09:39.64,1:09:41.70,Default,,0000,0000,0000,,come with some\Nnonsensical point that's
Dialogue: 0,1:09:41.70,1:09:43.29,Default,,0000,0000,0000,,not going to be on the sphere.
Dialogue: 0,1:09:43.29,1:09:45.86,Default,,0000,0000,0000,,This will be because I\Nplugged it in in my mind.
Dialogue: 0,1:09:45.86,1:09:50.23,Default,,0000,0000,0000,,I get 8 plus 8 is 16 last\Ntime I checked, right?
Dialogue: 0,1:09:50.23,1:09:54.98,Default,,0000,0000,0000,,So after we do this\Nwe take a break.
Dialogue: 0,1:09:54.98,1:09:58.28,Default,,0000,0000,0000,,Suppose that this is a\Nproblem on your midterm,
Dialogue: 0,1:09:58.28,1:10:00.74,Default,,0000,0000,0000,,or on your final or\Non your homework,
Dialogue: 0,1:10:00.74,1:10:04.33,Default,,0000,0000,0000,,or on somebody [? YouTubed it ?]\Nfor a lot of money,
Dialogue: 0,1:10:04.33,1:10:10.01,Default,,0000,0000,0000,,you asked them, $25 an hour\Nfor me to work that problem.
Dialogue: 0,1:10:10.01,1:10:10.57,Default,,0000,0000,0000,,That's good.
Dialogue: 0,1:10:10.57,1:10:16.73,Default,,0000,0000,0000,,I mean-- it's-- it's a\Nclass that you're taking
Dialogue: 0,1:10:16.73,1:10:20.03,Default,,0000,0000,0000,,for your general requirement\Nbecause your school wants you
Dialogue: 0,1:10:20.03,1:10:22.47,Default,,0000,0000,0000,,to take calc 3.
Dialogue: 0,1:10:22.47,1:10:25.57,Default,,0000,0000,0000,,But it gives you-- and\NI know from experience,
Dialogue: 0,1:10:25.57,1:10:27.67,Default,,0000,0000,0000,,some of my students came\Nback to me and said,
Dialogue: 0,1:10:27.67,1:10:30.16,Default,,0000,0000,0000,,after I took calc\N3, I understood it
Dialogue: 0,1:10:30.16,1:10:33.38,Default,,0000,0000,0000,,so well that I was able to\Ntutor calc 1, calc 2, calc 3,
Dialogue: 0,1:10:33.38,1:10:35.84,Default,,0000,0000,0000,,so I got a double job.
Dialogue: 0,1:10:35.84,1:10:38.06,Default,,0000,0000,0000,,Several hours a week,\Nthe tutoring center,
Dialogue: 0,1:10:38.06,1:10:39.56,Default,,0000,0000,0000,,math department,\Nand several hours
Dialogue: 0,1:10:39.56,1:10:40.64,Default,,0000,0000,0000,,at the [INAUDIBLE] center.
Dialogue: 0,1:10:40.64,1:10:42.67,Default,,0000,0000,0000,,You know what I'm talking about?
Dialogue: 0,1:10:42.67,1:10:46.22,Default,,0000,0000,0000,,So I've had students who did\Nwell and ended up liking this,
Dialogue: 0,1:10:46.22,1:10:49.28,Default,,0000,0000,0000,,and said I can tutor\Nthis in my sleep.
Dialogue: 0,1:10:49.28,1:10:53.76,Default,,0000,0000,0000,,So-- and also private tutoring\Nis always a possibility.
Dialogue: 0,1:10:53.76,1:10:55.20,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:10:55.20,1:10:58.67,Default,,0000,0000,0000,,Write total differential.
Dialogue: 0,1:10:58.67,1:11:04.26,Default,,0000,0000,0000,,df equals, and now\NI'll say at any point.
Dialogue: 0,1:11:04.26,1:11:06.96,Default,,0000,0000,0000,,So I don't care what\Nthe value will be.
Dialogue: 0,1:11:06.96,1:11:08.82,Default,,0000,0000,0000,,I didn't say at what point.
Dialogue: 0,1:11:08.82,1:11:09.92,Default,,0000,0000,0000,,It means in general.
Dialogue: 0,1:11:09.92,1:11:12.01,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,1:11:12.01,1:11:14.90,Default,,0000,0000,0000,,You tell me, you\Nknow that by now.
Dialogue: 0,1:11:14.90,1:11:18.41,Default,,0000,0000,0000,,2x times what?
Dialogue: 0,1:11:18.41,1:11:20.30,Default,,0000,0000,0000,,Now, you learned\Nyour lesson, you're
Dialogue: 0,1:11:20.30,1:11:21.98,Default,,0000,0000,0000,,never gonna make mistakes.
Dialogue: 0,1:11:21.98,1:11:25.49,Default,,0000,0000,0000,,2y plus 2z dz.
Dialogue: 0,1:11:25.49,1:11:26.45,Default,,0000,0000,0000,,That is very good.
Dialogue: 0,1:11:26.45,1:11:28.01,Default,,0000,0000,0000,,That's the total differential.
Dialogue: 0,1:11:28.01,1:11:33.96,Default,,0000,0000,0000,,Now, what is the equation\Nof the tangent plane?
Dialogue: 0,1:11:33.96,1:11:37.04,Default,,0000,0000,0000,,It's not gonna be that.
Dialogue: 0,1:11:37.04,1:11:40.67,Default,,0000,0000,0000,,Because I'm not\Nconsidering a graph.
Dialogue: 0,1:11:40.67,1:11:44.59,Default,,0000,0000,0000,,I'm considering an\Nimplicitly given surface
Dialogue: 0,1:11:44.59,1:11:52.72,Default,,0000,0000,0000,,by this implicit equation f of\Nx, y, z, equals c, your friend.
Dialogue: 0,1:11:52.72,1:11:57.73,Default,,0000,0000,0000,,So what was, in that case,\Nthe equation of the plane
Dialogue: 0,1:11:57.73,1:11:59.63,Default,,0000,0000,0000,,written as?
Dialogue: 0,1:11:59.63,1:12:02.48,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:12:02.48,1:12:05.13,Default,,0000,0000,0000,,PROFESSOR TODA: I'm--\Nyeah, you guys are smart.
Dialogue: 0,1:12:05.13,1:12:06.44,Default,,0000,0000,0000,,I mean, you are fast.
Dialogue: 0,1:12:06.44,1:12:07.79,Default,,0000,0000,0000,,Let's do it in general.
Dialogue: 0,1:12:07.79,1:12:11.64,Default,,0000,0000,0000,,F sub x-- we did that last\Ntime, [INAUDIBLE] times--
Dialogue: 0,1:12:11.64,1:12:14.26,Default,,0000,0000,0000,,do you guys remember?
Dialogue: 0,1:12:14.26,1:12:16.47,Default,,0000,0000,0000,,x minus x0.
Dialogue: 0,1:12:16.47,1:12:20.66,Default,,0000,0000,0000,,And this is at the point plus\Nbig F sub y at the point times
Dialogue: 0,1:12:20.66,1:12:25.59,Default,,0000,0000,0000,,y minus y0 plus big F sub\Nz at the point z minus z0.
Dialogue: 0,1:12:25.59,1:12:26.81,Default,,0000,0000,0000,,This is just review.
Dialogue: 0,1:12:26.81,1:12:27.99,Default,,0000,0000,0000,,Equals 0.
Dialogue: 0,1:12:27.99,1:12:28.49,Default,,0000,0000,0000,,Stop.
Dialogue: 0,1:12:28.49,1:12:31.47,Default,,0000,0000,0000,,Where do these guys come from?
Dialogue: 0,1:12:31.47,1:12:32.95,Default,,0000,0000,0000,,From the gradient.
Dialogue: 0,1:12:32.95,1:12:34.83,Default,,0000,0000,0000,,From the gradient.
Dialogue: 0,1:12:34.83,1:12:40.15,Default,,0000,0000,0000,,Which are the a,b,c, now I\Nknow my ABCs, from the normal.
Dialogue: 0,1:12:40.15,1:12:41.92,Default,,0000,0000,0000,,My ABCs from the normal.
Dialogue: 0,1:12:41.92,1:12:46.64,Default,,0000,0000,0000,,So in this case-- I\Ndon't want to erase
Dialogue: 0,1:12:46.64,1:12:49.02,Default,,0000,0000,0000,,this beautiful picture.
Dialogue: 0,1:12:49.02,1:12:54.91,Default,,0000,0000,0000,,The last thing I have to do\Nbefore the break is-- you
Dialogue: 0,1:12:54.91,1:12:56.90,Default,,0000,0000,0000,,said 0.
Dialogue: 0,1:12:56.90,1:12:58.96,Default,,0000,0000,0000,,I'm a lazy person by definition.
Dialogue: 0,1:12:58.96,1:13:02.99,Default,,0000,0000,0000,,Can you tell me why\Nyou said 0 times?
Dialogue: 0,1:13:02.99,1:13:05.00,Default,,0000,0000,0000,,STUDENT: Because the\Nx value is [INAUDIBLE]
Dialogue: 0,1:13:05.00,1:13:07.43,Default,,0000,0000,0000,,PROFESSOR TODA: You said\N2x, plug in and x equals 0
Dialogue: 0,1:13:07.43,1:13:10.05,Default,,0000,0000,0000,,from your point,\NMagdalena, so you don't
Dialogue: 0,1:13:10.05,1:13:12.41,Default,,0000,0000,0000,,have to write down everything.
Dialogue: 0,1:13:12.41,1:13:19.67,Default,,0000,0000,0000,,But I'm gonna write down 0\Ntimes x minus 0 plus-- what's
Dialogue: 0,1:13:19.67,1:13:20.61,Default,,0000,0000,0000,,next for me?
Dialogue: 0,1:13:20.61,1:13:21.71,Default,,0000,0000,0000,,STUDENT: 2 square root 8.
Dialogue: 0,1:13:21.71,1:13:23.63,Default,,0000,0000,0000,,PROFESSOR TODA: 2y, 2 root 8.
Dialogue: 0,1:13:23.63,1:13:26.19,Default,,0000,0000,0000,,Is root 8 beautiful?
Dialogue: 0,1:13:26.19,1:13:28.10,Default,,0000,0000,0000,,It looks like heck.
Dialogue: 0,1:13:28.10,1:13:32.87,Default,,0000,0000,0000,,At the end I'm gonna\Nbrush it up a little bit.
Dialogue: 0,1:13:32.87,1:13:39.12,Default,,0000,0000,0000,,This is the partial-- f sub y of\Nt times y minus-- who is y, z?
Dialogue: 0,1:13:39.12,1:13:40.78,Default,,0000,0000,0000,,Root 8.
Dialogue: 0,1:13:40.78,1:13:41.73,Default,,0000,0000,0000,,Do I like it?
Dialogue: 0,1:13:41.73,1:13:43.64,Default,,0000,0000,0000,,I hate it, but it\Ndoesn't matter.
Dialogue: 0,1:13:43.64,1:13:45.54,Default,,0000,0000,0000,,Because I'm gonna simplify.
Dialogue: 0,1:13:45.54,1:13:52.47,Default,,0000,0000,0000,,Plus again, 2 root 8, thank you.
Dialogue: 0,1:13:52.47,1:13:56.71,Default,,0000,0000,0000,,This is my c guy.
Dialogue: 0,1:13:56.71,1:14:02.44,Default,,0000,0000,0000,,Times z minus root 8 equals 0.
Dialogue: 0,1:14:02.44,1:14:05.43,Default,,0000,0000,0000,,I picked another example\Nfrom the one from the book,
Dialogue: 0,1:14:05.43,1:14:08.83,Default,,0000,0000,0000,,because you are gonna\Nread the book anyway.
Dialogue: 0,1:14:08.83,1:14:11.96,Default,,0000,0000,0000,,I'm gonna erase that.
Dialogue: 0,1:14:11.96,1:14:14.83,Default,,0000,0000,0000,,And I'm gonna brush\Nthis up because it
Dialogue: 0,1:14:14.83,1:14:17.49,Default,,0000,0000,0000,,looks horrible to me.
Dialogue: 0,1:14:17.49,1:14:19.89,Default,,0000,0000,0000,,Thank God this goes away.
Dialogue: 0,1:14:19.89,1:14:21.98,Default,,0000,0000,0000,,So the plane will\Nsimply be a combination
Dialogue: 0,1:14:21.98,1:14:24.25,Default,,0000,0000,0000,,of my y and z in a constant.
Dialogue: 0,1:14:24.25,1:14:28.00,Default,,0000,0000,0000,,And if I want to\Nmake my life easier,
Dialogue: 0,1:14:28.00,1:14:30.47,Default,,0000,0000,0000,,I'm gonna divide by what?
Dialogue: 0,1:14:30.47,1:14:32.28,Default,,0000,0000,0000,,By this.
Dialogue: 0,1:14:32.28,1:14:34.39,Default,,0000,0000,0000,,So in the end, it\Ndoesn't matter.
Dialogue: 0,1:14:34.39,1:14:35.92,Default,,0000,0000,0000,,Come on.
Dialogue: 0,1:14:35.92,1:14:42.30,Default,,0000,0000,0000,,I'll get y minus root 8 plus\Nc minus root 8 equals 0.
Dialogue: 0,1:14:42.30,1:14:44.02,Default,,0000,0000,0000,,Do I like it?
Dialogue: 0,1:14:44.02,1:14:44.77,Default,,0000,0000,0000,,I hate it.
Dialogue: 0,1:14:44.77,1:14:46.66,Default,,0000,0000,0000,,No, you know, I don't like it.
Dialogue: 0,1:14:46.66,1:14:49.04,Default,,0000,0000,0000,,Why don't I like it?
Dialogue: 0,1:14:49.04,1:14:50.43,Default,,0000,0000,0000,,It's not simplified.
Dialogue: 0,1:14:50.43,1:14:56.00,Default,,0000,0000,0000,,So in any case, if this\Nwere multiple choice,
Dialogue: 0,1:14:56.00,1:14:59.45,Default,,0000,0000,0000,,it would not be written\Nlike that, right?
Dialogue: 0,1:14:59.45,1:15:03.99,Default,,0000,0000,0000,,So what would be the\Nsimplified claim in this case?
Dialogue: 0,1:15:03.99,1:15:09.27,Default,,0000,0000,0000,,The way I would write\Nit-- a y plus a z minus--
Dialogue: 0,1:15:09.27,1:15:11.49,Default,,0000,0000,0000,,think, what is root 8?
Dialogue: 0,1:15:11.49,1:15:12.53,Default,,0000,0000,0000,,STUDENT: 2 root 2.
Dialogue: 0,1:15:12.53,1:15:13.74,Default,,0000,0000,0000,,PROFESSOR TODA: And 2 root 2.
Dialogue: 0,1:15:13.74,1:15:20.99,Default,,0000,0000,0000,,And 2 root 2, how\Nmuch-- minus 4 root 2.
Dialogue: 0,1:15:20.99,1:15:28.84,Default,,0000,0000,0000,,And this is how you are expected\Nto leave this answer boxed.
Dialogue: 0,1:15:28.84,1:15:37.81,Default,,0000,0000,0000,,This is that tangent\Nplane at the point.
Dialogue: 0,1:15:37.81,1:15:41.20,Default,,0000,0000,0000,,
Dialogue: 0,1:15:41.20,1:15:42.65,Default,,0000,0000,0000,,To the sphere.
Dialogue: 0,1:15:42.65,1:15:45.57,Default,,0000,0000,0000,,
Dialogue: 0,1:15:45.57,1:15:48.57,Default,,0000,0000,0000,,There are programs--\None time I was teaching
Dialogue: 0,1:15:48.57,1:15:53.97,Default,,0000,0000,0000,,advance geometry, 4331, and one\Nthing I gave my students to do,
Dialogue: 0,1:15:53.97,1:15:58.92,Default,,0000,0000,0000,,which was a lot of fun--\Nusing a parametrization,
Dialogue: 0,1:15:58.92,1:16:02.71,Default,,0000,0000,0000,,plot the entire\Nsphere with MathLab.
Dialogue: 0,1:16:02.71,1:16:04.06,Default,,0000,0000,0000,,We did it with MathLab.
Dialogue: 0,1:16:04.06,1:16:06.93,Default,,0000,0000,0000,,Some people said they know\N[INAUDIBLE] I didn't care.
Dialogue: 0,1:16:06.93,1:16:09.33,Default,,0000,0000,0000,,So MathLab for me\Nwas easier, so we
Dialogue: 0,1:16:09.33,1:16:11.80,Default,,0000,0000,0000,,plotted the sphere in MathLab.
Dialogue: 0,1:16:11.80,1:16:14.94,Default,,0000,0000,0000,,We picked a point,\Nand we drew-- well,
Dialogue: 0,1:16:14.94,1:16:21.96,Default,,0000,0000,0000,,we drew-- with MathLab we\Ndrew the tangent plane that
Dialogue: 0,1:16:21.96,1:16:25.95,Default,,0000,0000,0000,,was tangent to the\Nsphere at that point.
Dialogue: 0,1:16:25.95,1:16:27.22,Default,,0000,0000,0000,,And they liked it.
Dialogue: 0,1:16:27.22,1:16:29.55,Default,,0000,0000,0000,,It was-- you know\Nwhat this class is,
Dialogue: 0,1:16:29.55,1:16:31.75,Default,,0000,0000,0000,,is-- if you're math\Nmajors you take it.
Dialogue: 0,1:16:31.75,1:16:34.25,Default,,0000,0000,0000,,It's called advanced geometries.
Dialogue: 0,1:16:34.25,1:16:35.63,Default,,0000,0000,0000,,Mainly it's theoretical.
Dialogue: 0,1:16:35.63,1:16:38.54,Default,,0000,0000,0000,,It teaches you Euclidian\Naxioms and stuff,
Dialogue: 0,1:16:38.54,1:16:41.54,Default,,0000,0000,0000,,and then some\Nnon-Euclidian geometries.
Dialogue: 0,1:16:41.54,1:16:45.83,Default,,0000,0000,0000,,But I thought that I would\Ndo it into an honors class.
Dialogue: 0,1:16:45.83,1:16:49.27,Default,,0000,0000,0000,,And I put one third of that\Nlast class visualization
Dialogue: 0,1:16:49.27,1:16:50.85,Default,,0000,0000,0000,,with MathLab of geometry.
Dialogue: 0,1:16:50.85,1:16:54.02,Default,,0000,0000,0000,,And I think that was what\Nthey liked the most, not so
Dialogue: 0,1:16:54.02,1:16:56.07,Default,,0000,0000,0000,,much the axiomatic\Npart and the proofs,
Dialogue: 0,1:16:56.07,1:17:03.27,Default,,0000,0000,0000,,but the hands-on computation\Nand visualization in the lab.
Dialogue: 0,1:17:03.27,1:17:04.98,Default,,0000,0000,0000,,We have this lab, 113.
Dialogue: 0,1:17:04.98,1:17:07.34,Default,,0000,0000,0000,,We used to have two labs,\Nbut now we are poor,
Dialogue: 0,1:17:07.34,1:17:09.09,Default,,0000,0000,0000,,we only have one.
Dialogue: 0,1:17:09.09,1:17:10.51,Default,,0000,0000,0000,,No, we lost the lab.
Dialogue: 0,1:17:10.51,1:17:13.66,Default,,0000,0000,0000,,The undergraduate\Nlab-- 009, next to you,
Dialogue: 0,1:17:13.66,1:17:18.56,Default,,0000,0000,0000,,is lost because-- I used\Nto each calc 3 there.
Dialogue: 0,1:17:18.56,1:17:21.54,Default,,0000,0000,0000,,Not because-- that's\Nnot why we lost it.
Dialogue: 0,1:17:21.54,1:17:24.83,Default,,0000,0000,0000,,We lost it because we-- we\Nput some 20 graduate students
Dialogue: 0,1:17:24.83,1:17:25.33,Default,,0000,0000,0000,,there.
Dialogue: 0,1:17:25.33,1:17:26.69,Default,,0000,0000,0000,,We have no space.
Dialogue: 0,1:17:26.69,1:17:30.81,Default,,0000,0000,0000,,And we have 130 graduate\Nstudents in mathematics.
Dialogue: 0,1:17:30.81,1:17:32.43,Default,,0000,0000,0000,,Where do you put them?
Dialogue: 0,1:17:32.43,1:17:34.16,Default,,0000,0000,0000,,We just cram them into cubicles.
Dialogue: 0,1:17:34.16,1:17:37.59,Default,,0000,0000,0000,,So they made 20 cubicles\Nhere, and they put some,
Dialogue: 0,1:17:37.59,1:17:40.01,Default,,0000,0000,0000,,so we lost the lab.
Dialogue: 0,1:17:40.01,1:17:41.86,Default,,0000,0000,0000,,It's sad.
Dialogue: 0,1:17:41.86,1:17:42.73,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:17:42.73,1:17:45.09,Default,,0000,0000,0000,,So that's it for now.
Dialogue: 0,1:17:45.09,1:17:47.54,Default,,0000,0000,0000,,We are gonna take a\Nshort break, and we
Dialogue: 0,1:17:47.54,1:17:52.12,Default,,0000,0000,0000,,will continue for one more hour,\Nwhich is mostly application.
Dialogue: 0,1:17:52.12,1:17:54.66,Default,,0000,0000,0000,,I'm sort of done with 11.4.
Dialogue: 0,1:17:54.66,1:17:57.84,Default,,0000,0000,0000,,I'll jump into 11.5 next.
Dialogue: 0,1:17:57.84,1:18:00.83,Default,,0000,0000,0000,,Take a short break.
Dialogue: 0,1:18:00.83,1:18:02.83,Default,,0000,0000,0000,,Thanks for the attendance.
Dialogue: 0,1:18:02.83,1:18:04.82,Default,,0000,0000,0000,,Oh, and you did the calculus.
Dialogue: 0,1:18:04.82,1:18:05.82,Default,,0000,0000,0000,,Very good.
Dialogue: 0,1:18:05.82,1:19:51.58,Default,,0000,0000,0000,,
Dialogue: 0,1:19:51.58,1:19:55.08,Default,,0000,0000,0000,,Did this homework give you\Na lot of headaches, troubles
Dialogue: 0,1:19:55.08,1:19:56.08,Default,,0000,0000,0000,,or anything, or not?
Dialogue: 0,1:19:56.08,1:19:57.57,Default,,0000,0000,0000,,Not too much?
Dialogue: 0,1:19:57.57,1:19:59.07,Default,,0000,0000,0000,,It's a long homework.
Dialogue: 0,1:19:59.07,1:20:00.57,Default,,0000,0000,0000,,49 problems-- 42 problems.
Dialogue: 0,1:20:00.57,1:20:05.56,Default,,0000,0000,0000,,
Dialogue: 0,1:20:05.56,1:20:07.05,Default,,0000,0000,0000,,It wasn't bad?
Dialogue: 0,1:20:07.05,1:22:39.08,Default,,0000,0000,0000,,
Dialogue: 0,1:22:39.08,1:22:45.83,Default,,0000,0000,0000,,OK, questions from the-- what\Nwas it, the first part-- mainly
Dialogue: 0,1:22:45.83,1:22:47.62,Default,,0000,0000,0000,,the first part of chapter 11.
Dialogue: 0,1:22:47.62,1:22:49.51,Default,,0000,0000,0000,,This is where we are.
Dialogue: 0,1:22:49.51,1:22:56.69,Default,,0000,0000,0000,,Right now we hit the\Nhalf point because 11.8
Dialogue: 0,1:22:56.69,1:22:59.25,Default,,0000,0000,0000,,is the last section.
Dialogue: 0,1:22:59.25,1:23:03.31,Default,,0000,0000,0000,,And we will do that, that's\NLagrange multipliers.
Dialogue: 0,1:23:03.31,1:23:06.90,Default,,0000,0000,0000,,So, let's do a little\Nbit of a review.
Dialogue: 0,1:23:06.90,1:23:08.84,Default,,0000,0000,0000,,Questions about homework.
Dialogue: 0,1:23:08.84,1:23:11.16,Default,,0000,0000,0000,,Do you have them?
Dialogue: 0,1:23:11.16,1:23:13.99,Default,,0000,0000,0000,,Imagine this would\Nbe office hour.
Dialogue: 0,1:23:13.99,1:23:15.12,Default,,0000,0000,0000,,What would you ask?
Dialogue: 0,1:23:15.12,1:23:17.80,Default,,0000,0000,0000,,
Dialogue: 0,1:23:17.80,1:23:19.51,Default,,0000,0000,0000,,STUDENT: I know it's\Na stupid question,
Dialogue: 0,1:23:19.51,1:23:22.21,Default,,0000,0000,0000,,but my visualization [INAUDIBLE]\Ncoming along, and question
Dialogue: 0,1:23:22.21,1:23:26.91,Default,,0000,0000,0000,,three about the sphere passing\Nthe plane and passing the line.
Dialogue: 0,1:23:26.91,1:23:31.59,Default,,0000,0000,0000,,So you have a 3, 5,\Nand 4 x, y, and z,
Dialogue: 0,1:23:31.59,1:23:34.34,Default,,0000,0000,0000,,and you have a radius of 5.
Dialogue: 0,1:23:34.34,1:23:36.28,Default,,0000,0000,0000,,Is it passing the x, y plane?
Dialogue: 0,1:23:36.28,1:23:40.71,Default,,0000,0000,0000,,Is it passing [INAUDIBLE]\Nx plane and [INAUDIBLE]
Dialogue: 0,1:23:40.71,1:23:42.22,Default,,0000,0000,0000,,passing the other plane.
Dialogue: 0,1:23:42.22,1:23:44.01,Default,,0000,0000,0000,,PROFESSOR TODA: So-- say again.
Dialogue: 0,1:23:44.01,1:23:46.05,Default,,0000,0000,0000,,So you have 3 and 4 and 5--
Dialogue: 0,1:23:46.05,1:23:47.52,Default,,0000,0000,0000,,STUDENT: x minus-- yes.
Dialogue: 0,1:23:47.52,1:23:49.38,Default,,0000,0000,0000,,PROFESSOR TODA: What\Nare the coordinates?
Dialogue: 0,1:23:49.38,1:23:50.69,Default,,0000,0000,0000,,STUDENT: 3, 4, and 5.
Dialogue: 0,1:23:50.69,1:23:53.41,Default,,0000,0000,0000,,PROFESSOR TODA: 3, 4, and\N5, just as you said them.
Dialogue: 0,1:23:53.41,1:23:54.19,Default,,0000,0000,0000,,You can--
Dialogue: 0,1:23:54.19,1:23:55.73,Default,,0000,0000,0000,,STUDENT: And the radius is 5.
Dialogue: 0,1:23:55.73,1:23:56.84,Default,,0000,0000,0000,,PROFESSOR TODA: Radius of?
Dialogue: 0,1:23:56.84,1:23:57.34,Default,,0000,0000,0000,,STUDENT: 5.
Dialogue: 0,1:23:57.34,1:23:59.64,Default,,0000,0000,0000,,Radius is equal to 5.
Dialogue: 0,1:23:59.64,1:24:00.60,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:24:00.60,1:24:02.11,Default,,0000,0000,0000,,PROFESSOR TODA: Yeah, well, OK.
Dialogue: 0,1:24:02.11,1:24:07.78,Default,,0000,0000,0000,,So assume you have a\Nsphere of radius 5, which
Dialogue: 0,1:24:07.78,1:24:09.27,Default,,0000,0000,0000,,means you have 25.
Dialogue: 0,1:24:09.27,1:24:14.71,Default,,0000,0000,0000,,If you do the 3 squared plus\N4 squared plus 5 squared,
Dialogue: 0,1:24:14.71,1:24:16.46,Default,,0000,0000,0000,,what is that?
Dialogue: 0,1:24:16.46,1:24:17.09,Default,,0000,0000,0000,,For this point.
Dialogue: 0,1:24:17.09,1:24:18.75,Default,,0000,0000,0000,,You have two separate points.
Dialogue: 0,1:24:18.75,1:24:22.92,Default,,0000,0000,0000,,For this point you\Nhave 25 plus 25.
Dialogue: 0,1:24:22.92,1:24:24.88,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:24:24.88,1:24:30.25,Default,,0000,0000,0000,,So you have the\Nspecific x0, y0, z0.
Dialogue: 0,1:24:30.25,1:24:39.06,Default,,0000,0000,0000,,You do the sum of the\Nsquares, and you get 50.
Dialogue: 0,1:24:39.06,1:24:43.90,Default,,0000,0000,0000,,My question is, is this point\Noutside, inside the sphere
Dialogue: 0,1:24:43.90,1:24:45.37,Default,,0000,0000,0000,,or on the sphere?
Dialogue: 0,1:24:45.37,1:24:47.01,Default,,0000,0000,0000,,On the sphere,\Nobviously, it's not,
Dialogue: 0,1:24:47.01,1:24:54.14,Default,,0000,0000,0000,,because it does not verify the\Nequation of the sphere, right?
Dialogue: 0,1:24:54.14,1:24:59.14,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] those the\Nlocation of the center point.
Dialogue: 0,1:24:59.14,1:25:01.31,Default,,0000,0000,0000,,STUDENT: Where's the\Ncenter of the sphere?
Dialogue: 0,1:25:01.31,1:25:02.14,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:25:02.14,1:25:05.64,Default,,0000,0000,0000,,
Dialogue: 0,1:25:05.64,1:25:09.14,Default,,0000,0000,0000,,PROFESSOR TODA: The center\Nof the sphere would be at 0.
Dialogue: 0,1:25:09.14,1:25:11.64,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:25:11.64,1:25:13.52,Default,,0000,0000,0000,,PROFESSOR TODA: We are\Nmaking up a question.
Dialogue: 0,1:25:13.52,1:25:14.73,Default,,0000,0000,0000,,So, right?
Dialogue: 0,1:25:14.73,1:25:16.78,Default,,0000,0000,0000,,So practically, I am\Nmaking up a question.
Dialogue: 0,1:25:16.78,1:25:17.45,Default,,0000,0000,0000,,STUDENT: Oh, OK.
Dialogue: 0,1:25:17.45,1:25:22.93,Default,,0000,0000,0000,,PROFESSOR TODA: So I'm saying if\Nyou have a sphere of radius 5,
Dialogue: 0,1:25:22.93,1:25:27.18,Default,,0000,0000,0000,,and somebody gives you this\Npoint of coordinates 3, 4,
Dialogue: 0,1:25:27.18,1:25:29.24,Default,,0000,0000,0000,,and 5, where is the point?
Dialogue: 0,1:25:29.24,1:25:34.93,Default,,0000,0000,0000,,Is it inside the sphere, outside\Nthe sphere or on the sphere?
Dialogue: 0,1:25:34.93,1:25:37.10,Default,,0000,0000,0000,,On the sphere it cannot be\Nbecause it doesn't verify
Dialogue: 0,1:25:37.10,1:25:39.78,Default,,0000,0000,0000,,the sphere.
Dialogue: 0,1:25:39.78,1:25:44.58,Default,,0000,0000,0000,,Ah, it looks like a Mr. Egg.
Dialogue: 0,1:25:44.58,1:25:47.28,Default,,0000,0000,0000,,I don't like it.
Dialogue: 0,1:25:47.28,1:25:50.61,Default,,0000,0000,0000,,I'm sorry, it's a sphere.
Dialogue: 0,1:25:50.61,1:25:54.88,Default,,0000,0000,0000,,So a point on a sphere that\Nwill have-- that's a hint.
Dialogue: 0,1:25:54.88,1:25:58.47,Default,,0000,0000,0000,,A point on a sphere that\Nwill have coordinates 3 and 4
Dialogue: 0,1:25:58.47,1:26:02.49,Default,,0000,0000,0000,,would be exactly 3, 4, and 0.
Dialogue: 0,1:26:02.49,1:26:05.96,Default,,0000,0000,0000,,So it would be where?
Dialogue: 0,1:26:05.96,1:26:07.76,Default,,0000,0000,0000,,STUDENT: 16, 4.
Dialogue: 0,1:26:07.76,1:26:11.48,Default,,0000,0000,0000,,PROFESSOR TODA: 3 squared plus\N4 squared is 5 squared, right?
Dialogue: 0,1:26:11.48,1:26:13.30,Default,,0000,0000,0000,,So those are\NPythagorean numbers.
Dialogue: 0,1:26:13.30,1:26:15.16,Default,,0000,0000,0000,,That's the beauty of them.
Dialogue: 0,1:26:15.16,1:26:22.60,Default,,0000,0000,0000,,
Dialogue: 0,1:26:22.60,1:26:27.57,Default,,0000,0000,0000,,I'm trying to draw well.
Dialogue: 0,1:26:27.57,1:26:28.41,Default,,0000,0000,0000,,Right.
Dialogue: 0,1:26:28.41,1:26:29.98,Default,,0000,0000,0000,,This is the point a.
Dialogue: 0,1:26:29.98,1:26:33.24,Default,,0000,0000,0000,,
Dialogue: 0,1:26:33.24,1:26:36.62,Default,,0000,0000,0000,,You go up how many?
Dialogue: 0,1:26:36.62,1:26:38.94,Default,,0000,0000,0000,,You shift by 5.
Dialogue: 0,1:26:38.94,1:26:41.33,Default,,0000,0000,0000,,So are you inside or outside?
Dialogue: 0,1:26:41.33,1:26:42.31,Default,,0000,0000,0000,,STUDENT: Outside.
Dialogue: 0,1:26:42.31,1:26:43.18,Default,,0000,0000,0000,,PROFESSOR TODA: Yeah.
Dialogue: 0,1:26:43.18,1:26:50.13,Default,,0000,0000,0000,,
Dialogue: 0,1:26:50.13,1:26:55.02,Default,,0000,0000,0000,,STUDENT: Are you outside\Nor are you exactly on-- oh.
Dialogue: 0,1:26:55.02,1:26:55.77,Default,,0000,0000,0000,,Sorry, I thought--
Dialogue: 0,1:26:55.77,1:26:56.40,Default,,0000,0000,0000,,PROFESSOR TODA: You go--
Dialogue: 0,1:26:56.40,1:26:58.29,Default,,0000,0000,0000,,STUDENT: I thought you\Nwere saying point a.
Dialogue: 0,1:26:58.29,1:26:59.96,Default,,0000,0000,0000,,Point a is like\Nexactly-- [INAUDIBLE]
Dialogue: 0,1:26:59.96,1:27:00.81,Default,,0000,0000,0000,,PROFESSOR TODA: You\Nare on the equator,
Dialogue: 0,1:27:00.81,1:27:02.26,Default,,0000,0000,0000,,and from the Equator\Nof the Earth,
Dialogue: 0,1:27:02.26,1:27:05.75,Default,,0000,0000,0000,,you're going parallel to the\Nz-axis, then you stay outside.
Dialogue: 0,1:27:05.75,1:27:08.57,Default,,0000,0000,0000,,But the question is\Nmore subtle than that.
Dialogue: 0,1:27:08.57,1:27:12.00,Default,,0000,0000,0000,,This is pretty--\Nyou figured it out.
Dialogue: 0,1:27:12.00,1:27:15.31,Default,,0000,0000,0000,,1 point-- 0.5 extra credit.
Dialogue: 0,1:27:15.31,1:27:18.58,Default,,0000,0000,0000,,That we don't have--\NI wish we had-- maybe
Dialogue: 0,1:27:18.58,1:27:19.90,Default,,0000,0000,0000,,we'll find some time.
Dialogue: 0,1:27:19.90,1:27:23.03,Default,,0000,0000,0000,,When I-- when we rewrite the\Nbook, maybe we should do that.
Dialogue: 0,1:27:23.03,1:27:38.64,Default,,0000,0000,0000,,So express the points outside\Nthe sphere, inside the sphere,
Dialogue: 0,1:27:38.64,1:27:50.21,Default,,0000,0000,0000,,and on the sphere\Nusing exclusively
Dialogue: 0,1:27:50.21,1:27:51.64,Default,,0000,0000,0000,,equalities and inequalities.
Dialogue: 0,1:27:51.64,1:27:57.90,Default,,0000,0000,0000,,
Dialogue: 0,1:27:57.90,1:27:58.90,Default,,0000,0000,0000,,And that's extra credit.
Dialogue: 0,1:27:58.90,1:28:01.00,Default,,0000,0000,0000,,So, of course, the\N[INAUDIBLE] is obvious.
Dialogue: 0,1:28:01.00,1:28:06.80,Default,,0000,0000,0000,,The sphere is the set of\Nthe triples x, y, z in R3.
Dialogue: 0,1:28:06.80,1:28:09.68,Default,,0000,0000,0000,,
Dialogue: 0,1:28:09.68,1:28:13.48,Default,,0000,0000,0000,,OK, I'm teaching you a little\Nbit of mathematical language.
Dialogue: 0,1:28:13.48,1:28:19.56,Default,,0000,0000,0000,,x, y, z belongs to R3,\NR3 being the free space,
Dialogue: 0,1:28:19.56,1:28:23.81,Default,,0000,0000,0000,,with the property that x squared\Nplus y squared plus z squared
Dialogue: 0,1:28:23.81,1:28:26.72,Default,,0000,0000,0000,,equals given a squared.
Dialogue: 0,1:28:26.72,1:28:29.84,Default,,0000,0000,0000,,What if you have less than,\Nwhat if you have greater than?
Dialogue: 0,1:28:29.84,1:28:31.84,Default,,0000,0000,0000,,Ah, shut up, Magdalena.
Dialogue: 0,1:28:31.84,1:28:33.40,Default,,0000,0000,0000,,This is all up to you.
Dialogue: 0,1:28:33.40,1:28:35.91,Default,,0000,0000,0000,,You will figure\Nout how the points
Dialogue: 0,1:28:35.91,1:28:40.92,Default,,0000,0000,0000,,on the outside and the points\Non the inside are characterized.
Dialogue: 0,1:28:40.92,1:28:47.06,Default,,0000,0000,0000,,And unfortunately we don't\Nemphasize that in the textbook.
Dialogue: 0,1:28:47.06,1:28:49.51,Default,,0000,0000,0000,,I'll erase.
Dialogue: 0,1:28:49.51,1:28:51.68,Default,,0000,0000,0000,,You figured it out.
Dialogue: 0,1:28:51.68,1:28:53.26,Default,,0000,0000,0000,,And now I want to\Nmove on to something
Dialogue: 0,1:28:53.26,1:28:57.06,Default,,0000,0000,0000,,a little bit challenging,\Nbut not very challenging.
Dialogue: 0,1:28:57.06,1:29:11.20,Default,,0000,0000,0000,,
Dialogue: 0,1:29:11.20,1:29:12.49,Default,,0000,0000,0000,,STUDENT: Professor, [INAUDIBLE]
Dialogue: 0,1:29:12.49,1:29:19.83,Default,,0000,0000,0000,,
Dialogue: 0,1:29:19.83,1:29:21.33,Default,,0000,0000,0000,,PROFESSOR TODA: The\Nlast requirement
Dialogue: 0,1:29:21.33,1:29:22.63,Default,,0000,0000,0000,,on the extra credit?
Dialogue: 0,1:29:22.63,1:29:26.66,Default,,0000,0000,0000,,So I said the sphere\Nrepresents the set of all
Dialogue: 0,1:29:26.66,1:29:29.53,Default,,0000,0000,0000,,triples x, y, z in\NR3 with the property
Dialogue: 0,1:29:29.53,1:29:31.99,Default,,0000,0000,0000,,that x squared plus y squared\Nplus y squared plus z squared
Dialogue: 0,1:29:31.99,1:29:33.88,Default,,0000,0000,0000,,equals a squared.
Dialogue: 0,1:29:33.88,1:29:36.84,Default,,0000,0000,0000,,With the equality sign.
Dialogue: 0,1:29:36.84,1:29:40.02,Default,,0000,0000,0000,,Represent the points on\Nthe inside of the sphere
Dialogue: 0,1:29:40.02,1:29:44.56,Default,,0000,0000,0000,,and the outside of the sphere\Nusing just inequalities.
Dialogue: 0,1:29:44.56,1:29:45.28,Default,,0000,0000,0000,,Mathematics.
Dialogue: 0,1:29:45.28,1:29:48.71,Default,,0000,0000,0000,,No writing, no words,\Njust mathematics.
Dialogue: 0,1:29:48.71,1:29:50.15,Default,,0000,0000,0000,,In set theory symbols.
Dialogue: 0,1:29:50.15,1:29:54.82,Default,,0000,0000,0000,,Like, the set of points\Nwith braces like that.
Dialogue: 0,1:29:54.82,1:29:57.68,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:29:57.68,1:30:02.62,Default,,0000,0000,0000,,I'll help you review a little\Nbit of stuff from the chain
Dialogue: 0,1:30:02.62,1:30:12.15,Default,,0000,0000,0000,,rule in-- in chapter--\NI don't know, guys,
Dialogue: 0,1:30:12.15,1:30:14.77,Default,,0000,0000,0000,,it was a long time ago.
Dialogue: 0,1:30:14.77,1:30:15.73,Default,,0000,0000,0000,,Shame on me.
Dialogue: 0,1:30:15.73,1:30:19.32,Default,,0000,0000,0000,,Chapter 3, calc 1.
Dialogue: 0,1:30:19.32,1:30:38.18,Default,,0000,0000,0000,,Versus chain rule rules in\Ncalc in-- chapter 5 calc 3.
Dialogue: 0,1:30:38.18,1:30:40.55,Default,,0000,0000,0000,,This is a little\Nbit of a warmup.
Dialogue: 0,1:30:40.55,1:30:42.32,Default,,0000,0000,0000,,I don't want to\N[INAUDIBLE] again
Dialogue: 0,1:30:42.32,1:30:44.33,Default,,0000,0000,0000,,next time when we\Nmeet on Thursday.
Dialogue: 0,1:30:44.33,1:30:45.99,Default,,0000,0000,0000,,Bless you.
Dialogue: 0,1:30:45.99,1:30:48.92,Default,,0000,0000,0000,,The bless you was\Nout of the context.
Dialogue: 0,1:30:48.92,1:30:51.58,Default,,0000,0000,0000,,What was the chain rule?
Dialogue: 0,1:30:51.58,1:30:53.67,Default,,0000,0000,0000,,We did compositions\Nof functions,
Dialogue: 0,1:30:53.67,1:31:01.09,Default,,0000,0000,0000,,and we had a diagram that we\Ndon't show you, but we should.
Dialogue: 0,1:31:01.09,1:31:05.05,Default,,0000,0000,0000,,There is practically a function\Nthat comes from a set A
Dialogue: 0,1:31:05.05,1:31:08.49,Default,,0000,0000,0000,,to a set B to a set\NC. These are the sets.
Dialogue: 0,1:31:08.49,1:31:12.76,Default,,0000,0000,0000,,And we have g and an f.
Dialogue: 0,1:31:12.76,1:31:17.48,Default,,0000,0000,0000,,And we have g of f of t.
Dialogue: 0,1:31:17.48,1:31:22.45,Default,,0000,0000,0000,,t is your favorite letter here.
Dialogue: 0,1:31:22.45,1:31:26.79,Default,,0000,0000,0000,,How do you do the\Nderivative with respect
Dialogue: 0,1:31:26.79,1:31:28.94,Default,,0000,0000,0000,,to g composed with f?
Dialogue: 0,1:31:28.94,1:31:32.92,Default,,0000,0000,0000,,
Dialogue: 0,1:31:32.92,1:31:36.85,Default,,0000,0000,0000,,I asked the same question to\Nmy Calc 1 and Calc 2 students,
Dialogue: 0,1:31:36.85,1:31:42.47,Default,,0000,0000,0000,,and they really had a hard\Ntime expressing themselves,
Dialogue: 0,1:31:42.47,1:31:44.71,Default,,0000,0000,0000,,expressing the chain rule.
Dialogue: 0,1:31:44.71,1:31:46.53,Default,,0000,0000,0000,,And when I gave them\Nan example, they
Dialogue: 0,1:31:46.53,1:31:49.60,Default,,0000,0000,0000,,said, oh, I know how to\Ndo it on the example.
Dialogue: 0,1:31:49.60,1:31:55.09,Default,,0000,0000,0000,,I just don't know how to do it\Non the-- I like the numbers,
Dialogue: 0,1:31:55.09,1:31:57.51,Default,,0000,0000,0000,,but I don't like them letters.
Dialogue: 0,1:31:57.51,1:32:02.34,Default,,0000,0000,0000,,So how do we do\Nit in an example?
Dialogue: 0,1:32:02.34,1:32:05.34,Default,,0000,0000,0000,,
Dialogue: 0,1:32:05.34,1:32:09.14,Default,,0000,0000,0000,,I chose natural log,\Nwhich you find everywhere.
Dialogue: 0,1:32:09.14,1:32:14.44,Default,,0000,0000,0000,,So how do you do d\Ndt of this animal?
Dialogue: 0,1:32:14.44,1:32:15.89,Default,,0000,0000,0000,,It's an animal.
Dialogue: 0,1:32:15.89,1:32:18.31,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:32:18.31,1:32:21.49,Default,,0000,0000,0000,,PROFESSOR TODA: So the idea\Nis you go from the outside
Dialogue: 0,1:32:21.49,1:32:23.14,Default,,0000,0000,0000,,to the inside, one at a time.
Dialogue: 0,1:32:23.14,1:32:24.68,Default,,0000,0000,0000,,My students know that.
Dialogue: 0,1:32:24.68,1:32:27.48,Default,,0000,0000,0000,,You prime the function,\Nthe outer function,
Dialogue: 0,1:32:27.48,1:32:30.57,Default,,0000,0000,0000,,the last one you applied,\Nto the function inside.
Dialogue: 0,1:32:30.57,1:32:33.77,Default,,0000,0000,0000,,And you prime that with\Nrespect to the argument.
Dialogue: 0,1:32:33.77,1:32:37.04,Default,,0000,0000,0000,,This is called the\Nargument in that case.
Dialogue: 0,1:32:37.04,1:32:40.69,Default,,0000,0000,0000,,Derivative of natural\Nlog is 1 over what?
Dialogue: 0,1:32:40.69,1:32:43.61,Default,,0000,0000,0000,,The argument.
Dialogue: 0,1:32:43.61,1:32:46.32,Default,,0000,0000,0000,,And you cover up natural\Nlog with your hand,
Dialogue: 0,1:32:46.32,1:32:47.17,Default,,0000,0000,0000,,and you keep going.
Dialogue: 0,1:32:47.17,1:32:51.87,Default,,0000,0000,0000,,And you say, next I go,\Ntimes the derivative
Dialogue: 0,1:32:51.87,1:32:55.70,Default,,0000,0000,0000,,of this square, plus 1,\Nprime with respect to t.
Dialogue: 0,1:32:55.70,1:32:58.46,Default,,0000,0000,0000,,So I go times 2t.
Dialogue: 0,1:32:58.46,1:33:01.29,Default,,0000,0000,0000,,And that's what we have.
Dialogue: 0,1:33:01.29,1:33:04.62,Default,,0000,0000,0000,,And they say, when you explain\Nit like that, they said to me,
Dialogue: 0,1:33:04.62,1:33:06.35,Default,,0000,0000,0000,,I can understand it.
Dialogue: 0,1:33:06.35,1:33:09.05,Default,,0000,0000,0000,,But I'm having a\Nproblem understanding it
Dialogue: 0,1:33:09.05,1:33:12.96,Default,,0000,0000,0000,,when you express this diagram--\Nthat it throws me off.
Dialogue: 0,1:33:12.96,1:33:19.24,Default,,0000,0000,0000,,So in order to avoid that kind\Nof theoretical misconception,
Dialogue: 0,1:33:19.24,1:33:24.99,Default,,0000,0000,0000,,I'm saying, let us see\Nwhat the heck this is.
Dialogue: 0,1:33:24.99,1:33:32.80,Default,,0000,0000,0000,,d dt of g of f of t, because\Nthis is what you're doing,
Dialogue: 0,1:33:32.80,1:33:34.68,Default,,0000,0000,0000,,has to have some understanding.
Dialogue: 0,1:33:34.68,1:33:38.62,Default,,0000,0000,0000,,The problem is that Mister\Nf of t, that lives here,
Dialogue: 0,1:33:38.62,1:33:40.37,Default,,0000,0000,0000,,has a different argument.
Dialogue: 0,1:33:40.37,1:33:45.41,Default,,0000,0000,0000,,The letter in B should\Nbe, let's say, u.
Dialogue: 0,1:33:45.41,1:33:48.51,Default,,0000,0000,0000,,
Dialogue: 0,1:33:48.51,1:33:51.68,Default,,0000,0000,0000,,That doesn't say\Nanything practically.
Dialogue: 0,1:33:51.68,1:33:54.00,Default,,0000,0000,0000,,How do you differentiate\Nwith respect to what?
Dialogue: 0,1:33:54.00,1:33:56.24,Default,,0000,0000,0000,,You cannot say d dt here.
Dialogue: 0,1:33:56.24,1:34:00.94,Default,,0000,0000,0000,,So you have to call f\Nof t something generic.
Dialogue: 0,1:34:00.94,1:34:05.21,Default,,0000,0000,0000,,You have to have a\Ngeneric variable for that.
Dialogue: 0,1:34:05.21,1:34:13.69,Default,,0000,0000,0000,,So you have then dg du, at\Nwhat specific value of u?
Dialogue: 0,1:34:13.69,1:34:18.05,Default,,0000,0000,0000,,At the specific value of\Nu that we have as f of t.
Dialogue: 0,1:34:18.05,1:34:21.57,Default,,0000,0000,0000,,Do you understand the\Nspecificity of this?
Dialogue: 0,1:34:21.57,1:34:26.70,Default,,0000,0000,0000,,Times-- that's the chain\Nrule, the product coming
Dialogue: 0,1:34:26.70,1:34:31.76,Default,,0000,0000,0000,,from the chain rule-- df pt.
Dialogue: 0,1:34:31.76,1:34:33.89,Default,,0000,0000,0000,,You take du dt or d of dt.
Dialogue: 0,1:34:33.89,1:34:34.94,Default,,0000,0000,0000,,It is the same thing.
Dialogue: 0,1:34:34.94,1:34:36.88,Default,,0000,0000,0000,,Say it again, df dt.
Dialogue: 0,1:34:36.88,1:34:41.26,Default,,0000,0000,0000,,
Dialogue: 0,1:34:41.26,1:34:43.54,Default,,0000,0000,0000,,I had a student ask me,\Nwhat if I put du dt?
Dialogue: 0,1:34:43.54,1:34:44.79,Default,,0000,0000,0000,,Would it be wrong?
Dialogue: 0,1:34:44.79,1:34:50.08,Default,,0000,0000,0000,,No, as long as you understand\Nthat u is a-something,
Dialogue: 0,1:34:50.08,1:34:54.87,Default,,0000,0000,0000,,as the image of this t.
Dialogue: 0,1:34:54.87,1:34:55.95,Default,,0000,0000,0000,,Do you know what he liked?
Dialogue: 0,1:34:55.95,1:34:58.94,Default,,0000,0000,0000,,
Dialogue: 0,1:34:58.94,1:35:01.79,Default,,0000,0000,0000,,He said, do you know\Nwhat I like about that?
Dialogue: 0,1:35:01.79,1:35:07.16,Default,,0000,0000,0000,,I like that I can imagine\Nthat these are two cowboys-- I
Dialogue: 0,1:35:07.16,1:35:09.45,Default,,0000,0000,0000,,told the same thing to my son.
Dialogue: 0,1:35:09.45,1:35:12.51,Default,,0000,0000,0000,,He was so excited,\Nnot about that,
Dialogue: 0,1:35:12.51,1:35:14.60,Default,,0000,0000,0000,,but about these two cowboys.
Dialogue: 0,1:35:14.60,1:35:17.33,Default,,0000,0000,0000,,Of course, he is 10.
Dialogue: 0,1:35:17.33,1:35:18.24,Default,,0000,0000,0000,,These are the cowboys.
Dialogue: 0,1:35:18.24,1:35:20.42,Default,,0000,0000,0000,,They are across.
Dialogue: 0,1:35:20.42,1:35:22.64,Default,,0000,0000,0000,,One is on top of\Nthe building there,
Dialogue: 0,1:35:22.64,1:35:24.85,Default,,0000,0000,0000,,shooting at this\Nguy, who is here
Dialogue: 0,1:35:24.85,1:35:28.48,Default,,0000,0000,0000,,across the street on the bottom.
Dialogue: 0,1:35:28.48,1:35:31.39,Default,,0000,0000,0000,,So they are\Nannihilating each other.
Dialogue: 0,1:35:31.39,1:35:33.29,Default,,0000,0000,0000,,They shoot and they die.
Dialogue: 0,1:35:33.29,1:35:37.08,Default,,0000,0000,0000,,And they die, and\Nyou're left with 1/3.
Dialogue: 0,1:35:37.08,1:35:41.78,Default,,0000,0000,0000,,The same idea is that, actually,\Nthese guys do not simplify.
Dialogue: 0,1:35:41.78,1:35:46.10,Default,,0000,0000,0000,,du and-- [? du, ?] they're not\Ncowboys who shoot at each other
Dialogue: 0,1:35:46.10,1:35:48.58,Default,,0000,0000,0000,,at the same time and both\Ndie at the same time.
Dialogue: 0,1:35:48.58,1:35:53.22,Default,,0000,0000,0000,,It is not so romantic.
Dialogue: 0,1:35:53.22,1:35:59.64,Default,,0000,0000,0000,,But the idea of remembering\Nthis formula is the same.
Dialogue: 0,1:35:59.64,1:36:03.70,Default,,0000,0000,0000,,Because practically, if you want\Nto annihilate the two cowboys
Dialogue: 0,1:36:03.70,1:36:06.33,Default,,0000,0000,0000,,and put your hands over them\Nso you don't see them anymore,
Dialogue: 0,1:36:06.33,1:36:10.58,Default,,0000,0000,0000,,du dt, you would\Nhave to remember, oh,
Dialogue: 0,1:36:10.58,1:36:12.43,Default,,0000,0000,0000,,so that was the\Nderivative with respect
Dialogue: 0,1:36:12.43,1:36:15.93,Default,,0000,0000,0000,,to t that I initially\Nhave of the guy on top,
Dialogue: 0,1:36:15.93,1:36:19.11,Default,,0000,0000,0000,,which was g of f of\Nthe composed function.
Dialogue: 0,1:36:19.11,1:36:22.85,Default,,0000,0000,0000,,So if you view g of f of t\Nas the composed function,
Dialogue: 0,1:36:22.85,1:36:23.84,Default,,0000,0000,0000,,who is that?
Dialogue: 0,1:36:23.84,1:36:28.98,Default,,0000,0000,0000,,The composition g\Ncomposed with f of t
Dialogue: 0,1:36:28.98,1:36:31.90,Default,,0000,0000,0000,,is the function g of f of t.
Dialogue: 0,1:36:31.90,1:36:34.94,Default,,0000,0000,0000,,This is the function that\Nyou want to differentiate
Dialogue: 0,1:36:34.94,1:36:37.37,Default,,0000,0000,0000,,with respect to time, t.
Dialogue: 0,1:36:37.37,1:36:40.98,Default,,0000,0000,0000,,This is this, prime\Nwith respect to t.
Dialogue: 0,1:36:40.98,1:36:46.02,Default,,0000,0000,0000,,It's like they would be killing\Neach other, and you would die.
Dialogue: 0,1:36:46.02,1:36:48.20,Default,,0000,0000,0000,,And I liked this\Nidea, and I said,
Dialogue: 0,1:36:48.20,1:36:50.39,Default,,0000,0000,0000,,I should tell that to my\Nstudents and to my son.
Dialogue: 0,1:36:50.39,1:36:52.92,Default,,0000,0000,0000,,And, of course, my son\Nstarted jumping around
Dialogue: 0,1:36:52.92,1:36:56.48,Default,,0000,0000,0000,,and said that he understands\Nmultiplication of fractions
Dialogue: 0,1:36:56.48,1:36:57.89,Default,,0000,0000,0000,,better now.
Dialogue: 0,1:36:57.89,1:37:01.39,Default,,0000,0000,0000,,They don't learn about\Nsimplifications-- I don't
Dialogue: 0,1:37:01.39,1:37:03.04,Default,,0000,0000,0000,,know how they teach these kids.
Dialogue: 0,1:37:03.04,1:37:06.32,Default,,0000,0000,0000,,
Dialogue: 0,1:37:06.32,1:37:07.83,Default,,0000,0000,0000,,It became so complicated.
Dialogue: 0,1:37:07.83,1:37:10.90,Default,,0000,0000,0000,,It's as if mathematics--\Nmathematics is the same.
Dialogue: 0,1:37:10.90,1:37:12.11,Default,,0000,0000,0000,,It hasn't changed.
Dialogue: 0,1:37:12.11,1:37:14.31,Default,,0000,0000,0000,,It's the people\Nwho make the rules
Dialogue: 0,1:37:14.31,1:37:17.42,Default,,0000,0000,0000,,on how to teach it that change.
Dialogue: 0,1:37:17.42,1:37:21.53,Default,,0000,0000,0000,,So he simply doesn't see\Nthat this simplifies.
Dialogue: 0,1:37:21.53,1:37:24.69,Default,,0000,0000,0000,,And when I tell him simplify,\Nhe's like, what is simplify?
Dialogue: 0,1:37:24.69,1:37:25.85,Default,,0000,0000,0000,,What is this word simplify?
Dialogue: 0,1:37:25.85,1:37:27.24,Default,,0000,0000,0000,,My teacher doesn't use it.
Dialogue: 0,1:37:27.24,1:37:31.70,Default,,0000,0000,0000,,So I feel like sometimes\NI want to shoot myself.
Dialogue: 0,1:37:31.70,1:37:35.38,Default,,0000,0000,0000,,But he went over that and\Nhe understood about the idea
Dialogue: 0,1:37:35.38,1:37:37.42,Default,,0000,0000,0000,,of simplification.
Dialogue: 0,1:37:37.42,1:37:39.37,Default,,0000,0000,0000,,[? He ?] composing\Nsomething on top
Dialogue: 0,1:37:39.37,1:37:43.19,Default,,0000,0000,0000,,and the bottom finding the\Ncommon factors up and down,
Dialogue: 0,1:37:43.19,1:37:44.82,Default,,0000,0000,0000,,crossing them out, and so on.
Dialogue: 0,1:37:44.82,1:37:47.26,Default,,0000,0000,0000,,And so now he knows\Nwhat it means.
Dialogue: 0,1:37:47.26,1:37:50.58,Default,,0000,0000,0000,,But imagine going to\Ncollege without having
Dialogue: 0,1:37:50.58,1:37:51.45,Default,,0000,0000,0000,,this early knowledge.
Dialogue: 0,1:37:51.45,1:37:55.13,Default,,0000,0000,0000,,You come to college,\Nyou were good in school,
Dialogue: 0,1:37:55.13,1:37:57.09,Default,,0000,0000,0000,,and you've never learned\Nenough simplification.
Dialogue: 0,1:37:57.09,1:38:00.22,Default,,0000,0000,0000,,And then somebody like me,\Nand tells you simplification.
Dialogue: 0,1:38:00.22,1:38:03.01,Default,,0000,0000,0000,,You say, she is a foreigner.
Dialogue: 0,1:38:03.01,1:38:07.77,Default,,0000,0000,0000,,She has a language barrier\Nthat is [INAUDIBLE] she has
Dialogue: 0,1:38:07.77,1:38:10.05,Default,,0000,0000,0000,,that I've never heard before.
Dialogue: 0,1:38:10.05,1:38:15.25,Default,,0000,0000,0000,,So I wish the people who\Nreally re-conceive, re-write
Dialogue: 0,1:38:15.25,1:38:18.82,Default,,0000,0000,0000,,the curriculum for K12\Nwould be a little bit
Dialogue: 0,1:38:18.82,1:38:21.73,Default,,0000,0000,0000,,more respectful of the history.
Dialogue: 0,1:38:21.73,1:38:25.59,Default,,0000,0000,0000,,Imagine that I\Nwould teach calculus
Dialogue: 0,1:38:25.59,1:38:28.81,Default,,0000,0000,0000,,without ever telling you\Nanything about Leibniz, who
Dialogue: 0,1:38:28.81,1:38:31.20,Default,,0000,0000,0000,,was Leibniz, he doesn't exist.
Dialogue: 0,1:38:31.20,1:38:34.10,Default,,0000,0000,0000,,Or Euler, or one\Nof these fathers.
Dialogue: 0,1:38:34.10,1:38:37.66,Default,,0000,0000,0000,,They are the ones who\Ncreated these notations.
Dialogue: 0,1:38:37.66,1:38:42.63,Default,,0000,0000,0000,,And if we never tell you\Nabout them, that I guess,
Dialogue: 0,1:38:42.63,1:38:47.40,Default,,0000,0000,0000,,wherever they are, it is an\Ninjustice that we are doing.
Dialogue: 0,1:38:47.40,1:38:48.18,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:38:48.18,1:38:53.52,Default,,0000,0000,0000,,Chain rule in\NChapter 5 of Calc 3.
Dialogue: 0,1:38:53.52,1:38:56.31,Default,,0000,0000,0000,,This is a little bit\Nmore complicated,
Dialogue: 0,1:38:56.31,1:38:59.69,Default,,0000,0000,0000,,but I'm going to teach it\Nto you because I like it.
Dialogue: 0,1:38:59.69,1:39:05.58,Default,,0000,0000,0000,,Imagine that you have z equals\Nx squared plus y squared.
Dialogue: 0,1:39:05.58,1:39:06.57,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:39:06.57,1:39:08.05,Default,,0000,0000,0000,,It's an example of a graph.
Dialogue: 0,1:39:08.05,1:39:10.61,Default,,0000,0000,0000,,And I just taught\Nyou what a graph is.
Dialogue: 0,1:39:10.61,1:39:13.49,Default,,0000,0000,0000,,
Dialogue: 0,1:39:13.49,1:39:22.94,Default,,0000,0000,0000,,But imagine that\Nxy follow a curve.
Dialogue: 0,1:39:22.94,1:39:25.90,Default,,0000,0000,0000,,
Dialogue: 0,1:39:25.90,1:39:27.88,Default,,0000,0000,0000,,[INAUDIBLE] with\Nrespect to time.
Dialogue: 0,1:39:27.88,1:39:37.78,Default,,0000,0000,0000,,
Dialogue: 0,1:39:37.78,1:39:41.33,Default,,0000,0000,0000,,And you will say, Magdalena,\Ncan you draw that?
Dialogue: 0,1:39:41.33,1:39:45.67,Default,,0000,0000,0000,,What in the world do you mean\Nthat x and y follow a curve?
Dialogue: 0,1:39:45.67,1:39:46.77,Default,,0000,0000,0000,,I'll try to draw it.
Dialogue: 0,1:39:46.77,1:39:48.64,Default,,0000,0000,0000,,First of all, you are on a walk.
Dialogue: 0,1:39:48.64,1:39:50.48,Default,,0000,0000,0000,,You are in a beautiful valley.
Dialogue: 0,1:39:50.48,1:39:51.48,Default,,0000,0000,0000,,It's not a vase.
Dialogue: 0,1:39:51.48,1:39:57.09,Default,,0000,0000,0000,,It's a circular\Nparaboloid, as an example.
Dialogue: 0,1:39:57.09,1:40:00.94,Default,,0000,0000,0000,,
Dialogue: 0,1:40:00.94,1:40:01.91,Default,,0000,0000,0000,,It's like an egg shell.
Dialogue: 0,1:40:01.91,1:40:05.29,Default,,0000,0000,0000,,
Dialogue: 0,1:40:05.29,1:40:07.05,Default,,0000,0000,0000,,You have a curve on that.
Dialogue: 0,1:40:07.05,1:40:08.05,Default,,0000,0000,0000,,You draw that.
Dialogue: 0,1:40:08.05,1:40:10.34,Default,,0000,0000,0000,,You have nothing better\Nto do than decorating eggs
Dialogue: 0,1:40:10.34,1:40:10.96,Default,,0000,0000,0000,,for Easter.
Dialogue: 0,1:40:10.96,1:40:12.19,Default,,0000,0000,0000,,Hey, wait.
Dialogue: 0,1:40:12.19,1:40:14.70,Default,,0000,0000,0000,,Easter is far, far away.
Dialogue: 0,1:40:14.70,1:40:17.36,Default,,0000,0000,0000,,But let's say you want to\Ndecorate eggs for Easter.
Dialogue: 0,1:40:17.36,1:40:22.80,Default,,0000,0000,0000,,You take some color of paint\Nand put paint on the egg.
Dialogue: 0,1:40:22.80,1:40:28.40,Default,,0000,0000,0000,,You are actually describing\Nan arc of a curve.
Dialogue: 0,1:40:28.40,1:40:38.24,Default,,0000,0000,0000,,And x and y, their\Nprojection on the floor
Dialogue: 0,1:40:38.24,1:40:39.60,Default,,0000,0000,0000,,will be x of t, y of t.
Dialogue: 0,1:40:39.60,1:40:42.79,Default,,0000,0000,0000,,
Dialogue: 0,1:40:42.79,1:40:45.47,Default,,0000,0000,0000,,Because you paint in time.
Dialogue: 0,1:40:45.47,1:40:46.22,Default,,0000,0000,0000,,You paint in time.
Dialogue: 0,1:40:46.22,1:40:48.09,Default,,0000,0000,0000,,You describe this in time.
Dialogue: 0,1:40:48.09,1:40:54.09,Default,,0000,0000,0000,,Now, if x of ty of t is\Nbeing projected on the floor.
Dialogue: 0,1:40:54.09,1:40:58.82,Default,,0000,0000,0000,,Of course, you have a curve\Nhere as well, which is what?
Dialogue: 0,1:40:58.82,1:41:05.70,Default,,0000,0000,0000,,Which it will be x\Nof t, y of t, z of t.
Dialogue: 0,1:41:05.70,1:41:06.62,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,1:41:06.62,1:41:11.91,Default,,0000,0000,0000,,Yes, because the altitude also\Ndepends on the motion in time.
Dialogue: 0,1:41:11.91,1:41:13.81,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:41:13.81,1:41:16.46,Default,,0000,0000,0000,,So what's missing here?
Dialogue: 0,1:41:16.46,1:41:18.88,Default,,0000,0000,0000,,It's missing the third\Ncoordinate, duh, that's
Dialogue: 0,1:41:18.88,1:41:21.38,Default,,0000,0000,0000,,0 because I'm on the floor.
Dialogue: 0,1:41:21.38,1:41:26.50,Default,,0000,0000,0000,,I'm on the xy plane, which\Nis the floor z equals z.
Dialogue: 0,1:41:26.50,1:41:28.56,Default,,0000,0000,0000,,But now let's\Nsuppose that I want
Dialogue: 0,1:41:28.56,1:41:36.57,Default,,0000,0000,0000,,to say this is f of x and y,\Nand I want to differentiate
Dialogue: 0,1:41:36.57,1:41:39.40,Default,,0000,0000,0000,,f with respect to t.
Dialogue: 0,1:41:39.40,1:41:40.73,Default,,0000,0000,0000,,And you go, say what?
Dialogue: 0,1:41:40.73,1:41:41.44,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,1:41:41.44,1:41:42.48,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:41:42.48,1:41:45.84,Default,,0000,0000,0000,,I differentiate f\Nwith respect to t.
Dialogue: 0,1:41:45.84,1:41:48.73,Default,,0000,0000,0000,,By differentiating\Nf with respect to t,
Dialogue: 0,1:41:48.73,1:41:54.78,Default,,0000,0000,0000,,I mean that I have f of\Nx and y differentiated
Dialogue: 0,1:41:54.78,1:41:56.14,Default,,0000,0000,0000,,with respect to t.
Dialogue: 0,1:41:56.14,1:41:58.07,Default,,0000,0000,0000,,And you say, wait, Magdalena.
Dialogue: 0,1:41:58.07,1:41:59.82,Default,,0000,0000,0000,,This doesn't make any sense.
Dialogue: 0,1:41:59.82,1:42:03.64,Default,,0000,0000,0000,,And you would be right to say\Nit doesn't make any sense.
Dialogue: 0,1:42:03.64,1:42:07.28,Default,,0000,0000,0000,,Can somebody tell me why\Nit doesn't make any sense?
Dialogue: 0,1:42:07.28,1:42:13.58,Default,,0000,0000,0000,,It's not clear where in the\Nworld the variable t is inside.
Dialogue: 0,1:42:13.58,1:42:17.22,Default,,0000,0000,0000,,So I'm going to say, OK,\Nx are themselves functions
Dialogue: 0,1:42:17.22,1:42:19.50,Default,,0000,0000,0000,,of t, functions of that.
Dialogue: 0,1:42:19.50,1:42:21.34,Default,,0000,0000,0000,,x of t, y of t.
Dialogue: 0,1:42:21.34,1:42:23.95,Default,,0000,0000,0000,,If I don't do that,\Nit's not clear.
Dialogue: 0,1:42:23.95,1:42:27.72,Default,,0000,0000,0000,,So this is a composed\Nfunction just like this one.
Dialogue: 0,1:42:27.72,1:42:28.68,Default,,0000,0000,0000,,Look at the similarity.
Dialogue: 0,1:42:28.68,1:42:31.10,Default,,0000,0000,0000,,It's really beautiful.
Dialogue: 0,1:42:31.10,1:42:35.67,Default,,0000,0000,0000,,This is a function of\Na function, g of f.
Dialogue: 0,1:42:35.67,1:42:38.52,Default,,0000,0000,0000,,This is a function\Nof two functions.
Dialogue: 0,1:42:38.52,1:42:43.30,Default,,0000,0000,0000,,Say it again, f is a function\Nof two functions, x and y.
Dialogue: 0,1:42:43.30,1:42:45.23,Default,,0000,0000,0000,,This was a function\Nof a function of t.
Dialogue: 0,1:42:45.23,1:42:47.64,Default,,0000,0000,0000,,This was a function\Nof two functions of t.
Dialogue: 0,1:42:47.64,1:42:48.61,Default,,0000,0000,0000,,Oh, my God.
Dialogue: 0,1:42:48.61,1:42:52.47,Default,,0000,0000,0000,,
Dialogue: 0,1:42:52.47,1:42:55.08,Default,,0000,0000,0000,,How do we compute this?
Dialogue: 0,1:42:55.08,1:42:56.85,Default,,0000,0000,0000,,There is a rule.
Dialogue: 0,1:42:56.85,1:42:58.20,Default,,0000,0000,0000,,It can be proved.
Dialogue: 0,1:42:58.20,1:43:01.95,Default,,0000,0000,0000,,We will look a little bit into\Nthe theoretical justification
Dialogue: 0,1:43:01.95,1:43:03.40,Default,,0000,0000,0000,,of this proof later.
Dialogue: 0,1:43:03.40,1:43:05.72,Default,,0000,0000,0000,,But practically what\Nyou do, you say,
Dialogue: 0,1:43:05.72,1:43:07.96,Default,,0000,0000,0000,,I have to have some\Norder in my life.
Dialogue: 0,1:43:07.96,1:43:09.09,Default,,0000,0000,0000,,OK.?
Dialogue: 0,1:43:09.09,1:43:12.88,Default,,0000,0000,0000,,So the way we do that,\Nwe differentiate first
Dialogue: 0,1:43:12.88,1:43:17.15,Default,,0000,0000,0000,,with respect to the first\Nlocation, which is x.
Dialogue: 0,1:43:17.15,1:43:21.52,Default,,0000,0000,0000,,I go there, but I cannot write\Ndf dx because f is a mother
Dialogue: 0,1:43:21.52,1:43:23.11,Default,,0000,0000,0000,,of two babies.
Dialogue: 0,1:43:23.11,1:43:26.52,Default,,0000,0000,0000,,f is a function of two\Nvariables, x and y.
Dialogue: 0,1:43:26.52,1:43:28.80,Default,,0000,0000,0000,,She has to be a mother\Nto both of them;
Dialogue: 0,1:43:28.80,1:43:31.62,Default,,0000,0000,0000,,otherwise, they get\Njealous of one another.
Dialogue: 0,1:43:31.62,1:43:37.63,Default,,0000,0000,0000,,So I have to say, partial\Nof f with respect to x,
Dialogue: 0,1:43:37.63,1:43:38.86,Default,,0000,0000,0000,,I cannot use d.
Dialogue: 0,1:43:38.86,1:43:43.51,Default,,0000,0000,0000,,Like Leibniz, I have\Nto use del, d of dx.
Dialogue: 0,1:43:43.51,1:43:49.03,Default,,0000,0000,0000,,At the point x of dy of t,\Nthis is the location I have.
Dialogue: 0,1:43:49.03,1:43:50.63,Default,,0000,0000,0000,,Times what?
Dialogue: 0,1:43:50.63,1:43:51.97,Default,,0000,0000,0000,,I keep derivation.
Dialogue: 0,1:43:51.97,1:43:55.64,Default,,0000,0000,0000,,I keep derivating, like\Ndon't drink and derive.
Dialogue: 0,1:43:55.64,1:43:56.63,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:43:56.63,1:43:58.98,Default,,0000,0000,0000,,The chain rule.
Dialogue: 0,1:43:58.98,1:44:05.43,Default,,0000,0000,0000,,Prime again, this guy x\Nwith respect to t, dx dt.
Dialogue: 0,1:44:05.43,1:44:09.32,Default,,0000,0000,0000,,And then you go,\Nplus because she has
Dialogue: 0,1:44:09.32,1:44:11.57,Default,,0000,0000,0000,,to be a mother to both kids.
Dialogue: 0,1:44:11.57,1:44:14.67,Default,,0000,0000,0000,,The same thing for\Nthe second child.
Dialogue: 0,1:44:14.67,1:44:17.69,Default,,0000,0000,0000,,So you go, the derivative\Nof f with respect
Dialogue: 0,1:44:17.69,1:44:26.99,Default,,0000,0000,0000,,to y, add x of ty\Nof t times dy dt.
Dialogue: 0,1:44:26.99,1:44:30.23,Default,,0000,0000,0000,,
Dialogue: 0,1:44:30.23,1:44:35.44,Default,,0000,0000,0000,,So you see on the surface, x and\Ny are moving according to time.
Dialogue: 0,1:44:35.44,1:44:39.00,Default,,0000,0000,0000,,And somehow we want to\Nmeasure the derivative
Dialogue: 0,1:44:39.00,1:44:42.79,Default,,0000,0000,0000,,of the resulting function,\Nor composition function,
Dialogue: 0,1:44:42.79,1:44:44.61,Default,,0000,0000,0000,,with respect to time.
Dialogue: 0,1:44:44.61,1:44:46.33,Default,,0000,0000,0000,,This is a very\Nimportant chain rule
Dialogue: 0,1:44:46.33,1:44:50.02,Default,,0000,0000,0000,,that I would like\Nyou to memorize.
Dialogue: 0,1:44:50.02,1:44:53.43,Default,,0000,0000,0000,,A chain rule.
Dialogue: 0,1:44:53.43,1:44:54.06,Default,,0000,0000,0000,,Chain Rule No.
Dialogue: 0,1:44:54.06,1:44:54.56,Default,,0000,0000,0000,,1.
Dialogue: 0,1:44:54.56,1:44:58.72,Default,,0000,0000,0000,,
Dialogue: 0,1:44:58.72,1:44:59.92,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:44:59.92,1:45:01.49,Default,,0000,0000,0000,,No, but for me it was.
Dialogue: 0,1:45:01.49,1:45:04.69,Default,,0000,0000,0000,,When I was 21 and I saw\Nthat-- and, of course,
Dialogue: 0,1:45:04.69,1:45:06.02,Default,,0000,0000,0000,,my teacher was good.
Dialogue: 0,1:45:06.02,1:45:10.35,Default,,0000,0000,0000,,And he told me, Magdalena,\Nimagine that instead of del you
Dialogue: 0,1:45:10.35,1:45:13.53,Default,,0000,0000,0000,,would have d's.
Dialogue: 0,1:45:13.53,1:45:16.68,Default,,0000,0000,0000,,So you have d and d and d and d.
Dialogue: 0,1:45:16.68,1:45:21.30,Default,,0000,0000,0000,,The dx dx here, dy dy here,\Nthey should be in your mind.
Dialogue: 0,1:45:21.30,1:45:22.72,Default,,0000,0000,0000,,They are facing each other.
Dialogue: 0,1:45:22.72,1:45:25.85,Default,,0000,0000,0000,,They are across on a diagonal.
Dialogue: 0,1:45:25.85,1:45:29.14,Default,,0000,0000,0000,,And then, of course, I didn't\Ntell my teacher my idea
Dialogue: 0,1:45:29.14,1:45:31.77,Default,,0000,0000,0000,,with the cowboys,\Nbut it was funny.
Dialogue: 0,1:45:31.77,1:45:38.81,Default,,0000,0000,0000,,So this is the chain rule\Nthat re-makes, or generalizes
Dialogue: 0,1:45:38.81,1:45:42.87,Default,,0000,0000,0000,,this idea to two variables.
Dialogue: 0,1:45:42.87,1:45:47.97,Default,,0000,0000,0000,,Let's finish the example\Nbecause we didn't do it.
Dialogue: 0,1:45:47.97,1:45:53.31,Default,,0000,0000,0000,,What is the derivative\Nof f in our case?
Dialogue: 0,1:45:53.31,1:46:01.66,Default,,0000,0000,0000,,df dt will be-- oh, my god--\Nat any point p, how arbitary,
Dialogue: 0,1:46:01.66,1:46:03.85,Default,,0000,0000,0000,,would be what?
Dialogue: 0,1:46:03.85,1:46:07.64,Default,,0000,0000,0000,,First, you write\Nwith respect to x.
Dialogue: 0,1:46:07.64,1:46:10.50,Default,,0000,0000,0000,,2x, right?
Dialogue: 0,1:46:10.50,1:46:11.00,Default,,0000,0000,0000,,2x.
Dialogue: 0,1:46:11.00,1:46:16.90,Default,,0000,0000,0000,,But then you have to compute\Nthis dx, add the pair you give.
Dialogue: 0,1:46:16.90,1:46:19.65,Default,,0000,0000,0000,,And the pair they\Ngave you has a t.
Dialogue: 0,1:46:19.65,1:46:23.45,Default,,0000,0000,0000,,So 2x is add x of\Nty-- if you're going
Dialogue: 0,1:46:23.45,1:46:25.34,Default,,0000,0000,0000,,to write it first\Nlike that, you're
Dialogue: 0,1:46:25.34,1:46:29.73,Default,,0000,0000,0000,,going to find it weird-- times,\NI'm done with the first guy.
Dialogue: 0,1:46:29.73,1:46:32.80,Default,,0000,0000,0000,,Then I'm going to take\Nthe second guy in red,
Dialogue: 0,1:46:32.80,1:46:35.31,Default,,0000,0000,0000,,and I'll put it here.
Dialogue: 0,1:46:35.31,1:46:39.28,Default,,0000,0000,0000,,dx dt, but dx dt\Neverybody knows.
Dialogue: 0,1:46:39.28,1:46:45.08,Default,,0000,0000,0000,,[INAUDIBLE] Let me\Nwrite it like this.
Dialogue: 0,1:46:45.08,1:46:52.19,Default,,0000,0000,0000,,Plus [INAUDIBLE] that\Nguy again with green-- dy
Dialogue: 0,1:46:52.19,1:46:59.15,Default,,0000,0000,0000,,computed at the pair x\Nof dy of [? t ?] times,
Dialogue: 0,1:46:59.15,1:47:01.51,Default,,0000,0000,0000,,again, in red, dy dt.
Dialogue: 0,1:47:01.51,1:47:06.73,Default,,0000,0000,0000,,
Dialogue: 0,1:47:06.73,1:47:08.77,Default,,0000,0000,0000,,So how do we write\Nthe whole thing?
Dialogue: 0,1:47:08.77,1:47:10.99,Default,,0000,0000,0000,,Could I have written it\Nfrom the beginning better?
Dialogue: 0,1:47:10.99,1:47:11.49,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,1:47:11.49,1:47:20.63,Default,,0000,0000,0000,,2x of t, dx dt plus 2y of t dy.
Dialogue: 0,1:47:20.63,1:47:21.61,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:47:21.61,1:47:25.15,Default,,0000,0000,0000,,No, this is the idea.
Dialogue: 0,1:47:25.15,1:47:28.07,Default,,0000,0000,0000,,Let's have something\Nmore specific.
Dialogue: 0,1:47:28.07,1:47:30.23,Default,,0000,0000,0000,,I'm going to erase\Nthe whole thing.
Dialogue: 0,1:47:30.23,1:47:36.23,Default,,0000,0000,0000,,
Dialogue: 0,1:47:36.23,1:47:40.20,Default,,0000,0000,0000,,I'll give you a problem\Nthat we gave on the final
Dialogue: 0,1:47:40.20,1:47:41.63,Default,,0000,0000,0000,,a few years ago.
Dialogue: 0,1:47:41.63,1:47:44.89,Default,,0000,0000,0000,,And I'll show you how my\Nstudents cheated on that.
Dialogue: 0,1:47:44.89,1:47:53.39,Default,,0000,0000,0000,,And I let them cheat, in\Na way, because in the end
Dialogue: 0,1:47:53.39,1:47:54.06,Default,,0000,0000,0000,,they were smart.
Dialogue: 0,1:47:54.06,1:47:59.35,Default,,0000,0000,0000,,It didn't matter how they did\Nthe problem, as long as they
Dialogue: 0,1:47:59.35,1:48:01.79,Default,,0000,0000,0000,,got the correct answer.
Dialogue: 0,1:48:01.79,1:48:03.33,Default,,0000,0000,0000,,So the problem was like that.
Dialogue: 0,1:48:03.33,1:48:10.16,Default,,0000,0000,0000,,And my colleague did that many\Nyears ago, several years ago,
Dialogue: 0,1:48:10.16,1:48:11.98,Default,,0000,0000,0000,,did that several times.
Dialogue: 0,1:48:11.98,1:48:19.70,Default,,0000,0000,0000,,So he said, let's do f of\Nt, dt squared and g of t.
Dialogue: 0,1:48:19.70,1:48:27.00,Default,,0000,0000,0000,,I'll I'll do this\None, dq plus 1.
Dialogue: 0,1:48:27.00,1:48:42.98,Default,,0000,0000,0000,,And then let's\N[INAUDIBLE] the w of u
Dialogue: 0,1:48:42.98,1:48:54.10,Default,,0000,0000,0000,,and B, exactly the same thing I\Ngave you before, [INAUDIBLE] I
Dialogue: 0,1:48:54.10,1:48:56.04,Default,,0000,0000,0000,,remember that.
Dialogue: 0,1:48:56.04,1:49:05.71,Default,,0000,0000,0000,,And he said, compute the\Nderivative of w of f of t,
Dialogue: 0,1:49:05.71,1:49:10.28,Default,,0000,0000,0000,,and g of t with respect to t.
Dialogue: 0,1:49:10.28,1:49:12.25,Default,,0000,0000,0000,,And you will ask,\Nwait a minute here.
Dialogue: 0,1:49:12.25,1:49:14.56,Default,,0000,0000,0000,,Why do you put d and not del?
Dialogue: 0,1:49:14.56,1:49:17.85,Default,,0000,0000,0000,,Because this is a composed\Nfunction that in the end
Dialogue: 0,1:49:17.85,1:49:20.58,Default,,0000,0000,0000,,is a function of t only.
Dialogue: 0,1:49:20.58,1:49:22.68,Default,,0000,0000,0000,,So if you do it as\Na composed function,
Dialogue: 0,1:49:22.68,1:49:26.04,Default,,0000,0000,0000,,because this goes like this.
Dialogue: 0,1:49:26.04,1:49:31.56,Default,,0000,0000,0000,,t goes to two\Nfunctions, f of t and u.
Dialogue: 0,1:49:31.56,1:49:34.45,Default,,0000,0000,0000,,
Dialogue: 0,1:49:34.45,1:49:40.85,Default,,0000,0000,0000,,And there is a function w\Nthat takes both of them, that
Dialogue: 0,1:49:40.85,1:49:42.87,Default,,0000,0000,0000,,is a function of both of them.
Dialogue: 0,1:49:42.87,1:49:46.84,Default,,0000,0000,0000,,In the end, this composition\Nthat's straight from here
Dialogue: 0,1:49:46.84,1:49:50.83,Default,,0000,0000,0000,,to here, is a function\Nof one variable only.
Dialogue: 0,1:49:50.83,1:49:54.85,Default,,0000,0000,0000,,
Dialogue: 0,1:49:54.85,1:49:58.28,Default,,0000,0000,0000,,So my students then-- it was in\Nthe beginning of the examine,
Dialogue: 0,1:49:58.28,1:49:59.02,Default,,0000,0000,0000,,I remember.
Dialogue: 0,1:49:59.02,1:50:02.30,Default,,0000,0000,0000,,And they said, well,\NI forgot, they said.
Dialogue: 0,1:50:02.30,1:50:03.86,Default,,0000,0000,0000,,I stayed up almost all night.
Dialogue: 0,1:50:03.86,1:50:05.43,Default,,0000,0000,0000,,Don't do that.
Dialogue: 0,1:50:05.43,1:50:06.39,Default,,0000,0000,0000,,Don't do what they did.
Dialogue: 0,1:50:06.39,1:50:08.33,Default,,0000,0000,0000,,Many of my students\Nstay up all night
Dialogue: 0,1:50:08.33,1:50:11.09,Default,,0000,0000,0000,,before the final because\NI think I scare people,
Dialogue: 0,1:50:11.09,1:50:12.70,Default,,0000,0000,0000,,and that's not what I mean.
Dialogue: 0,1:50:12.70,1:50:15.48,Default,,0000,0000,0000,,I just want you to study.
Dialogue: 0,1:50:15.48,1:50:18.67,Default,,0000,0000,0000,,But they stay up before\Nthe final and the next day,
Dialogue: 0,1:50:18.67,1:50:19.38,Default,,0000,0000,0000,,I'm a vegetable.
Dialogue: 0,1:50:19.38,1:50:21.41,Default,,0000,0000,0000,,I don't even remember\Nthe chain rule.
Dialogue: 0,1:50:21.41,1:50:23.45,Default,,0000,0000,0000,,So they did not\Nremember the chain rule
Dialogue: 0,1:50:23.45,1:50:25.00,Default,,0000,0000,0000,,that I've just wrote.
Dialogue: 0,1:50:25.00,1:50:28.49,Default,,0000,0000,0000,,And they said, oh, but I\Nthink I know how to do it.
Dialogue: 0,1:50:28.49,1:50:29.99,Default,,0000,0000,0000,,And I said, shh.
Dialogue: 0,1:50:29.99,1:50:31.98,Default,,0000,0000,0000,,Just don't say anything.
Dialogue: 0,1:50:31.98,1:50:34.84,Default,,0000,0000,0000,,Let me show you how the\Ncourse coordinator wanted
Dialogue: 0,1:50:34.84,1:50:37.17,Default,,0000,0000,0000,,that done several years ago.
Dialogue: 0,1:50:37.17,1:50:40.16,Default,,0000,0000,0000,,So he wanted it done\Nby the chain rule.
Dialogue: 0,1:50:40.16,1:50:41.55,Default,,0000,0000,0000,,He didn't say how you do it.
Dialogue: 0,1:50:41.55,1:50:42.05,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:50:42.05,1:50:44.16,Default,,0000,0000,0000,,He said just get to\Nthe right answer.
Dialogue: 0,1:50:44.16,1:50:45.61,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,1:50:45.61,1:50:46.78,Default,,0000,0000,0000,,He wanted it done like that.
Dialogue: 0,1:50:46.78,1:50:55.70,Default,,0000,0000,0000,,He said, dw of f of tg\Nof p with respect to t,
Dialogue: 0,1:50:55.70,1:51:06.74,Default,,0000,0000,0000,,would be dw du, instead\Nof u you have f of t.
Dialogue: 0,1:51:06.74,1:51:16.73,Default,,0000,0000,0000,,f of tg of t times df\Ndt plus dw with respect
Dialogue: 0,1:51:16.73,1:51:18.94,Default,,0000,0000,0000,,to the second variable.
Dialogue: 0,1:51:18.94,1:51:25.14,Default,,0000,0000,0000,,So this would be u, and\Nthis would be v with respect
Dialogue: 0,1:51:25.14,1:51:27.30,Default,,0000,0000,0000,,to the variable v,\Nthe second variable
Dialogue: 0,1:51:27.30,1:51:30.73,Default,,0000,0000,0000,,where [? measure ?]\Nthat f of dg of t.
Dialogue: 0,1:51:30.73,1:51:39.41,Default,,0000,0000,0000,,Evaluate it there times dg dt.
Dialogue: 0,1:51:39.41,1:51:46.14,Default,,0000,0000,0000,,So it's like dv dt, which is dg\Ndt. [INAUDIBLE] So he did that,
Dialogue: 0,1:51:46.14,1:51:48.04,Default,,0000,0000,0000,,and he expected\Npeople to do what?
Dialogue: 0,1:51:48.04,1:51:51.19,Default,,0000,0000,0000,,He expected people to take\Na u squared the same 2 times
Dialogue: 0,1:51:51.19,1:51:54.25,Default,,0000,0000,0000,,u, just like you\Ndid before, 2 times.
Dialogue: 0,1:51:54.25,1:51:57.72,Default,,0000,0000,0000,,And instead of u, since u is\Nf of t to [INAUDIBLE] puts
Dialogue: 0,1:51:57.72,1:52:13.17,Default,,0000,0000,0000,,2f of t, this is the first\Nsquiggly thing, times v of dt.
Dialogue: 0,1:52:13.17,1:52:19.93,Default,,0000,0000,0000,,2t is this smiley face.
Dialogue: 0,1:52:19.93,1:52:31.20,Default,,0000,0000,0000,,This is 2t plus--\Nwhat is the f dv?
Dialogue: 0,1:52:31.20,1:52:37.60,Default,,0000,0000,0000,,Dw with respect to dv is\Ngoing to be 2v 2 time gf t.
Dialogue: 0,1:52:37.60,1:52:46.79,Default,,0000,0000,0000,,When I evaluate add gf\Nt, this funny fellow
Dialogue: 0,1:52:46.79,1:52:57.58,Default,,0000,0000,0000,,with this funny fellow, times qg\Nd, which, with your permission
Dialogue: 0,1:52:57.58,1:53:00.89,Default,,0000,0000,0000,,I'm going to erase\Nand write 3p squared.
Dialogue: 0,1:53:00.89,1:53:04.03,Default,,0000,0000,0000,,
Dialogue: 0,1:53:04.03,1:53:07.34,Default,,0000,0000,0000,,And the last row he expected\Nmy students to write
Dialogue: 0,1:53:07.34,1:53:22.14,Default,,0000,0000,0000,,was 2t squared times 2t plus\N2pq plus 1, times 3t squared.
Dialogue: 0,1:53:22.14,1:53:27.58,Default,,0000,0000,0000,,
Dialogue: 0,1:53:27.58,1:53:31.54,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:53:31.54,1:53:43.21,Default,,0000,0000,0000,,So [INAUDIBLE] 2t 2x\N2t squared, correct.
Dialogue: 0,1:53:43.21,1:53:49.73,Default,,0000,0000,0000,,I forgot to identify\Nthis as that.
Dialogue: 0,1:53:49.73,1:53:50.23,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:53:50.23,1:53:52.65,Default,,0000,0000,0000,,So in the end, the answer\Nis a simplified answer.
Dialogue: 0,1:53:52.65,1:53:53.93,Default,,0000,0000,0000,,Can you tell me what it is?
Dialogue: 0,1:53:53.93,1:53:55.25,Default,,0000,0000,0000,,I'm too lazy to write it down.
Dialogue: 0,1:53:55.25,1:53:56.94,Default,,0000,0000,0000,,You compute it.
Dialogue: 0,1:53:56.94,1:53:58.93,Default,,0000,0000,0000,,How much is it simplified?
Dialogue: 0,1:53:58.93,1:54:00.42,Default,,0000,0000,0000,,Find it as a polynomial.
Dialogue: 0,1:54:00.42,1:54:01.42,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:54:01.42,1:54:04.40,Default,,0000,0000,0000,,
Dialogue: 0,1:54:04.40,1:54:08.87,Default,,0000,0000,0000,,PROFESSOR TODA:\NSo you have 6, 6--
Dialogue: 0,1:54:08.87,1:54:10.15,Default,,0000,0000,0000,,STUDENT: 16 cubed plus 3--
Dialogue: 0,1:54:10.15,1:54:15.30,Default,,0000,0000,0000,,PROFESSOR TODA: T\Nto the 5th plus--
Dialogue: 0,1:54:15.30,1:54:17.24,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:54:17.24,1:54:19.35,Default,,0000,0000,0000,,PROFESSOR TODA: In\Norder, in order.
Dialogue: 0,1:54:19.35,1:54:20.42,Default,,0000,0000,0000,,What's the next guy?
Dialogue: 0,1:54:20.42,1:54:21.64,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:54:21.64,1:54:22.83,Default,,0000,0000,0000,,PROFESSOR TODA: 4t cubed.
Dialogue: 0,1:54:22.83,1:54:23.98,Default,,0000,0000,0000,,And the last guy--
Dialogue: 0,1:54:23.98,1:54:24.92,Default,,0000,0000,0000,,STUDENT: 6t squared.
Dialogue: 0,1:54:24.92,1:54:26.05,Default,,0000,0000,0000,,PROFESSOR TODA: 6t squared.
Dialogue: 0,1:54:26.05,1:54:31.22,Default,,0000,0000,0000,,
Dialogue: 0,1:54:31.22,1:54:31.72,Default,,0000,0000,0000,,Yes?
Dialogue: 0,1:54:31.72,1:54:33.32,Default,,0000,0000,0000,,Did you get the same thing?
Dialogue: 0,1:54:33.32,1:54:34.22,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:54:34.22,1:54:37.14,Default,,0000,0000,0000,,Now, how did my students do it?
Dialogue: 0,1:54:37.14,1:54:37.64,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:54:37.64,1:54:40.17,Default,,0000,0000,0000,,
Dialogue: 0,1:54:40.17,1:54:41.42,Default,,0000,0000,0000,,Did they apply the chain rule?
Dialogue: 0,1:54:41.42,1:54:41.92,Default,,0000,0000,0000,,No.
Dialogue: 0,1:54:41.92,1:54:44.10,Default,,0000,0000,0000,,They said OK, this\Nis how it goes.
Dialogue: 0,1:54:44.10,1:54:46.96,Default,,0000,0000,0000,,
Dialogue: 0,1:54:46.96,1:54:58.21,Default,,0000,0000,0000,,W of U of T and V of T is U is\NF. So this guy is T squared,
Dialogue: 0,1:54:58.21,1:55:01.89,Default,,0000,0000,0000,,T squared squared,\Nplus this guy is T
Dialogue: 0,1:55:01.89,1:55:08.96,Default,,0000,0000,0000,,cubed plus 1 taken and\Nshaken and squared.
Dialogue: 0,1:55:08.96,1:55:13.71,Default,,0000,0000,0000,,And then when I do the\Nwhole thing, derivative
Dialogue: 0,1:55:13.71,1:55:22.64,Default,,0000,0000,0000,,of this with respect\Nto T, I get--
Dialogue: 0,1:55:22.64,1:55:27.57,Default,,0000,0000,0000,,I'm too lazy-- T to the\N4 prime is 40 cubed.
Dialogue: 0,1:55:27.57,1:55:28.87,Default,,0000,0000,0000,,I'm not going to do on the map.
Dialogue: 0,1:55:28.87,1:55:37.38,Default,,0000,0000,0000,,2 out T cubed plus 1 times\Nchain rule, 3t squared.
Dialogue: 0,1:55:37.38,1:55:49.61,Default,,0000,0000,0000,,40 cubed plus 16 to the 5 plus--\N[INAUDIBLE] 2 and 6t squared.
Dialogue: 0,1:55:49.61,1:55:56.45,Default,,0000,0000,0000,,So you realize that I\Nhave to give them 100%.
Dialogue: 0,1:55:56.45,1:55:59.46,Default,,0000,0000,0000,,Although they were very\Nhonest and said, we blanked.
Dialogue: 0,1:55:59.46,1:56:01.13,Default,,0000,0000,0000,,We don't remember\Nthe chain rule.
Dialogue: 0,1:56:01.13,1:56:02.93,Default,,0000,0000,0000,,We don't remember the formula.
Dialogue: 0,1:56:02.93,1:56:03.55,Default,,0000,0000,0000,,So that's fine.
Dialogue: 0,1:56:03.55,1:56:05.08,Default,,0000,0000,0000,,Do whatever you can.
Dialogue: 0,1:56:05.08,1:56:06.92,Default,,0000,0000,0000,,So I gave them 100% for that.
Dialogue: 0,1:56:06.92,1:56:11.28,Default,,0000,0000,0000,,But realize that the\Nauthor of the problem
Dialogue: 0,1:56:11.28,1:56:14.08,Default,,0000,0000,0000,,was a little bit naive.
Dialogue: 0,1:56:14.08,1:56:16.52,Default,,0000,0000,0000,,Because you could have\Ndone this differently.
Dialogue: 0,1:56:16.52,1:56:22.19,Default,,0000,0000,0000,,I mean if you wanted to\Nactually test the whole thing,
Dialogue: 0,1:56:22.19,1:56:26.17,Default,,0000,0000,0000,,you wouldn't have given-- let's\Nsay you wouldn't have given
Dialogue: 0,1:56:26.17,1:56:32.40,Default,,0000,0000,0000,,the actual-- yeah, you wouldn't\Nhave given the actual functions
Dialogue: 0,1:56:32.40,1:56:37.53,Default,,0000,0000,0000,,and say write the chain\Nformula symbolically
Dialogue: 0,1:56:37.53,1:56:44.88,Default,,0000,0000,0000,,for this function applied\Nfor F of T and G of T.
Dialogue: 0,1:56:44.88,1:56:49.34,Default,,0000,0000,0000,,So it was-- they\Nwere just lucky.
Dialogue: 0,1:56:49.34,1:56:52.47,Default,,0000,0000,0000,,Remember that you need\Nto know this chain rule.
Dialogue: 0,1:56:52.47,1:56:53.97,Default,,0000,0000,0000,,It's going to be\None of the problems
Dialogue: 0,1:56:53.97,1:56:56.90,Default,,0000,0000,0000,,to be emphasized in the exams.
Dialogue: 0,1:56:56.90,1:57:02.41,Default,,0000,0000,0000,,Maybe one of the top 15 or\N16 most important topics.
Dialogue: 0,1:57:02.41,1:57:07.17,Default,,0000,0000,0000,,
Dialogue: 0,1:57:07.17,1:57:07.90,Default,,0000,0000,0000,,Is that OK?
Dialogue: 0,1:57:07.90,1:57:09.40,Default,,0000,0000,0000,,Can I erase the whole thing?
Dialogue: 0,1:57:09.40,1:57:09.90,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:57:09.90,1:57:11.39,Default,,0000,0000,0000,,Let me erase the whole thing.
Dialogue: 0,1:57:11.39,1:57:44.33,Default,,0000,0000,0000,,
Dialogue: 0,1:57:44.33,1:57:44.83,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:57:44.83,1:57:45.82,Default,,0000,0000,0000,,Any other questions?
Dialogue: 0,1:57:45.82,1:58:02.29,Default,,0000,0000,0000,,
Dialogue: 0,1:58:02.29,1:58:03.66,Default,,0000,0000,0000,,No?
Dialogue: 0,1:58:03.66,1:58:05.28,Default,,0000,0000,0000,,I'm not going to let\Nyou go right away,
Dialogue: 0,1:58:05.28,1:58:07.78,Default,,0000,0000,0000,,we're going to work one\Nmore problem or two more
Dialogue: 0,1:58:07.78,1:58:08.78,Default,,0000,0000,0000,,simple problems.
Dialogue: 0,1:58:08.78,1:58:10.77,Default,,0000,0000,0000,,And then we are going to go.
Dialogue: 0,1:58:10.77,1:58:11.27,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:58:11.27,1:58:22.75,Default,,0000,0000,0000,,
Dialogue: 0,1:58:22.75,1:58:26.48,Default,,0000,0000,0000,,So question.
Dialogue: 0,1:58:26.48,1:58:27.99,Default,,0000,0000,0000,,A question.
Dialogue: 0,1:58:27.99,1:58:32.95,Default,,0000,0000,0000,,
Dialogue: 0,1:58:32.95,1:58:39.89,Default,,0000,0000,0000,,What do you think the\Ngradient is good at?
Dialogue: 0,1:58:39.89,1:58:49.31,Default,,0000,0000,0000,,
Dialogue: 0,1:58:49.31,1:58:50.98,Default,,0000,0000,0000,,Two reasons, right.
Dialogue: 0,1:58:50.98,1:58:54.34,Default,,0000,0000,0000,,Review number one.
Dialogue: 0,1:58:54.34,1:58:59.12,Default,,0000,0000,0000,,If you have an increasingly\Ndefined function,
Dialogue: 0,1:58:59.12,1:59:02.86,Default,,0000,0000,0000,,then the gradient of F was what?
Dialogue: 0,1:59:02.86,1:59:21.90,Default,,0000,0000,0000,,Equals direction of the\Nnormal to the surface S--
Dialogue: 0,1:59:21.90,1:59:26.40,Default,,0000,0000,0000,,let's say S is given\Nincreasingly at the point
Dialogue: 0,1:59:26.40,1:59:27.39,Default,,0000,0000,0000,,with [INAUDIBLE].
Dialogue: 0,1:59:27.39,1:59:31.88,Default,,0000,0000,0000,,
Dialogue: 0,1:59:31.88,1:59:33.38,Default,,0000,0000,0000,,But any other reason?
Dialogue: 0,1:59:33.38,2:00:00.33,Default,,0000,0000,0000,,
Dialogue: 0,2:00:00.33,2:00:01.82,Default,,0000,0000,0000,,Let's take that again.
Dialogue: 0,2:00:01.82,2:00:05.82,Default,,0000,0000,0000,,Z equals x squared\Nplus y squared.
Dialogue: 0,2:00:05.82,2:00:07.81,Default,,0000,0000,0000,,Let's compute a few\Npartial derivatives.
Dialogue: 0,2:00:07.81,2:00:09.31,Default,,0000,0000,0000,,Let's compute the gradient.
Dialogue: 0,2:00:09.31,2:00:21.12,Default,,0000,0000,0000,,The gradient is Fs of x, Fs\Nof y, where this is F of xy
Dialogue: 0,2:00:21.12,2:00:24.89,Default,,0000,0000,0000,,or Fs of xi plus Fs of yj.
Dialogue: 0,2:00:24.89,2:00:28.88,Default,,0000,0000,0000,,
Dialogue: 0,2:00:28.88,2:00:31.38,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,2:00:31.38,2:00:34.37,Default,,0000,0000,0000,,And we drew it.
Dialogue: 0,2:00:34.37,2:00:42.35,Default,,0000,0000,0000,,I drew this case, and we also\Ndrew another related example,
Dialogue: 0,2:00:42.35,2:00:45.85,Default,,0000,0000,0000,,where we took Z equals 1 minus\Nx squared minus y squared.
Dialogue: 0,2:00:45.85,2:00:46.84,Default,,0000,0000,0000,,And we went skiing.
Dialogue: 0,2:00:46.84,2:00:52.33,Default,,0000,0000,0000,,And we were so happy last week\Nto go skiing, because we still
Dialogue: 0,2:00:52.33,2:00:57.65,Default,,0000,0000,0000,,had snow in New\NMexico, and we-- and we
Dialogue: 0,2:00:57.65,2:01:02.55,Default,,0000,0000,0000,,said now we computed the\NZ to be minus 2x minus 2y.
Dialogue: 0,2:01:02.55,2:01:06.02,Default,,0000,0000,0000,,
Dialogue: 0,2:01:06.02,2:01:09.60,Default,,0000,0000,0000,,And we said, I'm\Nlooking at the slopes.
Dialogue: 0,2:01:09.60,2:01:12.94,Default,,0000,0000,0000,,This is the x duration\Nand the y duration.
Dialogue: 0,2:01:12.94,2:01:18.67,Default,,0000,0000,0000,,And I'm looking at the slopes of\Nthe lines of these two curves.
Dialogue: 0,2:01:18.67,2:01:23.63,Default,,0000,0000,0000,,So one that goes\Ndown, like that.
Dialogue: 0,2:01:23.63,2:01:25.01,Default,,0000,0000,0000,,So this was for what?
Dialogue: 0,2:01:25.01,2:01:27.54,Default,,0000,0000,0000,,For y equals 0.
Dialogue: 0,2:01:27.54,2:01:32.19,Default,,0000,0000,0000,,And this was for x equals 0.
Dialogue: 0,2:01:32.19,2:01:36.64,Default,,0000,0000,0000,,
Dialogue: 0,2:01:36.64,2:01:39.58,Default,,0000,0000,0000,,Curve, x equals\N0 curve in plane.
Dialogue: 0,2:01:39.58,2:01:40.36,Default,,0000,0000,0000,,Right?
Dialogue: 0,2:01:40.36,2:01:42.74,Default,,0000,0000,0000,,We just cross-section\Nour surface,
Dialogue: 0,2:01:42.74,2:01:43.95,Default,,0000,0000,0000,,and we have this [INAUDIBLE].
Dialogue: 0,2:01:43.95,2:01:51.59,Default,,0000,0000,0000,,And then we have the two\Ntangents, two slopes.
Dialogue: 0,2:01:51.59,2:01:54.06,Default,,0000,0000,0000,,And we computed them everywhere.
Dialogue: 0,2:01:54.06,2:02:00.49,Default,,0000,0000,0000,,
Dialogue: 0,2:02:00.49,2:02:01.97,Default,,0000,0000,0000,,At every point.
Dialogue: 0,2:02:01.97,2:02:06.91,Default,,0000,0000,0000,,
Dialogue: 0,2:02:06.91,2:02:10.84,Default,,0000,0000,0000,,But realize that to go\Nup or down these hills,
Dialogue: 0,2:02:10.84,2:02:15.10,Default,,0000,0000,0000,,I can go on a curve\Nlike that, or I
Dialogue: 0,2:02:15.10,2:02:17.95,Default,,0000,0000,0000,,can go-- remember the\Ntrain of Mickey Mouse going
Dialogue: 0,2:02:17.95,2:02:20.18,Default,,0000,0000,0000,,on the hilly point on the hill?
Dialogue: 0,2:02:20.18,2:02:22.17,Default,,0000,0000,0000,,We try to take different paths.
Dialogue: 0,2:02:22.17,2:02:24.17,Default,,0000,0000,0000,,We are going hiking.
Dialogue: 0,2:02:24.17,2:02:28.65,Default,,0000,0000,0000,,We are going hiking, and we'll\Ntake hiking through the pass.
Dialogue: 0,2:02:28.65,2:02:38.61,Default,,0000,0000,0000,,
Dialogue: 0,2:02:38.61,2:02:41.10,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:02:41.10,2:03:01.42,Default,,0000,0000,0000,,How do we get the maximum\Nrate of change of the function
Dialogue: 0,2:03:01.42,2:03:03.60,Default,,0000,0000,0000,,Z equals F of x1?
Dialogue: 0,2:03:03.60,2:03:05.87,Default,,0000,0000,0000,,So now I'm\Nanticipating something.
Dialogue: 0,2:03:05.87,2:03:10.68,Default,,0000,0000,0000,,I'd like to see your intuition,\Nyour inborn sense of I
Dialogue: 0,2:03:10.68,2:03:12.30,Default,,0000,0000,0000,,know what's going to happen.
Dialogue: 0,2:03:12.30,2:03:14.09,Default,,0000,0000,0000,,And you know what\Nthat from Mister--
Dialogue: 0,2:03:14.09,2:03:14.84,Default,,0000,0000,0000,,STUDENT: Heinrich.
Dialogue: 0,2:03:14.84,2:03:17.59,Default,,0000,0000,0000,,PROFESSOR TODA: [? Heinrich ?]\Nfrom high school.
Dialogue: 0,2:03:17.59,2:03:21.28,Default,,0000,0000,0000,,So I'm asking-- let me\Nrephrase the question
Dialogue: 0,2:03:21.28,2:03:23.13,Default,,0000,0000,0000,,like a non-mathematician.
Dialogue: 0,2:03:23.13,2:03:24.23,Default,,0000,0000,0000,,Let's go hiking.
Dialogue: 0,2:03:24.23,2:03:30.27,Default,,0000,0000,0000,,This is [INAUDIBLE] we\Ngo to the lighthouse.
Dialogue: 0,2:03:30.27,2:03:33.79,Default,,0000,0000,0000,,Which path shall I take\Non my mountain, my hill,
Dialogue: 0,2:03:33.79,2:03:37.57,Default,,0000,0000,0000,,my god knows what\Ngeography, in order
Dialogue: 0,2:03:37.57,2:03:40.44,Default,,0000,0000,0000,,to obtain the maximum\Nrate of change?
Dialogue: 0,2:03:40.44,2:03:43.52,Default,,0000,0000,0000,,That means the\Nhighest derivative.
Dialogue: 0,2:03:43.52,2:03:46.47,Default,,0000,0000,0000,,In what direction do I get\Nthe highest derivative?
Dialogue: 0,2:03:46.47,2:03:49.11,Default,,0000,0000,0000,,STUDENT: In what direction you\Nget the highest derivative--
Dialogue: 0,2:03:49.11,2:03:50.74,Default,,0000,0000,0000,,PROFESSOR TODA: So\Nin which direction--
Dialogue: 0,2:03:50.74,2:03:53.33,Default,,0000,0000,0000,,in which direction\Non this hill do
Dialogue: 0,2:03:53.33,2:03:55.10,Default,,0000,0000,0000,,I get the highest derivative?
Dialogue: 0,2:03:55.10,2:03:57.01,Default,,0000,0000,0000,,The highest rate of change.
Dialogue: 0,2:03:57.01,2:04:03.74,Default,,0000,0000,0000,,Rate of change means I want to\Nget the fastest possible way
Dialogue: 0,2:04:03.74,2:04:04.85,Default,,0000,0000,0000,,somewhere.
Dialogue: 0,2:04:04.85,2:04:08.23,Default,,0000,0000,0000,,STUDENT: The shortest slope?
Dialogue: 0,2:04:08.23,2:04:09.86,Default,,0000,0000,0000,,Along just the straight line up.
Dialogue: 0,2:04:09.86,2:04:10.83,Default,,0000,0000,0000,,PROFESSOR TODA: Along--
Dialogue: 0,2:04:10.83,2:04:12.29,Default,,0000,0000,0000,,STUDENT: You don't want\Nto take any [INAUDIBLE].
Dialogue: 0,2:04:12.29,2:04:13.27,Default,,0000,0000,0000,,PROFESSOR TODA: Right.
Dialogue: 0,2:04:13.27,2:04:13.91,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,2:04:13.91,2:04:15.14,Default,,0000,0000,0000,,It could be along any axis.
Dialogue: 0,2:04:15.14,2:04:17.71,Default,,0000,0000,0000,,PROFESSOR TODA: So could\Nyou see which direction
Dialogue: 0,2:04:17.71,2:04:19.07,Default,,0000,0000,0000,,those are-- very good.
Dialogue: 0,2:04:19.07,2:04:21.23,Default,,0000,0000,0000,,Actually you were getting\Nto the same direction.
Dialogue: 0,2:04:21.23,2:04:24.37,Default,,0000,0000,0000,,So [INAUDIBLE] says\NMagdalena, don't be silly.
Dialogue: 0,2:04:24.37,2:04:28.30,Default,,0000,0000,0000,,The actual maximum rate of\Nchange for the function Z
Dialogue: 0,2:04:28.30,2:04:31.07,Default,,0000,0000,0000,,is obviously, because\Nit is common sense,
Dialogue: 0,2:04:31.07,2:04:36.63,Default,,0000,0000,0000,,it's obviously happening if\Nyou take the so-called-- what
Dialogue: 0,2:04:36.63,2:04:37.88,Default,,0000,0000,0000,,are these guys?
Dialogue: 0,2:04:37.88,2:04:40.81,Default,,0000,0000,0000,,[INAUDIBLE], not meridians.
Dialogue: 0,2:04:40.81,2:04:42.28,Default,,0000,0000,0000,,STUDENT: Longtitudes?
Dialogue: 0,2:04:42.28,2:04:43.26,Default,,0000,0000,0000,,PROFESSOR TODA: OK.
Dialogue: 0,2:04:43.26,2:04:44.73,Default,,0000,0000,0000,,That is-- OK.
Dialogue: 0,2:04:44.73,2:04:47.81,Default,,0000,0000,0000,,Suppose that we don't hike,\Nbecause it's too tiring.
Dialogue: 0,2:04:47.81,2:04:51.17,Default,,0000,0000,0000,,We go down from the\Ntop of the hill.
Dialogue: 0,2:04:51.17,2:04:53.31,Default,,0000,0000,0000,,Ah, there's also very good idea.
Dialogue: 0,2:04:53.31,2:04:58.87,Default,,0000,0000,0000,,So when you let yourself\Ngo down on a sleigh,
Dialogue: 0,2:04:58.87,2:05:02.56,Default,,0000,0000,0000,,don't think bobsled or\Nanything-- just a sleigh,
Dialogue: 0,2:05:02.56,2:05:04.11,Default,,0000,0000,0000,,think of a child's sleigh.
Dialogue: 0,2:05:04.11,2:05:07.68,Default,,0000,0000,0000,,No, take a plastic bag\Nand put your butt in it
Dialogue: 0,2:05:07.68,2:05:10.59,Default,,0000,0000,0000,,and let yourself go.
Dialogue: 0,2:05:10.59,2:05:14.14,Default,,0000,0000,0000,,What is their\Ndirection actually?
Dialogue: 0,2:05:14.14,2:05:19.92,Default,,0000,0000,0000,,Your body will find the\Nfastest way to get down.
Dialogue: 0,2:05:19.92,2:05:23.06,Default,,0000,0000,0000,,The fastest way to get\Ndown will happen exactly
Dialogue: 0,2:05:23.06,2:05:27.71,Default,,0000,0000,0000,,in the same\Ndirections going down
Dialogue: 0,2:05:27.71,2:05:29.60,Default,,0000,0000,0000,,in the directions\Nof these meridians.
Dialogue: 0,2:05:29.60,2:05:34.10,Default,,0000,0000,0000,,
Dialogue: 0,2:05:34.10,2:05:35.51,Default,,0000,0000,0000,,OK?
Dialogue: 0,2:05:35.51,2:05:37.00,Default,,0000,0000,0000,,And now, [INAUDIBLE].
Dialogue: 0,2:05:37.00,2:05:46.42,Default,,0000,0000,0000,,
Dialogue: 0,2:05:46.42,2:05:58.58,Default,,0000,0000,0000,,The maximum rate of\Nchange will always
Dialogue: 0,2:05:58.58,2:06:07.26,Default,,0000,0000,0000,,happen in the direction\Nof the gradient.
Dialogue: 0,2:06:07.26,2:06:14.70,Default,,0000,0000,0000,,
Dialogue: 0,2:06:14.70,2:06:18.95,Default,,0000,0000,0000,,You can get a little\Nbit ahead of time
Dialogue: 0,2:06:18.95,2:06:21.86,Default,,0000,0000,0000,,by just-- I would like this\Nto [INAUDIBLE] in your heads
Dialogue: 0,2:06:21.86,2:06:23.86,Default,,0000,0000,0000,,until we get to that section.
Dialogue: 0,2:06:23.86,2:06:26.88,Default,,0000,0000,0000,,In one section we will be there.
Dialogue: 0,2:06:26.88,2:06:40.43,Default,,0000,0000,0000,,We also-- it's also reformulated\Nas the highest, the steepest,
Dialogue: 0,2:06:40.43,2:06:42.08,Default,,0000,0000,0000,,ascent or descent.
Dialogue: 0,2:06:42.08,2:06:44.57,Default,,0000,0000,0000,,The steepest.
Dialogue: 0,2:06:44.57,2:06:58.55,Default,,0000,0000,0000,,The steepest ascent or\Nthe steepest descent
Dialogue: 0,2:06:58.55,2:07:09.52,Default,,0000,0000,0000,,always happens in the\Ndirection of the gradient.
Dialogue: 0,2:07:09.52,2:07:14.55,Default,,0000,0000,0000,,
Dialogue: 0,2:07:14.55,2:07:17.11,Default,,0000,0000,0000,,Ascent is when you hike\Nto the top of the hill.
Dialogue: 0,2:07:17.11,2:07:21.45,Default,,0000,0000,0000,,Descent is when you let yourself\Ngo in the plastic [INAUDIBLE]
Dialogue: 0,2:07:21.45,2:07:25.27,Default,,0000,0000,0000,,bag in the snow.
Dialogue: 0,2:07:25.27,2:07:26.08,Default,,0000,0000,0000,,Right?
Dialogue: 0,2:07:26.08,2:07:30.03,Default,,0000,0000,0000,,Can you verify this happens\Njust on this example?
Dialogue: 0,2:07:30.03,2:07:32.54,Default,,0000,0000,0000,,It's true in general,\Nfor any smooth function.
Dialogue: 0,2:07:32.54,2:07:36.01,Default,,0000,0000,0000,,Our smooth function is\Na really nice function.
Dialogue: 0,2:07:36.01,2:07:39.82,Default,,0000,0000,0000,,So what is the gradient?
Dialogue: 0,2:07:39.82,2:07:42.72,Default,,0000,0000,0000,,Well again, it was 2x 2y, right?
Dialogue: 0,2:07:42.72,2:07:45.85,Default,,0000,0000,0000,,
Dialogue: 0,2:07:45.85,2:07:50.51,Default,,0000,0000,0000,,And that means at a certain\Npoint, x0 y0, whenever you are,
Dialogue: 0,2:07:50.51,2:07:52.30,Default,,0000,0000,0000,,guys you don't\Nnecessarily have to start
Dialogue: 0,2:07:52.30,2:07:54.75,Default,,0000,0000,0000,,from the top of the hill.
Dialogue: 0,2:07:54.75,2:07:58.87,Default,,0000,0000,0000,,You can be-- OK,\Nthis is your cabin.
Dialogue: 0,2:07:58.87,2:08:01.97,Default,,0000,0000,0000,,And here you are with\Nfriends, or with mom and dad,
Dialogue: 0,2:08:01.97,2:08:05.11,Default,,0000,0000,0000,,or whoever, on the hill.
Dialogue: 0,2:08:05.11,2:08:09.03,Default,,0000,0000,0000,,You get out, you take the\Nsleigh, and you go down.
Dialogue: 0,2:08:09.03,2:08:14.32,Default,,0000,0000,0000,,So no matter where\Nyou are, there you go.
Dialogue: 0,2:08:14.32,2:08:22.93,Default,,0000,0000,0000,,You have 2x0 times\Ni plus 2y0 times j.
Dialogue: 0,2:08:22.93,2:08:31.64,Default,,0000,0000,0000,,And the direction of the\Ngradient will be 2x0 2y0.
Dialogue: 0,2:08:31.64,2:08:34.52,Default,,0000,0000,0000,,Do you like this one?
Dialogue: 0,2:08:34.52,2:08:39.24,Default,,0000,0000,0000,,Well in this case,\Nif you were-- suppose
Dialogue: 0,2:08:39.24,2:08:42.30,Default,,0000,0000,0000,,you were at the\Npoint [INAUDIBLE].
Dialogue: 0,2:08:42.30,2:08:49.23,Default,,0000,0000,0000,,
Dialogue: 0,2:08:49.23,2:08:53.60,Default,,0000,0000,0000,,You are at the point\Nof coordinates--
Dialogue: 0,2:08:53.60,2:08:55.13,Default,,0000,0000,0000,,do you want to be here?
Dialogue: 0,2:08:55.13,2:08:57.00,Default,,0000,0000,0000,,You want to be here, right?
Dialogue: 0,2:08:57.00,2:08:58.77,Default,,0000,0000,0000,,So we've done that before.
Dialogue: 0,2:08:58.77,2:09:02.56,Default,,0000,0000,0000,,I'll take it as 1 over\N[? square root of ?]
Dialogue: 0,2:09:02.56,2:09:09.62,Default,,0000,0000,0000,,2-- I'm trying to be creative\Ntoday-- [INAUDIBLE] y equals 0,
Dialogue: 0,2:09:09.62,2:09:14.58,Default,,0000,0000,0000,,and Z equals-- what's left?
Dialogue: 0,2:09:14.58,2:09:16.46,Default,,0000,0000,0000,,1/2, right?
Dialogue: 0,2:09:16.46,2:09:17.75,Default,,0000,0000,0000,,Where am I?
Dialogue: 0,2:09:17.75,2:09:20.39,Default,,0000,0000,0000,,Guys, do you realize where I am?
Dialogue: 0,2:09:20.39,2:09:21.64,Default,,0000,0000,0000,,I'll [? take a ?] [INAUDIBLE].
Dialogue: 0,2:09:21.64,2:09:24.34,Default,,0000,0000,0000,,
Dialogue: 0,2:09:24.34,2:09:25.14,Default,,0000,0000,0000,,y0.
Dialogue: 0,2:09:25.14,2:09:28.66,Default,,0000,0000,0000,,So I need to be on this\Nmeridian on the red thingy.
Dialogue: 0,2:09:28.66,2:09:33.92,Default,,0000,0000,0000,,
Dialogue: 0,2:09:33.92,2:09:37.36,Default,,0000,0000,0000,,And somewhere here.
Dialogue: 0,2:09:37.36,2:09:40.25,Default,,0000,0000,0000,,
Dialogue: 0,2:09:40.25,2:09:43.38,Default,,0000,0000,0000,,What's the duration\Nof the gradient here?
Dialogue: 0,2:09:43.38,2:09:45.83,Default,,0000,0000,0000,,Delta z at this p.
Dialogue: 0,2:09:45.83,2:09:56.61,Default,,0000,0000,0000,,
Dialogue: 0,2:09:56.61,2:09:58.76,Default,,0000,0000,0000,,Then you say ah,\Nwell, I don't get it.
Dialogue: 0,2:09:58.76,2:10:04.04,Default,,0000,0000,0000,,I have-- the second guy will\Nbecome 0, because y0 is 0.
Dialogue: 0,2:10:04.04,2:10:06.98,Default,,0000,0000,0000,,The first guy will become\N1 over square root of 2.
Dialogue: 0,2:10:06.98,2:10:15.28,Default,,0000,0000,0000,,So I have 2 times 1 over square\Nroot of 2 times i plus 0j.
Dialogue: 0,2:10:15.28,2:10:29.42,Default,,0000,0000,0000,,It means in the direction of i--\Nin the direction of i-- from p,
Dialogue: 0,2:10:29.42,2:10:39.20,Default,,0000,0000,0000,,I have the fastest-- fastest,\NMagdalena, fastest-- descent
Dialogue: 0,2:10:39.20,2:10:39.70,Default,,0000,0000,0000,,possible.
Dialogue: 0,2:10:39.70,2:10:43.01,Default,,0000,0000,0000,,
Dialogue: 0,2:10:43.01,2:10:46.73,Default,,0000,0000,0000,,But we don't say in\Nthe direction of i
Dialogue: 0,2:10:46.73,2:10:49.64,Default,,0000,0000,0000,,in our everyday life, right?
Dialogue: 0,2:10:49.64,2:10:53.27,Default,,0000,0000,0000,,Let's say geographic points.
Dialogue: 0,2:10:53.27,2:10:58.61,Default,,0000,0000,0000,,We are-- I'm a bug,\Nand this is north.
Dialogue: 0,2:10:58.61,2:11:00.09,Default,,0000,0000,0000,,This is south.
Dialogue: 0,2:11:00.09,2:11:05.01,Default,,0000,0000,0000,,
Dialogue: 0,2:11:05.01,2:11:05.99,Default,,0000,0000,0000,,This is east.
Dialogue: 0,2:11:05.99,2:11:08.96,Default,,0000,0000,0000,,
Dialogue: 0,2:11:08.96,2:11:11.47,Default,,0000,0000,0000,,And this is west.
Dialogue: 0,2:11:11.47,2:11:18.14,Default,,0000,0000,0000,,So if I go east, going east\Nmeans going in the direction i.
Dialogue: 0,2:11:18.14,2:11:23.04,Default,,0000,0000,0000,,
Dialogue: 0,2:11:23.04,2:11:25.51,Default,,0000,0000,0000,,Now suppose-- I'm going\Nto finish with this one.
Dialogue: 0,2:11:25.51,2:11:28.87,Default,,0000,0000,0000,,Suppose that my house\Nis not on the prairie
Dialogue: 0,2:11:28.87,2:11:31.71,Default,,0000,0000,0000,,but my house is here.
Dialogue: 0,2:11:31.71,2:11:34.40,Default,,0000,0000,0000,,House, h.
Dialogue: 0,2:11:34.40,2:11:37.83,Default,,0000,0000,0000,,Find me a wood\Npoint to be there.
Dialogue: 0,2:11:37.83,2:11:39.74,Default,,0000,0000,0000,,STUDENT: Northeast.
Dialogue: 0,2:11:39.74,2:11:41.17,Default,,0000,0000,0000,,Or to get further down.
Dialogue: 0,2:11:41.17,2:11:45.05,Default,,0000,0000,0000,,PROFESSOR TODA: Anything, what\Nwould look like why I'm here?
Dialogue: 0,2:11:45.05,2:11:48.04,Default,,0000,0000,0000,,x0, y0, z0.
Dialogue: 0,2:11:48.04,2:11:50.04,Default,,0000,0000,0000,,Hm.
Dialogue: 0,2:11:50.04,2:11:57.80,Default,,0000,0000,0000,,1/2, 1/2, and I\Nneed the minimum.
Dialogue: 0,2:11:57.80,2:12:02.58,Default,,0000,0000,0000,,So I want to be on the\Nbisecting plane between the two.
Dialogue: 0,2:12:02.58,2:12:03.42,Default,,0000,0000,0000,,You understand?
Dialogue: 0,2:12:03.42,2:12:04.40,Default,,0000,0000,0000,,This is my quarter.
Dialogue: 0,2:12:04.40,2:12:06.87,Default,,0000,0000,0000,,And I want to be in\Nthis bisecting plane.
Dialogue: 0,2:12:06.87,2:12:10.42,Default,,0000,0000,0000,,So I'll take 1/2, 1/2, and\Nwhat results from here?
Dialogue: 0,2:12:10.42,2:12:11.54,Default,,0000,0000,0000,,I have to do math.
Dialogue: 0,2:12:11.54,2:12:16.11,Default,,0000,0000,0000,,1 minus 1/4 minus 1/4 is 1/2.
Dialogue: 0,2:12:16.11,2:12:17.90,Default,,0000,0000,0000,,Right?
Dialogue: 0,2:12:17.90,2:12:19.74,Default,,0000,0000,0000,,1/2, 1/2, 1/2.
Dialogue: 0,2:12:19.74,2:12:22.44,Default,,0000,0000,0000,,This is where my house\Nis [? and so on. ?]
Dialogue: 0,2:12:22.44,2:12:24.11,Default,,0000,0000,0000,,And this is full of smoke.
Dialogue: 0,2:12:24.11,2:12:29.94,Default,,0000,0000,0000,,And what is the\Nmaximum rate of change?
Dialogue: 0,2:12:29.94,2:12:34.50,Default,,0000,0000,0000,,What is the steepest\Ndescent is the trajectory
Dialogue: 0,2:12:34.50,2:12:37.92,Default,,0000,0000,0000,,that my body will take\Nwhen I let myself go down
Dialogue: 0,2:12:37.92,2:12:39.40,Default,,0000,0000,0000,,on the sleigh.
Dialogue: 0,2:12:39.40,2:12:40.88,Default,,0000,0000,0000,,How do I compute that?
Dialogue: 0,2:12:40.88,2:12:43.62,Default,,0000,0000,0000,,I will just do the same thing.
Dialogue: 0,2:12:43.62,2:12:49.90,Default,,0000,0000,0000,,Delta z at the point x0\Nequals 1/2, y0 equals 1/2,
Dialogue: 0,2:12:49.90,2:12:52.10,Default,,0000,0000,0000,,z0 equals 1/2.
Dialogue: 0,2:12:52.10,2:12:54.06,Default,,0000,0000,0000,,Well what do I get as direction?
Dialogue: 0,2:12:54.06,2:12:57.49,Default,,0000,0000,0000,,That will be the\Ndirection of the gradient.
Dialogue: 0,2:12:57.49,2:13:02.92,Default,,0000,0000,0000,,2 times 1/2-- you\Nguys with me still?
Dialogue: 0,2:13:02.92,2:13:09.24,Default,,0000,0000,0000,,i plus 2 times 1/2 with j.
Dialogue: 0,2:13:09.24,2:13:14.31,Default,,0000,0000,0000,,And there is no Mr.\Nz0 In the picture.
Dialogue: 0,2:13:14.31,2:13:14.81,Default,,0000,0000,0000,,Why?
Dialogue: 0,2:13:14.81,2:13:17.10,Default,,0000,0000,0000,,Because that will\Ngive me the direction
Dialogue: 0,2:13:17.10,2:13:22.00,Default,,0000,0000,0000,,like on-- in a geographic way.
Dialogue: 0,2:13:22.00,2:13:24.42,Default,,0000,0000,0000,,North, west, east, south.
Dialogue: 0,2:13:24.42,2:13:26.49,Default,,0000,0000,0000,,These are the\Ndirection in plane.
Dialogue: 0,2:13:26.49,2:13:28.12,Default,,0000,0000,0000,,I'm not talking\Ndirections on the hill,
Dialogue: 0,2:13:28.12,2:13:31.50,Default,,0000,0000,0000,,I'm talking\Ndirections on the map.
Dialogue: 0,2:13:31.50,2:13:33.53,Default,,0000,0000,0000,,These are directions on the map.
Dialogue: 0,2:13:33.53,2:13:35.93,Default,,0000,0000,0000,,So what is the direction\Ni plus j on the map?
Dialogue: 0,2:13:35.93,2:13:39.82,Default,,0000,0000,0000,,If you show this to a\Ngeography major and say,
Dialogue: 0,2:13:39.82,2:13:43.34,Default,,0000,0000,0000,,I'm going in the direction\Ni plus j on the map,
Dialogue: 0,2:13:43.34,2:13:45.70,Default,,0000,0000,0000,,he will say you are crazy.
Dialogue: 0,2:13:45.70,2:13:47.98,Default,,0000,0000,0000,,He doesn't understand the thing.
Dialogue: 0,2:13:47.98,2:13:50.28,Default,,0000,0000,0000,,But you know what you mean.
Dialogue: 0,2:13:50.28,2:13:54.10,Default,,0000,0000,0000,,East for you is the\Ndirection of i in the x-axis.
Dialogue: 0,2:13:54.10,2:13:56.21,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,2:13:56.21,2:13:58.43,Default,,0000,0000,0000,,And this is north.
Dialogue: 0,2:13:58.43,2:13:59.63,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,2:13:59.63,2:14:01.79,Default,,0000,0000,0000,,The y direction is north.
Dialogue: 0,2:14:01.79,2:14:06.41,Default,,0000,0000,0000,,So I'm going perfectly\Nnortheast at a 45-degree angle.
Dialogue: 0,2:14:06.41,2:14:08.09,Default,,0000,0000,0000,,If I tell the\Ngeography major I'm
Dialogue: 0,2:14:08.09,2:14:10.72,Default,,0000,0000,0000,,going northeast perfectly in\Nthe middle, he will say I know.
Dialogue: 0,2:14:10.72,2:14:13.56,Default,,0000,0000,0000,,But you will know that\Nfor you, that is i plus j.
Dialogue: 0,2:14:13.56,2:14:15.65,Default,,0000,0000,0000,,Because you are\Nthe mathematician.
Dialogue: 0,2:14:15.65,2:14:17.10,Default,,0000,0000,0000,,Right?
Dialogue: 0,2:14:17.10,2:14:18.98,Default,,0000,0000,0000,,So you go down.
Dialogue: 0,2:14:18.98,2:14:20.78,Default,,0000,0000,0000,,And this is where you are.
Dialogue: 0,2:14:20.78,2:14:22.50,Default,,0000,0000,0000,,And you're on the meridian.
Dialogue: 0,2:14:22.50,2:14:25.09,Default,,0000,0000,0000,,This is the direction i plus j.
Dialogue: 0,2:14:25.09,2:14:29.61,Default,,0000,0000,0000,,So if I want to project my\Ntrajectory-- I went down
Dialogue: 0,2:14:29.61,2:14:33.26,Default,,0000,0000,0000,,with the sleigh, all the way\Ndown-- project the trajectory,
Dialogue: 0,2:14:33.26,2:14:36.81,Default,,0000,0000,0000,,my trajectory is a\Nbody on the snow.
Dialogue: 0,2:14:36.81,2:14:39.32,Default,,0000,0000,0000,,Projecting it on the\Nground is this one.
Dialogue: 0,2:14:39.32,2:14:43.80,Default,,0000,0000,0000,,So it is exactly the\Ndirection i plus j.
Dialogue: 0,2:14:43.80,2:14:44.32,Default,,0000,0000,0000,,Right, guys?
Dialogue: 0,2:14:44.32,2:14:48.17,Default,,0000,0000,0000,,So exactly northeast\Nperfectly at 45-degree angles.
Dialogue: 0,2:14:48.17,2:14:51.15,Default,,0000,0000,0000,,Now one caveat.
Dialogue: 0,2:14:51.15,2:14:53.46,Default,,0000,0000,0000,,One caveat, because\Nwhen we get there,
Dialogue: 0,2:14:53.46,2:14:59.06,Default,,0000,0000,0000,,you should be ready\Nalready, in 11.6 and 11.7.
Dialogue: 0,2:14:59.06,2:15:02.93,Default,,0000,0000,0000,,When we will say direction,\Nwe are also crazy people.
Dialogue: 0,2:15:02.93,2:15:04.92,Default,,0000,0000,0000,,I told you, mathematicians\Nare not normal.
Dialogue: 0,2:15:04.92,2:15:07.11,Default,,0000,0000,0000,,You have to be a\Nlittle bit crazy
Dialogue: 0,2:15:07.11,2:15:11.46,Default,,0000,0000,0000,,to want to do all the stuff\Nin your head like that.
Dialogue: 0,2:15:11.46,2:15:16.42,Default,,0000,0000,0000,,i plus j for us is not a\Ndirection most of the time.
Dialogue: 0,2:15:16.42,2:15:20.02,Default,,0000,0000,0000,,When we say direction, we mean\Nwe normalize that direction.
Dialogue: 0,2:15:20.02,2:15:23.06,Default,,0000,0000,0000,,We take the unit\Nvector, which is unique,
Dialogue: 0,2:15:23.06,2:15:25.94,Default,,0000,0000,0000,,for responding to i plus j.
Dialogue: 0,2:15:25.94,2:15:28.89,Default,,0000,0000,0000,,So what is that\Nunique unit vector?
Dialogue: 0,2:15:28.89,2:15:32.51,Default,,0000,0000,0000,,You learned in Chapter 9\Neverything is connected.
Dialogue: 0,2:15:32.51,2:15:33.89,Default,,0000,0000,0000,,It's a big circle.
Dialogue: 0,2:15:33.89,2:15:35.34,Default,,0000,0000,0000,,i plus j, very good.
Dialogue: 0,2:15:35.34,2:15:40.19,Default,,0000,0000,0000,,So direction is a unit vector\Nfor most mathematicians,
Dialogue: 0,2:15:40.19,2:15:45.39,Default,,0000,0000,0000,,which means you will be i\Nplus j over square root of 2.
Dialogue: 0,2:15:45.39,2:15:51.97,Default,,0000,0000,0000,,So in Chapter 5, please\Nremember, unlike Chapter 9,
Dialogue: 0,2:15:51.97,2:15:55.61,Default,,0000,0000,0000,,direction is a unit vector.
Dialogue: 0,2:15:55.61,2:15:59.61,Default,,0000,0000,0000,,In Chapter 9, Chapter 10,\Nit said direction lmn,
Dialogue: 0,2:15:59.61,2:16:00.65,Default,,0000,0000,0000,,direction god knows what.
Dialogue: 0,2:16:00.65,2:16:05.90,Default,,0000,0000,0000,,But in Chapter 11, direction\Nis a vector in plane,
Dialogue: 0,2:16:05.90,2:16:07.86,Default,,0000,0000,0000,,like this one, i\Nplus [INAUDIBLE]
Dialogue: 0,2:16:07.86,2:16:12.02,Default,,0000,0000,0000,,has to be a unique\Nnormal-- a unique vector.
Dialogue: 0,2:16:12.02,2:16:12.51,Default,,0000,0000,0000,,OK?
Dialogue: 0,2:16:12.51,2:16:14.32,Default,,0000,0000,0000,,And we-- keep that in mind.
Dialogue: 0,2:16:14.32,2:16:16.05,Default,,0000,0000,0000,,Next time, when we\Nmeet on Thursday,
Dialogue: 0,2:16:16.05,2:16:19.96,Default,,0000,0000,0000,,you will understand why\Nwe need to normalize it.
Dialogue: 0,2:16:19.96,2:16:23.22,Default,,0000,0000,0000,,Now can we say goodbye to\Nthe snow and everything?
Dialogue: 0,2:16:23.22,2:16:25.81,Default,,0000,0000,0000,,It's not going to\Nshow up much anymore.
Dialogue: 0,2:16:25.81,2:16:27.56,Default,,0000,0000,0000,,Remember this example.
Dialogue: 0,2:16:27.56,2:16:30.66,Default,,0000,0000,0000,,But we will start with\Nflowers next time.
Dialogue: 0,2:16:30.66,2:16:31.26,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:16:31.26,2:16:32.76,Default,,0000,0000,0000,,Have a nice day.
Dialogue: 0,2:16:32.76,2:16:33.96,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,2:16:33.96,2:16:36.41,Default,,0000,0000,0000,,Let me stop the video.
Dialogue: 0,2:16:36.41,2:16:37.24,Default,,0000,0000,0000,,