[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.93,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.93,0:00:05.67,Default,,0000,0000,0000,,PROFESSOR: So let's\Nforget about this example
Dialogue: 0,0:00:05.67,0:00:10.91,Default,,0000,0000,0000,,and review what we learned\Nin 11.3, chapter 11.
Dialogue: 0,0:00:10.91,0:00:17.16,Default,,0000,0000,0000,,Chapter 11, again, was\Nfunctions of several variables.
Dialogue: 0,0:00:17.16,0:00:21.03,Default,,0000,0000,0000,,In our case, I'll say\Nfunctions of two variables.
Dialogue: 0,0:00:21.03,0:00:27.01,Default,,0000,0000,0000,,
Dialogue: 0,0:00:27.01,0:00:30.40,Default,,0000,0000,0000,,11.3 taught you, what?
Dialogue: 0,0:00:30.40,0:00:32.10,Default,,0000,0000,0000,,Taught you some\Nbeautiful things.
Dialogue: 0,0:00:32.10,0:00:34.98,Default,,0000,0000,0000,,Practically, if you\Nunderstand this picture,
Dialogue: 0,0:00:34.98,0:00:37.50,Default,,0000,0000,0000,,you will remember everything.
Dialogue: 0,0:00:37.50,0:00:43.63,Default,,0000,0000,0000,,This picture is going to\Ntry and [INAUDIBLE] a graph
Dialogue: 0,0:00:43.63,0:00:45.54,Default,,0000,0000,0000,,that's sitting\Nabove here somewhere
Dialogue: 0,0:00:45.54,0:00:50.87,Default,,0000,0000,0000,,in Euclidean free space,\Ndimensional space.
Dialogue: 0,0:00:50.87,0:00:53.18,Default,,0000,0000,0000,,You have the origin.
Dialogue: 0,0:00:53.18,0:00:57.15,Default,,0000,0000,0000,,And you say I want markers.
Dialogue: 0,0:00:57.15,0:00:59.12,Default,,0000,0000,0000,,No, you don't say\NI want markers.
Dialogue: 0,0:00:59.12,0:01:02.74,Default,,0000,0000,0000,,I say I want markers.
Dialogue: 0,0:01:02.74,0:01:09.38,Default,,0000,0000,0000,,We want to fix a point\Nx0, y0 on the surface,
Dialogue: 0,0:01:09.38,0:01:11.16,Default,,0000,0000,0000,,assuming the surface is smooth.
Dialogue: 0,0:01:11.16,0:01:14.03,Default,,0000,0000,0000,,
Dialogue: 0,0:01:14.03,0:01:17.26,Default,,0000,0000,0000,,That x0 of mine\Nshould be projected.
Dialogue: 0,0:01:17.26,0:01:20.25,Default,,0000,0000,0000,,I'm going to try to draw\Nbetter than I did last time.
Dialogue: 0,0:01:20.25,0:01:23.41,Default,,0000,0000,0000,,X0, y0 corresponds\Nto a certain altitude
Dialogue: 0,0:01:23.41,0:01:26.56,Default,,0000,0000,0000,,z0 that is projected like that.
Dialogue: 0,0:01:26.56,0:01:29.05,Default,,0000,0000,0000,,And this is my\N[INAUDIBLE] 0 here.
Dialogue: 0,0:01:29.05,0:01:31.31,Default,,0000,0000,0000,,But I don't care much\Nabout that right now.
Dialogue: 0,0:01:31.31,0:01:34.85,Default,,0000,0000,0000,,I care about the\Nfact that locally, I
Dialogue: 0,0:01:34.85,0:01:40.76,Default,,0000,0000,0000,,represent the function\Nas a graph-- z of f-- f
Dialogue: 0,0:01:40.76,0:01:43.65,Default,,0000,0000,0000,,of x and y defined\Nover a domain.
Dialogue: 0,0:01:43.65,0:01:47.63,Default,,0000,0000,0000,,I have a domain\Nthat is an open set.
Dialogue: 0,0:01:47.63,0:01:50.99,Default,,0000,0000,0000,,And you connect to-- that's\Nmore than you need to know.
Dialogue: 0,0:01:50.99,0:01:52.04,Default,,0000,0000,0000,,Could be anything.
Dialogue: 0,0:01:52.04,0:01:56.34,Default,,0000,0000,0000,,Could be a square, could be\Na-- this could be something,
Dialogue: 0,0:01:56.34,0:01:58.34,Default,,0000,0000,0000,,a nice patch of them like.
Dialogue: 0,0:01:58.34,0:02:01.04,Default,,0000,0000,0000,,
Dialogue: 0,0:02:01.04,0:02:05.35,Default,,0000,0000,0000,,So the projection of my\Npoint here is x0, y0.
Dialogue: 0,0:02:05.35,0:02:08.15,Default,,0000,0000,0000,,I'm going to draw these\Nparallels as well as I can.
Dialogue: 0,0:02:08.15,0:02:10.66,Default,,0000,0000,0000,,But I cannot draw very well.
Dialogue: 0,0:02:10.66,0:02:12.13,Default,,0000,0000,0000,,But I'm trying.
Dialogue: 0,0:02:12.13,0:02:18.22,Default,,0000,0000,0000,,X0 and y0-- and\Nremember from last time.
Dialogue: 0,0:02:18.22,0:02:19.89,Default,,0000,0000,0000,,What did we say?
Dialogue: 0,0:02:19.89,0:02:25.12,Default,,0000,0000,0000,,I'm going to draw a plane\Nof equation x equals x0.
Dialogue: 0,0:02:25.12,0:02:27.73,Default,,0000,0000,0000,,
Dialogue: 0,0:02:27.73,0:02:30.20,Default,,0000,0000,0000,,All right, I'll try.
Dialogue: 0,0:02:30.20,0:02:33.75,Default,,0000,0000,0000,,I'll try and do a good job--\Nx equals x0 is this plane.
Dialogue: 0,0:02:33.75,0:02:35.24,Default,,0000,0000,0000,,STUDENT: Don't you\Nhave the x amount
Dialogue: 0,0:02:35.24,0:02:36.71,Default,,0000,0000,0000,,and the y amounts backward?
Dialogue: 0,0:02:36.71,0:02:38.88,Default,,0000,0000,0000,,Or [INAUDIBLE]
Dialogue: 0,0:02:38.88,0:02:39.46,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,0:02:39.46,0:02:40.43,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:02:40.43,0:02:44.28,Default,,0000,0000,0000,,PROFESSOR: x is this 1\Ncoming towards you like that.
Dialogue: 0,0:02:44.28,0:02:47.28,Default,,0000,0000,0000,,And I also think about\Nthat always, Ryan.
Dialogue: 0,0:02:47.28,0:02:48.73,Default,,0000,0000,0000,,Do I have them backward?
Dialogue: 0,0:02:48.73,0:02:49.93,Default,,0000,0000,0000,,This time, I was lucky.
Dialogue: 0,0:02:49.93,0:02:51.37,Default,,0000,0000,0000,,I didn't have them backward.
Dialogue: 0,0:02:51.37,0:02:52.91,Default,,0000,0000,0000,,So y goes this way.
Dialogue: 0,0:02:52.91,0:02:57.92,Default,,0000,0000,0000,,For y0, let me pick another\Ncolor, a more beautiful color.
Dialogue: 0,0:02:57.92,0:03:01.18,Default,,0000,0000,0000,,
Dialogue: 0,0:03:01.18,0:03:07.18,Default,,0000,0000,0000,,For y0, my video is not\Ngoing to see the y0.
Dialogue: 0,0:03:07.18,0:03:10.76,Default,,0000,0000,0000,,But hopefully, it's going to\Nsee it, this beautiful line.
Dialogue: 0,0:03:10.76,0:03:12.47,Default,,0000,0000,0000,,Spring is coming.
Dialogue: 0,0:03:12.47,0:03:15.49,Default,,0000,0000,0000,,So this is going\Nto be the plane.
Dialogue: 0,0:03:15.49,0:03:19.86,Default,,0000,0000,0000,,
Dialogue: 0,0:03:19.86,0:03:24.16,Default,,0000,0000,0000,,Label it [INAUDIBLE]\Ny equals 0y.
Dialogue: 0,0:03:24.16,0:03:30.69,Default,,0000,0000,0000,,Now, the green plane cuts the\Nsurface into a plane curve,
Dialogue: 0,0:03:30.69,0:03:33.03,Default,,0000,0000,0000,,of course, because\Nteasing the plane
Dialogue: 0,0:03:33.03,0:03:35.73,Default,,0000,0000,0000,,that I drew with the line.
Dialogue: 0,0:03:35.73,0:03:41.08,Default,,0000,0000,0000,,And in the plane that I drew\Nwith red-- was it red or pink?
Dialogue: 0,0:03:41.08,0:03:42.43,Default,,0000,0000,0000,,It's Valentine's Day.
Dialogue: 0,0:03:42.43,0:03:43.04,Default,,0000,0000,0000,,It's pink.
Dialogue: 0,0:03:43.04,0:03:47.16,Default,,0000,0000,0000,,OK, so I have it like that.
Dialogue: 0,0:03:47.16,0:03:48.63,Default,,0000,0000,0000,,So what is the pink curve?
Dialogue: 0,0:03:48.63,0:03:53.68,Default,,0000,0000,0000,,The pink curve is the\Nintersection between z
Dialogue: 0,0:03:53.68,0:03:58.08,Default,,0000,0000,0000,,equals-- x equals x0\Nplane with my surface.
Dialogue: 0,0:03:58.08,0:04:00.11,Default,,0000,0000,0000,,My surface is black.
Dialogue: 0,0:04:00.11,0:04:03.80,Default,,0000,0000,0000,,I'm going to say s on surface.
Dialogue: 0,0:04:03.80,0:04:05.77,Default,,0000,0000,0000,,And then I have a pink curve.
Dialogue: 0,0:04:05.77,0:04:06.85,Default,,0000,0000,0000,,Let's call it c1.
Dialogue: 0,0:04:06.85,0:04:11.10,Default,,0000,0000,0000,,Because you cannot see\Npink on your notes.
Dialogue: 0,0:04:11.10,0:04:14.59,Default,,0000,0000,0000,,You only can imagine that\Nit's not the same thing.
Dialogue: 0,0:04:14.59,0:04:24.40,Default,,0000,0000,0000,,C2 is y equals y0 plane\Nintersected with s.
Dialogue: 0,0:04:24.40,0:04:26.38,Default,,0000,0000,0000,,And what have we\Nlearned last time?
Dialogue: 0,0:04:26.38,0:04:32.72,Default,,0000,0000,0000,,Last time, we learned that\Nwe introduce some derivatives
Dialogue: 0,0:04:32.72,0:04:37.24,Default,,0000,0000,0000,,at the point at 0, y is\N0, so that they represent
Dialogue: 0,0:04:37.24,0:04:40.40,Default,,0000,0000,0000,,those partial derivatives of\Nthe function z with respect
Dialogue: 0,0:04:40.40,0:04:42.01,Default,,0000,0000,0000,,to x and y.
Dialogue: 0,0:04:42.01,0:04:56.55,Default,,0000,0000,0000,,So we have the partial z sub x\Nat x0, y0 and the partial and z
Dialogue: 0,0:04:56.55,0:05:00.96,Default,,0000,0000,0000,,sub y at x0, y0.
Dialogue: 0,0:05:00.96,0:05:04.36,Default,,0000,0000,0000,,Do we have a more\Nelegant definition?
Dialogue: 0,0:05:04.36,0:05:06.86,Default,,0000,0000,0000,,That's elegant enough for\Nme, thank you very much.
Dialogue: 0,0:05:06.86,0:05:09.81,Default,,0000,0000,0000,,But if I wanted to give the\Noriginal definition, what
Dialogue: 0,0:05:09.81,0:05:11.18,Default,,0000,0000,0000,,was that?
Dialogue: 0,0:05:11.18,0:05:19.74,Default,,0000,0000,0000,,That is d of bx at x0, y0, which\Nis a limit of the difference
Dialogue: 0,0:05:19.74,0:05:20.24,Default,,0000,0000,0000,,quotient.
Dialogue: 0,0:05:20.24,0:05:23.00,Default,,0000,0000,0000,,And this time, we're going to--\Nnot going to do the x of y.
Dialogue: 0,0:05:23.00,0:05:25.02,Default,,0000,0000,0000,,I I'm different today.
Dialogue: 0,0:05:25.02,0:05:27.47,Default,,0000,0000,0000,,So I do h goes to 0.
Dialogue: 0,0:05:27.47,0:05:31.04,Default,,0000,0000,0000,,H is my smallest\Ndisplacement of [INAUDIBLE].
Dialogue: 0,0:05:31.04,0:05:34.33,Default,,0000,0000,0000,,Here, I have f of-- now,\Nwho is the variable?
Dialogue: 0,0:05:34.33,0:05:34.85,Default,,0000,0000,0000,,X.
Dialogue: 0,0:05:34.85,0:05:38.94,Default,,0000,0000,0000,,So who is going to say fixed?
Dialogue: 0,0:05:38.94,0:05:47.73,Default,,0000,0000,0000,,Y. So I'm going to say I'm\Ndisplacing mister x0 with an h.
Dialogue: 0,0:05:47.73,0:05:55.32,Default,,0000,0000,0000,,And y0 will be fixed minus\Nf of x0, y0, all over h.
Dialogue: 0,0:05:55.32,0:06:02.31,Default,,0000,0000,0000,,So again, instead of-- instead\Nof a delta x, I call the h.
Dialogue: 0,0:06:02.31,0:06:06.01,Default,,0000,0000,0000,,And the derivative\Nwith respect to y
Dialogue: 0,0:06:06.01,0:06:11.32,Default,,0000,0000,0000,,will assume that\Nx0 is a constant.
Dialogue: 0,0:06:11.32,0:06:13.75,Default,,0000,0000,0000,,I saw how well\N[INAUDIBLE] explained that
Dialogue: 0,0:06:13.75,0:06:15.88,Default,,0000,0000,0000,,and I'm ambitious.
Dialogue: 0,0:06:15.88,0:06:18.04,Default,,0000,0000,0000,,I want to do an even better\Njob than [INAUDIBLE].
Dialogue: 0,0:06:18.04,0:06:20.37,Default,,0000,0000,0000,,Hopefully, I might manage.
Dialogue: 0,0:06:20.37,0:06:32.82,Default,,0000,0000,0000,,D of ty equals [INAUDIBLE] h\Ngoing to 0 of that CF of-- now,
Dialogue: 0,0:06:32.82,0:06:34.91,Default,,0000,0000,0000,,who's telling me what we have?
Dialogue: 0,0:06:34.91,0:06:37.97,Default,,0000,0000,0000,,Of course, mister\Nx, y and y, yy.
Dialogue: 0,0:06:37.97,0:06:43.62,Default,,0000,0000,0000,,F of x0, y0 is their constant\Nwaiting for his turn.
Dialogue: 0,0:06:43.62,0:06:46.66,Default,,0000,0000,0000,,H is your parameter.
Dialogue: 0,0:06:46.66,0:06:50.94,Default,,0000,0000,0000,,And then you'll have, what?
Dialogue: 0,0:06:50.94,0:06:51.99,Default,,0000,0000,0000,,H0 is fixed, right?
Dialogue: 0,0:06:51.99,0:06:53.28,Default,,0000,0000,0000,,STUDENT: So h0 is--
Dialogue: 0,0:06:53.28,0:06:54.28,Default,,0000,0000,0000,,PROFESSOR: --fixed.
Dialogue: 0,0:06:54.28,0:06:56.26,Default,,0000,0000,0000,,Y is the variable.
Dialogue: 0,0:06:56.26,0:07:01.91,Default,,0000,0000,0000,,So I go into the direction\Nof y starting from y0.
Dialogue: 0,0:07:01.91,0:07:05.65,Default,,0000,0000,0000,,And I displace that with\Na small quantity, right?
Dialogue: 0,0:07:05.65,0:07:09.01,Default,,0000,0000,0000,,So these are my\Npartial velocity--
Dialogue: 0,0:07:09.01,0:07:11.89,Default,,0000,0000,0000,,my partial derivatives, I'm\Nsorry, not partial velocities.
Dialogue: 0,0:07:11.89,0:07:12.76,Default,,0000,0000,0000,,Forget what I said.
Dialogue: 0,0:07:12.76,0:07:17.66,Default,,0000,0000,0000,,I said something that\Nyou will learn later.
Dialogue: 0,0:07:17.66,0:07:18.91,Default,,0000,0000,0000,,What are those?
Dialogue: 0,0:07:18.91,0:07:29.70,Default,,0000,0000,0000,,Those are the slopes at x0, y0\Nof the tangents at the point
Dialogue: 0,0:07:29.70,0:07:31.33,Default,,0000,0000,0000,,here, OK?
Dialogue: 0,0:07:31.33,0:07:36.87,Default,,0000,0000,0000,,The tangents to the two curves,\Nthe pink one-- the pink one
Dialogue: 0,0:07:36.87,0:07:42.70,Default,,0000,0000,0000,,and the green one, all right?
Dialogue: 0,0:07:42.70,0:07:48.67,Default,,0000,0000,0000,,For the pink one, for the pink\Ncurve, what is the variable?
Dialogue: 0,0:07:48.67,0:07:51.11,Default,,0000,0000,0000,,The variable is the y, right?
Dialogue: 0,0:07:51.11,0:07:56.88,Default,,0000,0000,0000,,So this is c1 is a\Ncurve that depends on y.
Dialogue: 0,0:07:56.88,0:07:59.67,Default,,0000,0000,0000,,And c2 is a curve\Nthat depends on x.
Dialogue: 0,0:07:59.67,0:08:03.53,Default,,0000,0000,0000,,So this comes with x0 fixed.
Dialogue: 0,0:08:03.53,0:08:05.56,Default,,0000,0000,0000,,I better write it like that.
Dialogue: 0,0:08:05.56,0:08:08.39,Default,,0000,0000,0000,,F of x is 0.
Dialogue: 0,0:08:08.39,0:08:15.20,Default,,0000,0000,0000,,Y, instead of c2 of x, I'll\Nsay f of y-- f of x and yz.
Dialogue: 0,0:08:15.20,0:08:17.97,Default,,0000,0000,0000,,
Dialogue: 0,0:08:17.97,0:08:22.30,Default,,0000,0000,0000,,So, which slope is which?
Dialogue: 0,0:08:22.30,0:08:26.28,Default,,0000,0000,0000,,
Dialogue: 0,0:08:26.28,0:08:29.89,Default,,0000,0000,0000,,The d of dy at this point\Nis the slope to this one.
Dialogue: 0,0:08:29.89,0:08:31.34,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:08:31.34,0:08:33.61,Default,,0000,0000,0000,,The slope of that tangent.
Dialogue: 0,0:08:33.61,0:08:36.76,Default,,0000,0000,0000,,Considered in the\Nplane where it is.
Dialogue: 0,0:08:36.76,0:08:38.74,Default,,0000,0000,0000,,How about the other one?
Dialogue: 0,0:08:38.74,0:08:47.63,Default,,0000,0000,0000,,S of x will be the slope of this\Nline in the green plane, OK?
Dialogue: 0,0:08:47.63,0:08:52.18,Default,,0000,0000,0000,,That is considered as a\Nplane of axis of coordinates.
Dialogue: 0,0:08:52.18,0:08:57.26,Default,,0000,0000,0000,,Good, good-- so we\Nknow what they are.
Dialogue: 0,0:08:57.26,0:09:04.66,Default,,0000,0000,0000,,A quick example to review--\NI've given you some really ugly,
Dialogue: 0,0:09:04.66,0:09:06.45,Default,,0000,0000,0000,,nasty functions today.
Dialogue: 0,0:09:06.45,0:09:08.90,Default,,0000,0000,0000,,The last time, you\Ndid a good job.
Dialogue: 0,0:09:08.90,0:09:11.73,Default,,0000,0000,0000,,So today, I'm not\Nchallenging you anymore.
Dialogue: 0,0:09:11.73,0:09:16.16,Default,,0000,0000,0000,,I'm just going to give\Nyou one simple example.
Dialogue: 0,0:09:16.16,0:09:19.37,Default,,0000,0000,0000,,And I'm asked you, what\Ndoes this guy look like
Dialogue: 0,0:09:19.37,0:09:25.28,Default,,0000,0000,0000,,and what will the meanings\Nof z sub x and z sub y be?
Dialogue: 0,0:09:25.28,0:09:26.52,Default,,0000,0000,0000,,What will they be at?
Dialogue: 0,0:09:26.52,0:09:33.13,Default,,0000,0000,0000,,Let's say I think I know what I\Nwant to take at the point 0, 0.
Dialogue: 0,0:09:33.13,0:09:40.40,Default,,0000,0000,0000,,And maybe you're going to\Ntell me what else it will be.
Dialogue: 0,0:09:40.40,0:09:42.83,Default,,0000,0000,0000,,And eventually at\Nanother point like z
Dialogue: 0,0:09:42.83,0:09:55.44,Default,,0000,0000,0000,,sub a, so coordinates, 1\Nover square root of 2 and 1
Dialogue: 0,0:09:55.44,0:09:58.39,Default,,0000,0000,0000,,over square root of 2.
Dialogue: 0,0:09:58.39,0:10:02.82,Default,,0000,0000,0000,,And v sub y is same-- 1\Nover square root of 2,
Dialogue: 0,0:10:02.82,0:10:05.00,Default,,0000,0000,0000,,1 over square root of 2.
Dialogue: 0,0:10:05.00,0:10:10.15,Default,,0000,0000,0000,,Can one draw them and have\Na geometric explanation
Dialogue: 0,0:10:10.15,0:10:13.58,Default,,0000,0000,0000,,of what's going on?
Dialogue: 0,0:10:13.58,0:10:16.59,Default,,0000,0000,0000,,Well, I don't want you to\Nforget the definitions,
Dialogue: 0,0:10:16.59,0:10:19.99,Default,,0000,0000,0000,,but since you absorbed them\Nwith your mind hopefully
Dialogue: 0,0:10:19.99,0:10:23.76,Default,,0000,0000,0000,,and with your eyes, you're not\Ngoing to need them anymore.
Dialogue: 0,0:10:23.76,0:10:27.05,Default,,0000,0000,0000,,We should be able to draw\Nthis quadric that you love.
Dialogue: 0,0:10:27.05,0:10:29.25,Default,,0000,0000,0000,,I'm sure you love it.
Dialogue: 0,0:10:29.25,0:10:31.63,Default,,0000,0000,0000,,When it's-- what\Ndoes it look like?
Dialogue: 0,0:10:31.63,0:10:35.100,Default,,0000,0000,0000,,
Dialogue: 0,0:10:35.100,0:10:37.45,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:10:37.45,0:10:40.66,Default,,0000,0000,0000,,PROFESSOR: Wait a minute, you're\Nnot awake or I'm not awake.
Dialogue: 0,0:10:40.66,0:10:44.15,Default,,0000,0000,0000,,So if you do x squared\Nplus y squared,
Dialogue: 0,0:10:44.15,0:10:46.88,Default,,0000,0000,0000,,don't write it down please.
Dialogue: 0,0:10:46.88,0:10:49.18,Default,,0000,0000,0000,,It would be that.
Dialogue: 0,0:10:49.18,0:10:50.68,Default,,0000,0000,0000,,And what is this?
Dialogue: 0,0:10:50.68,0:10:51.96,Default,,0000,0000,0000,,STUDENT: That's a [INAUDIBLE].
Dialogue: 0,0:10:51.96,0:10:54.83,Default,,0000,0000,0000,,PROFESSOR: A circular\Nparaboloid-- you are correct.
Dialogue: 0,0:10:54.83,0:10:57.42,Default,,0000,0000,0000,,We've done that before.
Dialogue: 0,0:10:57.42,0:10:59.66,Default,,0000,0000,0000,,I'd say it looks\Nlike an egg shell,
Dialogue: 0,0:10:59.66,0:11:02.52,Default,,0000,0000,0000,,but it's actually--\Nthis is a parabola
Dialogue: 0,0:11:02.52,0:11:04.82,Default,,0000,0000,0000,,if it's going to infinity.
Dialogue: 0,0:11:04.82,0:11:06.27,Default,,0000,0000,0000,,And you said a bunch of circles.
Dialogue: 0,0:11:06.27,0:11:06.77,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:11:06.77,0:11:08.77,Default,,0000,0000,0000,,STUDENT: So is it an\Nupside down graph?
Dialogue: 0,0:11:08.77,0:11:10.83,Default,,0000,0000,0000,,PROFESSOR: It's an\Nupside down paraboloid.
Dialogue: 0,0:11:10.83,0:11:11.66,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:11:11.66,0:11:13.58,Default,,0000,0000,0000,,PROFESSOR: So, very\Ngood-- how do we do that?
Dialogue: 0,0:11:13.58,0:11:16.29,Default,,0000,0000,0000,,We make this guy\Nlook in the mirror.
Dialogue: 0,0:11:16.29,0:11:17.26,Default,,0000,0000,0000,,This is the lake.
Dialogue: 0,0:11:17.26,0:11:18.46,Default,,0000,0000,0000,,The lake is xy plane.
Dialogue: 0,0:11:18.46,0:11:21.47,Default,,0000,0000,0000,,So this guy is\Nlooking in the mirror.
Dialogue: 0,0:11:21.47,0:11:25.16,Default,,0000,0000,0000,,Take his image and\Nshift it just like he
Dialogue: 0,0:11:25.16,0:11:29.27,Default,,0000,0000,0000,,said-- shift it one unit up.
Dialogue: 0,0:11:29.27,0:11:30.75,Default,,0000,0000,0000,,This is one.
Dialogue: 0,0:11:30.75,0:11:34.02,Default,,0000,0000,0000,,You're going to have\Nanother paraboloid.
Dialogue: 0,0:11:34.02,0:11:39.37,Default,,0000,0000,0000,,So from this construction,\NI'm going to draw.
Dialogue: 0,0:11:39.37,0:11:42.04,Default,,0000,0000,0000,,And he's going to look\Nlike you took a cup
Dialogue: 0,0:11:42.04,0:11:44.16,Default,,0000,0000,0000,,and you put it upside down.
Dialogue: 0,0:11:44.16,0:11:45.77,Default,,0000,0000,0000,,But it's more like\Nan eggshell, right?
Dialogue: 0,0:11:45.77,0:11:48.95,Default,,0000,0000,0000,,It's not a cup because\Na cup is supposed
Dialogue: 0,0:11:48.95,0:11:50.77,Default,,0000,0000,0000,,to have a flat bottom, right?
Dialogue: 0,0:11:50.77,0:11:54.69,Default,,0000,0000,0000,,But this is like an eggshell.
Dialogue: 0,0:11:54.69,0:11:58.09,Default,,0000,0000,0000,,And I'll draw.
Dialogue: 0,0:11:58.09,0:12:01.80,Default,,0000,0000,0000,,And for this fellow, we\Nhave a beautiful picture
Dialogue: 0,0:12:01.80,0:12:09.94,Default,,0000,0000,0000,,that looks like this hopefully\NBut I'm going to try and draw.
Dialogue: 0,0:12:09.94,0:12:12.56,Default,,0000,0000,0000,,STUDENT: Are you looking\Nfrom a top to bottom?
Dialogue: 0,0:12:12.56,0:12:16.47,Default,,0000,0000,0000,,PROFESSOR: We can look it\Nwhatever you want to look.
Dialogue: 0,0:12:16.47,0:12:19.30,Default,,0000,0000,0000,,That's a very good thing.
Dialogue: 0,0:12:19.30,0:12:23.05,Default,,0000,0000,0000,,You're getting too close\Nto what I wanted to go.
Dialogue: 0,0:12:23.05,0:12:25.38,Default,,0000,0000,0000,,We'll discuss in one minute.
Dialogue: 0,0:12:25.38,0:12:29.80,Default,,0000,0000,0000,,So you can imagine this\Nis a hill full of snow.
Dialogue: 0,0:12:29.80,0:12:33.27,Default,,0000,0000,0000,,Although in two days,\Nwe have Valentine's Day
Dialogue: 0,0:12:33.27,0:12:35.31,Default,,0000,0000,0000,,and there is no snow.
Dialogue: 0,0:12:35.31,0:12:38.61,Default,,0000,0000,0000,,But assume that we\Ngo to New Mexico
Dialogue: 0,0:12:38.61,0:12:41.38,Default,,0000,0000,0000,,and we find a hill\Nthat more or less looks
Dialogue: 0,0:12:41.38,0:12:44.51,Default,,0000,0000,0000,,like a perfect hill like that.
Dialogue: 0,0:12:44.51,0:12:50.43,Default,,0000,0000,0000,,And we start thinking\Nof skiing down the hill.
Dialogue: 0,0:12:50.43,0:12:52.57,Default,,0000,0000,0000,,Where am I at 0, 0?
Dialogue: 0,0:12:52.57,0:12:55.09,Default,,0000,0000,0000,,I am on top of the hill.
Dialogue: 0,0:12:55.09,0:12:59.42,Default,,0000,0000,0000,,I'm on top of the\Nhill and I decide
Dialogue: 0,0:12:59.42,0:13:03.74,Default,,0000,0000,0000,,to analyze the slope\Nof the tangents
Dialogue: 0,0:13:03.74,0:13:08.10,Default,,0000,0000,0000,,to the surface in the\Ndirection of-- who is this?
Dialogue: 0,0:13:08.10,0:13:10.29,Default,,0000,0000,0000,,Like now, and you\Nmake me nervous.
Dialogue: 0,0:13:10.29,0:13:13.55,Default,,0000,0000,0000,,So in the direction of\Ny, I have one slope.
Dialogue: 0,0:13:13.55,0:13:17.62,Default,,0000,0000,0000,,In the direction of x, I have\Nanother slope in general.
Dialogue: 0,0:13:17.62,0:13:21.05,Default,,0000,0000,0000,,Only in this case, they\Nare the same slope.
Dialogue: 0,0:13:21.05,0:13:25.49,Default,,0000,0000,0000,,And what is that same slope if\NI'm here on top of the hill?
Dialogue: 0,0:13:25.49,0:13:30.18,Default,,0000,0000,0000,,This is me-- well, I don't\Nknow, one of you guys.
Dialogue: 0,0:13:30.18,0:13:34.00,Default,,0000,0000,0000,,
Dialogue: 0,0:13:34.00,0:13:37.28,Default,,0000,0000,0000,,That looks horrible.
Dialogue: 0,0:13:37.28,0:13:38.24,Default,,0000,0000,0000,,What's going to happen?
Dialogue: 0,0:13:38.24,0:13:39.82,Default,,0000,0000,0000,,We don't want to think about it.
Dialogue: 0,0:13:39.82,0:13:42.35,Default,,0000,0000,0000,,But it definitely is too steep.
Dialogue: 0,0:13:42.35,0:13:46.03,Default,,0000,0000,0000,,So this will be the slope\Nof the line in the direction
Dialogue: 0,0:13:46.03,0:13:46.95,Default,,0000,0000,0000,,with respect to y.
Dialogue: 0,0:13:46.95,0:13:50.08,Default,,0000,0000,0000,,So I'm going to think\Nf sub y and f sub
Dialogue: 0,0:13:50.08,0:13:55.93,Default,,0000,0000,0000,,x if I change my skis go\Nthis direction and I go down.
Dialogue: 0,0:13:55.93,0:14:01.73,Default,,0000,0000,0000,,So I could go down this\Nway and break my neck.
Dialogue: 0,0:14:01.73,0:14:07.53,Default,,0000,0000,0000,,Or I could go down this way\Nand break my neck as well.
Dialogue: 0,0:14:07.53,0:14:15.67,Default,,0000,0000,0000,,OK, it has to go like-- right?
Dialogue: 0,0:14:15.67,0:14:17.94,Default,,0000,0000,0000,,Can you tell me what\Nthese guys will be?
Dialogue: 0,0:14:17.94,0:14:20.27,Default,,0000,0000,0000,,I'm going to put them in pink\Nbecause they're beautiful.
Dialogue: 0,0:14:20.27,0:14:21.12,Default,,0000,0000,0000,,STUDENT: 0 [INAUDIBLE].
Dialogue: 0,0:14:21.12,0:14:22.83,Default,,0000,0000,0000,,PROFESSOR: Thank God,\Nthey are beautiful.
Dialogue: 0,0:14:22.83,0:14:23.90,Default,,0000,0000,0000,,Larry, what does it mean?
Dialogue: 0,0:14:23.90,0:14:28.07,Default,,0000,0000,0000,,That means that the two\Ntangents, the tangents
Dialogue: 0,0:14:28.07,0:14:33.71,Default,,0000,0000,0000,,to the curves, are horizontal.
Dialogue: 0,0:14:33.71,0:14:38.10,Default,,0000,0000,0000,,And if I were to draw the plane\Nbetween those two tangents--
Dialogue: 0,0:14:38.10,0:14:44.57,Default,,0000,0000,0000,,one tangent is in pink\Npen, our is in green.
Dialogue: 0,0:14:44.57,0:14:46.11,Default,,0000,0000,0000,,Today, I'm all about colors.
Dialogue: 0,0:14:46.11,0:14:47.09,Default,,0000,0000,0000,,I'm in a good mood.
Dialogue: 0,0:14:47.09,0:14:50.69,Default,,0000,0000,0000,,
Dialogue: 0,0:14:50.69,0:14:54.97,Default,,0000,0000,0000,,And that's going to be the\Nso-called tangent plane--
Dialogue: 0,0:14:54.97,0:15:06.98,Default,,0000,0000,0000,,tangent plane to the surface\Nat x0, y0, which is the origin.
Dialogue: 0,0:15:06.98,0:15:09.29,Default,,0000,0000,0000,,That was a nice point.
Dialogue: 0,0:15:09.29,0:15:10.68,Default,,0000,0000,0000,,That is a nice point.
Dialogue: 0,0:15:10.68,0:15:12.55,Default,,0000,0000,0000,,Not all the points\Nwill be [INAUDIBLE]
Dialogue: 0,0:15:12.55,0:15:14.68,Default,,0000,0000,0000,,and nice but beautiful.
Dialogue: 0,0:15:14.68,0:15:18.30,Default,,0000,0000,0000,,[INAUDIBLE] I take the\Nnice-- well, not so nice,
Dialogue: 0,0:15:18.30,0:15:19.15,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:15:19.15,0:15:21.53,Default,,0000,0000,0000,,You'll have to figure it out.
Dialogue: 0,0:15:21.53,0:15:26.53,Default,,0000,0000,0000,,How do I get-- well, first\Nof all, where is this point?
Dialogue: 0,0:15:26.53,0:15:31.72,Default,,0000,0000,0000,,If I take x to be 1 over square\N2 and y to be 1 over square 2
Dialogue: 0,0:15:31.72,0:15:33.87,Default,,0000,0000,0000,,and I plug them in,\Nwhat's z going to be?
Dialogue: 0,0:15:33.87,0:15:35.01,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:15:35.01,0:15:37.64,Default,,0000,0000,0000,,PROFESSOR: 0, and I\Ndid that on purpose.
Dialogue: 0,0:15:37.64,0:15:41.02,Default,,0000,0000,0000,,Because in that case, I'm\Ngoing to be on flat line again.
Dialogue: 0,0:15:41.02,0:15:42.25,Default,,0000,0000,0000,,This look like [INAUDIBLE].
Dialogue: 0,0:15:42.25,0:15:45.09,Default,,0000,0000,0000,,Except [INAUDIBLE]\Nis not z equal 0.
Dialogue: 0,0:15:45.09,0:15:47.49,Default,,0000,0000,0000,,What is [INAUDIBLE] like?
Dialogue: 0,0:15:47.49,0:15:48.39,Default,,0000,0000,0000,,z equals--
Dialogue: 0,0:15:48.39,0:15:49.32,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:15:49.32,0:15:49.95,Default,,0000,0000,0000,,PROFESSOR: Huh?
Dialogue: 0,0:15:49.95,0:15:51.48,Default,,0000,0000,0000,,STUDENT: I don't know.
Dialogue: 0,0:15:51.48,0:15:54.08,Default,,0000,0000,0000,,PROFESSOR: Do you want to\Ngo in meters or in feet?
Dialogue: 0,0:15:54.08,0:15:56.28,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:15:56.28,0:15:57.60,Default,,0000,0000,0000,,It's about a mile.
Dialogue: 0,0:15:57.60,0:15:59.08,Default,,0000,0000,0000,,PROFESSOR: Yes, I don't know.
Dialogue: 0,0:15:59.08,0:16:02.84,Default,,0000,0000,0000,,I thought it's about one\Nkilometer, 1,000 something
Dialogue: 0,0:16:02.84,0:16:03.82,Default,,0000,0000,0000,,meters.
Dialogue: 0,0:16:03.82,0:16:05.87,Default,,0000,0000,0000,,But somebody said it's more so.
Dialogue: 0,0:16:05.87,0:16:10.04,Default,,0000,0000,0000,,It's flat land, and I'd say\Nabout a mile above the sea
Dialogue: 0,0:16:10.04,0:16:11.09,Default,,0000,0000,0000,,level.
Dialogue: 0,0:16:11.09,0:16:20.41,Default,,0000,0000,0000,,All right, now, I am going to\Nbe in flat land right here,
Dialogue: 0,0:16:20.41,0:16:24.77,Default,,0000,0000,0000,,1 over a root 2, 1\Nover a root 2, and 0.
Dialogue: 0,0:16:24.77,0:16:26.06,Default,,0000,0000,0000,,What happened here?
Dialogue: 0,0:16:26.06,0:16:28.99,Default,,0000,0000,0000,,Here, I just already\Nbroke my neck, you know.
Dialogue: 0,0:16:28.99,0:16:32.29,Default,,0000,0000,0000,,Well, if I came\Nin this direction,
Dialogue: 0,0:16:32.29,0:16:36.62,Default,,0000,0000,0000,,I would need to draw a\Nprospective trajectory that
Dialogue: 0,0:16:36.62,0:16:38.88,Default,,0000,0000,0000,,was hopefully not mine.
Dialogue: 0,0:16:38.88,0:16:43.70,Default,,0000,0000,0000,,And the tangent would--\Nthe tangent, the slope
Dialogue: 0,0:16:43.70,0:16:45.65,Default,,0000,0000,0000,,of the tangent, would be funny.
Dialogue: 0,0:16:45.65,0:16:47.74,Default,,0000,0000,0000,,Let's see what you need to do.
Dialogue: 0,0:16:47.74,0:16:52.54,Default,,0000,0000,0000,,You need to say, OK, prime\Nwith respect to x, minus 2x.
Dialogue: 0,0:16:52.54,0:16:56.99,Default,,0000,0000,0000,,And then at the point x\Nequals 1 over [INAUDIBLE] 2 y
Dialogue: 0,0:16:56.99,0:16:59.95,Default,,0000,0000,0000,,equals 1 over 2,\Nyou just plug in.
Dialogue: 0,0:16:59.95,0:17:00.92,Default,,0000,0000,0000,,And what do you have?
Dialogue: 0,0:17:00.92,0:17:02.42,Default,,0000,0000,0000,,STUDENT: Square\Nroot of [INAUDIBLE].
Dialogue: 0,0:17:02.42,0:17:05.00,Default,,0000,0000,0000,,PROFESSOR: Negative\Nsquare root of 2-- my god
Dialogue: 0,0:17:05.00,0:17:08.94,Default,,0000,0000,0000,,that is really bad as a slope.
Dialogue: 0,0:17:08.94,0:17:11.00,Default,,0000,0000,0000,,It's a steep slope.
Dialogue: 0,0:17:11.00,0:17:14.01,Default,,0000,0000,0000,,And this one-- how\Nabout this one?
Dialogue: 0,0:17:14.01,0:17:16.10,Default,,0000,0000,0000,,Same idea, symmetric function.
Dialogue: 0,0:17:16.10,0:17:20.34,Default,,0000,0000,0000,,And it's going to be exactly\Nthe same-- very steep slope.
Dialogue: 0,0:17:20.34,0:17:22.82,Default,,0000,0000,0000,,Why are they negative numbers?
Dialogue: 0,0:17:22.82,0:17:27.47,Default,,0000,0000,0000,,Because the slope is\Ngoing down, right?
Dialogue: 0,0:17:27.47,0:17:31.99,Default,,0000,0000,0000,,That's the kind of slope I\Nhave in both directions--
Dialogue: 0,0:17:31.99,0:17:38.57,Default,,0000,0000,0000,,one and-- all right.
Dialogue: 0,0:17:38.57,0:17:48.31,Default,,0000,0000,0000,,If I were to draw\Nthis thing continuing,
Dialogue: 0,0:17:48.31,0:17:51.80,Default,,0000,0000,0000,,how would I represent\Nthose slopes?
Dialogue: 0,0:17:51.80,0:17:56.68,Default,,0000,0000,0000,,This circle-- this circle is\Njust making my life harder.
Dialogue: 0,0:17:56.68,0:17:59.60,Default,,0000,0000,0000,,But I would need to imagine\Nthose slopes as being
Dialogue: 0,0:17:59.60,0:18:02.33,Default,,0000,0000,0000,,like I'm here, all right?
Dialogue: 0,0:18:02.33,0:18:03.70,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:18:03.70,0:18:12.36,Default,,0000,0000,0000,,And I will need to draw\Nx0-- well, what is that?
Dialogue: 0,0:18:12.36,0:18:24.78,Default,,0000,0000,0000,,1 over root 2 and 1 over root\N2 And I would draw two planes.
Dialogue: 0,0:18:24.78,0:18:30.08,Default,,0000,0000,0000,,And I would have two curves.
Dialogue: 0,0:18:30.08,0:18:33.22,Default,,0000,0000,0000,,And when you slice\Nup, imagine this
Dialogue: 0,0:18:33.22,0:18:34.49,Default,,0000,0000,0000,,would be a piece of cheese.
Dialogue: 0,0:18:34.49,0:18:35.32,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:18:35.32,0:18:35.96,Default,,0000,0000,0000,,PROFESSOR: And you cut--
Dialogue: 0,0:18:35.96,0:18:36.79,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:18:36.79,0:18:41.78,Default,,0000,0000,0000,,
Dialogue: 0,0:18:41.78,0:18:42.49,Default,,0000,0000,0000,,PROFESSOR: Right?
Dialogue: 0,0:18:42.49,0:18:43.07,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:18:43.07,0:18:44.97,Default,,0000,0000,0000,,PROFESSOR: And you cut\Nin this other side.
Dialogue: 0,0:18:44.97,0:18:48.81,Default,,0000,0000,0000,,Well, this is the one\Nthat's facing you.
Dialogue: 0,0:18:48.81,0:18:49.65,Default,,0000,0000,0000,,You cut like that.
Dialogue: 0,0:18:49.65,0:18:51.36,Default,,0000,0000,0000,,And when you cut like\Nthis, it's facing--
Dialogue: 0,0:18:51.36,0:18:52.19,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:18:52.19,0:18:54.85,Default,,0000,0000,0000,,
Dialogue: 0,0:18:54.85,0:18:56.85,Default,,0000,0000,0000,,PROFESSOR: Hm?
Dialogue: 0,0:18:56.85,0:18:59.34,Default,,0000,0000,0000,,But anyway, let's not\Ndraw the other one.
Dialogue: 0,0:18:59.34,0:19:00.84,Default,,0000,0000,0000,,It's hard, right?
Dialogue: 0,0:19:00.84,0:19:04.08,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] angle\Nlike this-- just the piece
Dialogue: 0,0:19:04.08,0:19:05.62,Default,,0000,0000,0000,,of the corner of the cheese.
Dialogue: 0,0:19:05.62,0:19:06.33,Default,,0000,0000,0000,,PROFESSOR: Right.
Dialogue: 0,0:19:06.33,0:19:08.35,Default,,0000,0000,0000,,STUDENT: The corner\Nis facing you.
Dialogue: 0,0:19:08.35,0:19:13.25,Default,,0000,0000,0000,,PROFESSOR: So yeah, so it's--\Nthe corner is facing you.
Dialogue: 0,0:19:13.25,0:19:17.00,Default,,0000,0000,0000,,STUDENT: So basically,\Nyou [INAUDIBLE] this.
Dialogue: 0,0:19:17.00,0:19:18.60,Default,,0000,0000,0000,,PROFESSOR: But-- exactly, but--
Dialogue: 0,0:19:18.60,0:19:20.02,Default,,0000,0000,0000,,STUDENT: Like this.
Dialogue: 0,0:19:20.02,0:19:22.89,Default,,0000,0000,0000,,PROFESSOR: Yeah, well--\Nyeah, it's hard to draw.
Dialogue: 0,0:19:22.89,0:19:25.71,Default,,0000,0000,0000,,So practically,\Nthis is what you're
Dialogue: 0,0:19:25.71,0:19:29.45,Default,,0000,0000,0000,,looking at it is slope that's\Nnegative in both directions.
Dialogue: 0,0:19:29.45,0:19:34.76,Default,,0000,0000,0000,,So you're going to go\Nthis way or this way.
Dialogue: 0,0:19:34.76,0:19:39.14,Default,,0000,0000,0000,,And it's much steeper than\Nyou imagine [INAUDIBLE].
Dialogue: 0,0:19:39.14,0:19:43.00,Default,,0000,0000,0000,,OK, they are equal.
Dialogue: 0,0:19:43.00,0:19:44.52,Default,,0000,0000,0000,,I'm trying to draw them equal.
Dialogue: 0,0:19:44.52,0:19:48.13,Default,,0000,0000,0000,,I don't know how\Nequal they can be.
Dialogue: 0,0:19:48.13,0:19:55.78,Default,,0000,0000,0000,,One belongs to one plane\Njust like you said.
Dialogue: 0,0:19:55.78,0:19:58.32,Default,,0000,0000,0000,,This belongs to this plane.
Dialogue: 0,0:19:58.32,0:20:07.48,Default,,0000,0000,0000,,And the green one belongs to\Nthe plane that's facing you.
Dialogue: 0,0:20:07.48,0:20:08.78,Default,,0000,0000,0000,,So the slope goes this way.
Dialogue: 0,0:20:08.78,0:20:11.17,Default,,0000,0000,0000,,But the two slopes are equal.
Dialogue: 0,0:20:11.17,0:20:13.48,Default,,0000,0000,0000,,You have to have a little\Nbit of imagination.
Dialogue: 0,0:20:13.48,0:20:16.65,Default,,0000,0000,0000,,We would need some cheese\Nto make a mountain of cheese
Dialogue: 0,0:20:16.65,0:20:18.40,Default,,0000,0000,0000,,and cut them and slice them.
Dialogue: 0,0:20:18.40,0:20:21.64,Default,,0000,0000,0000,,We'll eat everything\Nafter, yeah.
Dialogue: 0,0:20:21.64,0:20:29.03,Default,,0000,0000,0000,,All right, let's move on to\Nsomething more challenging
Dialogue: 0,0:20:29.03,0:20:31.78,Default,,0000,0000,0000,,now that we got to\Nthe tangent plane.
Dialogue: 0,0:20:31.78,0:20:34.20,Default,,0000,0000,0000,,So if somebody would\Nsay, wait a minute,
Dialogue: 0,0:20:34.20,0:20:38.23,Default,,0000,0000,0000,,you said this is the tangent\Nplane to the surface.
Dialogue: 0,0:20:38.23,0:20:40.37,Default,,0000,0000,0000,,You just introduced\Na new notion.
Dialogue: 0,0:20:40.37,0:20:41.45,Default,,0000,0000,0000,,You were fooling us.
Dialogue: 0,0:20:41.45,0:20:43.82,Default,,0000,0000,0000,,I'm fooling you guys.
Dialogue: 0,0:20:43.82,0:20:48.59,Default,,0000,0000,0000,,It's not April 1, but this\Nkind of a not a neat thing.
Dialogue: 0,0:20:48.59,0:20:58.50,Default,,0000,0000,0000,,I just tried to introduce\Nyou into the section 11.4.
Dialogue: 0,0:20:58.50,0:21:02.71,Default,,0000,0000,0000,,So if you have a piece\Nof a curve that's smooth
Dialogue: 0,0:21:02.71,0:21:08.01,Default,,0000,0000,0000,,and you have a point\Nx0, y0, can you
Dialogue: 0,0:21:08.01,0:21:12.86,Default,,0000,0000,0000,,find out the equation\Nof the tangent plane?
Dialogue: 0,0:21:12.86,0:21:16.61,Default,,0000,0000,0000,,Pi, and this is s form surface.
Dialogue: 0,0:21:16.61,0:21:20.82,Default,,0000,0000,0000,,How can I find the equation\Nof the tangent plane?
Dialogue: 0,0:21:20.82,0:21:27.18,Default,,0000,0000,0000,,
Dialogue: 0,0:21:27.18,0:21:33.21,Default,,0000,0000,0000,,That x0, y0-- 12 is\Ngoing to be also z0.
Dialogue: 0,0:21:33.21,0:21:44.88,Default,,0000,0000,0000,,But what I mean that x0,\Ny0 is in on the floor
Dialogue: 0,0:21:44.88,0:21:47.21,Default,,0000,0000,0000,,as a projection.
Dialogue: 0,0:21:47.21,0:21:50.50,Default,,0000,0000,0000,,So I'm always\Nlooking at the graph.
Dialogue: 0,0:21:50.50,0:21:52.50,Default,,0000,0000,0000,,And that's why.
Dialogue: 0,0:21:52.50,0:21:54.30,Default,,0000,0000,0000,,The moment I stop\Nlooking at the graph,
Dialogue: 0,0:21:54.30,0:21:55.34,Default,,0000,0000,0000,,things will be different.
Dialogue: 0,0:21:55.34,0:21:59.77,Default,,0000,0000,0000,,But I'm looking at the graph\Nof independent variables x, y.
Dialogue: 0,0:21:59.77,0:22:02.50,Default,,0000,0000,0000,,And that's why those guys\Nare always on the floor.
Dialogue: 0,0:22:02.50,0:22:07.56,Default,,0000,0000,0000,,A and z would be a function\Nto keep in the variable.
Dialogue: 0,0:22:07.56,0:22:09.30,Default,,0000,0000,0000,,Now, does anybody know?
Dialogue: 0,0:22:09.30,0:22:12.61,Default,,0000,0000,0000,,Because I know you guys\Nare reading in advance
Dialogue: 0,0:22:12.61,0:22:16.26,Default,,0000,0000,0000,,and you have better\Nteachers than me.
Dialogue: 0,0:22:16.26,0:22:17.37,Default,,0000,0000,0000,,You have the internet.
Dialogue: 0,0:22:17.37,0:22:18.16,Default,,0000,0000,0000,,You have the links.
Dialogue: 0,0:22:18.16,0:22:18.96,Default,,0000,0000,0000,,You have YouTube.
Dialogue: 0,0:22:18.96,0:22:20.62,Default,,0000,0000,0000,,You have Khan Academy.
Dialogue: 0,0:22:20.62,0:22:24.53,Default,,0000,0000,0000,,I know from a bunch of you\Nthat you have already gone
Dialogue: 0,0:22:24.53,0:22:27.24,Default,,0000,0000,0000,,over half of the chapter 11.
Dialogue: 0,0:22:27.24,0:22:30.84,Default,,0000,0000,0000,,I just hope that now you\Ncan compare what you learned
Dialogue: 0,0:22:30.84,0:22:34.11,Default,,0000,0000,0000,,with what I'm teaching\Nyou, And I'm not
Dialogue: 0,0:22:34.11,0:22:36.84,Default,,0000,0000,0000,,expecting you to go in\Nadvance, but several of you
Dialogue: 0,0:22:36.84,0:22:38.59,Default,,0000,0000,0000,,already know this formula.
Dialogue: 0,0:22:38.59,0:22:43.62,Default,,0000,0000,0000,,We talked about it in\Noffice hours on yesterday.
Dialogue: 0,0:22:43.62,0:22:45.99,Default,,0000,0000,0000,,Because Tuesday, I\Ndidn't have office hours.
Dialogue: 0,0:22:45.99,0:22:49.79,Default,,0000,0000,0000,,I had a coordinator meeting.
Dialogue: 0,0:22:49.79,0:22:55.94,Default,,0000,0000,0000,,So what equation corresponds\Nto the tangent plate?
Dialogue: 0,0:22:55.94,0:22:56.78,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:22:56.78,0:23:00.42,Default,,0000,0000,0000,,
Dialogue: 0,0:23:00.42,0:23:02.12,Default,,0000,0000,0000,,PROFESSOR: Several\Nof you know it.
Dialogue: 0,0:23:02.12,0:23:03.97,Default,,0000,0000,0000,,You know what I hated?
Dialogue: 0,0:23:03.97,0:23:05.12,Default,,0000,0000,0000,,It's fine that you know it.
Dialogue: 0,0:23:05.12,0:23:08.25,Default,,0000,0000,0000,,I'm proud of you guys\Nand I'll write it.
Dialogue: 0,0:23:08.25,0:23:12.32,Default,,0000,0000,0000,,But when I was a freshman--\Nor what the heck was I?
Dialogue: 0,0:23:12.32,0:23:17.24,Default,,0000,0000,0000,,A sophomore I think-- no, I was\Na freshman when they fed me.
Dialogue: 0,0:23:17.24,0:23:19.36,Default,,0000,0000,0000,,They spoon-fed me this equation.
Dialogue: 0,0:23:19.36,0:23:22.38,Default,,0000,0000,0000,,And I didn't understand\Nanything at the time.
Dialogue: 0,0:23:22.38,0:23:25.91,Default,,0000,0000,0000,,I hated the fact that\Nthe Professor painted it
Dialogue: 0,0:23:25.91,0:23:30.86,Default,,0000,0000,0000,,on the board just like\Nthat out of the blue.
Dialogue: 0,0:23:30.86,0:23:33.70,Default,,0000,0000,0000,,I want to see a proof.
Dialogue: 0,0:23:33.70,0:23:39.63,Default,,0000,0000,0000,,And he was able to-- I think\Nhe could have done a good job.
Dialogue: 0,0:23:39.63,0:23:42.73,Default,,0000,0000,0000,,But he didn't.
Dialogue: 0,0:23:42.73,0:23:46.79,Default,,0000,0000,0000,,He showed us a bunch\Nof justifications
Dialogue: 0,0:23:46.79,0:23:53.40,Default,,0000,0000,0000,,like if you generally have\Na surface in implicit form,
Dialogue: 0,0:23:53.40,0:23:58.10,Default,,0000,0000,0000,,I told you that\Nthe gradient of F
Dialogue: 0,0:23:58.10,0:24:01.53,Default,,0000,0000,0000,,represents the normal\Nconnection, right?
Dialogue: 0,0:24:01.53,0:24:06.51,Default,,0000,0000,0000,,And he prepared us pretty\Ngood for what could
Dialogue: 0,0:24:06.51,0:24:08.85,Default,,0000,0000,0000,,have been the proof of that.
Dialogue: 0,0:24:08.85,0:24:10.20,Default,,0000,0000,0000,,He said, OK, guys.
Dialogue: 0,0:24:10.20,0:24:12.47,Default,,0000,0000,0000,,You know the duration\Nof the normal
Dialogue: 0,0:24:12.47,0:24:16.00,Default,,0000,0000,0000,,as even as the gradient over\Nthe next of the gradient,
Dialogue: 0,0:24:16.00,0:24:17.95,Default,,0000,0000,0000,,if you want unit normal.
Dialogue: 0,0:24:17.95,0:24:19.34,Default,,0000,0000,0000,,How did he do that?
Dialogue: 0,0:24:19.34,0:24:21.90,Default,,0000,0000,0000,,Well, he had a\Nbunch of examples.
Dialogue: 0,0:24:21.90,0:24:23.44,Default,,0000,0000,0000,,He had the sphere.
Dialogue: 0,0:24:23.44,0:24:25.80,Default,,0000,0000,0000,,He showed us that\Nfor the sphere,
Dialogue: 0,0:24:25.80,0:24:29.88,Default,,0000,0000,0000,,you have the normal,\Nwhich is the continuation
Dialogue: 0,0:24:29.88,0:24:31.40,Default,,0000,0000,0000,,of the position vector.
Dialogue: 0,0:24:31.40,0:24:34.26,Default,,0000,0000,0000,,Then he said, OK, you\Ncan have approximations
Dialogue: 0,0:24:34.26,0:24:39.37,Default,,0000,0000,0000,,of a surface that is smooth and\Nround with oscillating spheres
Dialogue: 0,0:24:39.37,0:24:45.29,Default,,0000,0000,0000,,just the way you have for a\Ncurve, a resonating circle,
Dialogue: 0,0:24:45.29,0:24:49.56,Default,,0000,0000,0000,,a resonating circle-- that's\Ncalled oscillating circle.
Dialogue: 0,0:24:49.56,0:24:53.07,Default,,0000,0000,0000,,Resonating circle-- in that\Ncase, what will the normal be?
Dialogue: 0,0:24:53.07,0:24:55.57,Default,,0000,0000,0000,,Well, the normal\Nwill have to depend
Dialogue: 0,0:24:55.57,0:24:57.12,Default,,0000,0000,0000,,on the radius of the circle.
Dialogue: 0,0:24:57.12,0:25:01.94,Default,,0000,0000,0000,,So you have a principal normal\Nor a normal if it's a plane
Dialogue: 0,0:25:01.94,0:25:02.79,Default,,0000,0000,0000,,curve.
Dialogue: 0,0:25:02.79,0:25:05.85,Default,,0000,0000,0000,,And it's easy to\Nunderstand that's the same
Dialogue: 0,0:25:05.85,0:25:07.21,Default,,0000,0000,0000,,as the gradient.
Dialogue: 0,0:25:07.21,0:25:12.68,Default,,0000,0000,0000,,So we have enough\Njustification for the direction
Dialogue: 0,0:25:12.68,0:25:16.23,Default,,0000,0000,0000,,of the gradient of such\Na function is always
Dialogue: 0,0:25:16.23,0:25:19.87,Default,,0000,0000,0000,,normal-- normal to the\Nsurface, normal to all
Dialogue: 0,0:25:19.87,0:25:22.73,Default,,0000,0000,0000,,the curves on the surface.
Dialogue: 0,0:25:22.73,0:25:26.66,Default,,0000,0000,0000,,If we want to find\Nthat without swallowing
Dialogue: 0,0:25:26.66,0:25:32.41,Default,,0000,0000,0000,,this like I had to when I\Nwas a student, it's not hard.
Dialogue: 0,0:25:32.41,0:25:35.32,Default,,0000,0000,0000,,And let me show\Nyou how we do it.
Dialogue: 0,0:25:35.32,0:25:37.33,Default,,0000,0000,0000,,We start from the graph, right?
Dialogue: 0,0:25:37.33,0:25:40.80,Default,,0000,0000,0000,,Z equals f of x and y.
Dialogue: 0,0:25:40.80,0:25:44.20,Default,,0000,0000,0000,,And we say, well, Magdalena,\Nbut this is a graph.
Dialogue: 0,0:25:44.20,0:25:47.55,Default,,0000,0000,0000,,It's not an implicit equation.
Dialogue: 0,0:25:47.55,0:25:50.32,Default,,0000,0000,0000,,And I'll say, yes it is.
Dialogue: 0,0:25:50.32,0:25:53.98,Default,,0000,0000,0000,,Let me show you how I make\Nit an implicit equation.
Dialogue: 0,0:25:53.98,0:25:56.30,Default,,0000,0000,0000,,I move z to the other side.
Dialogue: 0,0:25:56.30,0:26:00.63,Default,,0000,0000,0000,,I put 0 equals f of xy minus z.
Dialogue: 0,0:26:00.63,0:26:04.27,Default,,0000,0000,0000,,Now it is an implicit equation.
Dialogue: 0,0:26:04.27,0:26:05.72,Default,,0000,0000,0000,,So you say you cheated.
Dialogue: 0,0:26:05.72,0:26:07.53,Default,,0000,0000,0000,,Yes, I did.
Dialogue: 0,0:26:07.53,0:26:08.19,Default,,0000,0000,0000,,I have cheated.
Dialogue: 0,0:26:08.19,0:26:10.75,Default,,0000,0000,0000,,
Dialogue: 0,0:26:10.75,0:26:14.88,Default,,0000,0000,0000,,It's funny that whenever\Nsomebody gives you a graph,
Dialogue: 0,0:26:14.88,0:26:16.92,Default,,0000,0000,0000,,you can rewrite that\Ngraph immediately
Dialogue: 0,0:26:16.92,0:26:18.71,Default,,0000,0000,0000,,as an implicit equation.
Dialogue: 0,0:26:18.71,0:26:23.76,Default,,0000,0000,0000,,So that implicit equation\Nis of the form big F of xyz
Dialogue: 0,0:26:23.76,0:26:27.47,Default,,0000,0000,0000,,now equals a\Nconstant, which is 0.
Dialogue: 0,0:26:27.47,0:26:32.33,Default,,0000,0000,0000,,F of xy is your old\Nfriend and minus z.
Dialogue: 0,0:26:32.33,0:26:36.34,Default,,0000,0000,0000,,Now, can you tell me what is\Nthe normal to this surface?
Dialogue: 0,0:26:36.34,0:26:40.01,Default,,0000,0000,0000,,Yeah, give me a splash\Nin a minute like that.
Dialogue: 0,0:26:40.01,0:26:43.19,Default,,0000,0000,0000,,So what is the gradient of f?
Dialogue: 0,0:26:43.19,0:26:44.100,Default,,0000,0000,0000,,Gradient of f will\Nbe the normal.
Dialogue: 0,0:26:44.100,0:26:47.11,Default,,0000,0000,0000,,I don't care if\Nit's unit or not.
Dialogue: 0,0:26:47.11,0:26:49.46,Default,,0000,0000,0000,,To heck with the unit or normal.
Dialogue: 0,0:26:49.46,0:26:54.18,Default,,0000,0000,0000,,I'm going to say I wanted\Nprime with respect to x, y,
Dialogue: 0,0:26:54.18,0:26:58.03,Default,,0000,0000,0000,,and z respectively.
Dialogue: 0,0:26:58.03,0:26:59.76,Default,,0000,0000,0000,,And what is the gradient?
Dialogue: 0,0:26:59.76,0:27:01.79,Default,,0000,0000,0000,,Is the vector.
Dialogue: 0,0:27:01.79,0:27:06.92,Default,,0000,0000,0000,,Big F sub x comma big F\Nsub y comma big F sub z.
Dialogue: 0,0:27:06.92,0:27:08.90,Default,,0000,0000,0000,,We see that last time.
Dialogue: 0,0:27:08.90,0:27:13.62,Default,,0000,0000,0000,,So the gradient of a\Nfunction is the vector
Dialogue: 0,0:27:13.62,0:27:17.13,Default,,0000,0000,0000,,whose coordinates are\Nthe partial velocity--
Dialogue: 0,0:27:17.13,0:27:19.56,Default,,0000,0000,0000,,your friends form last time.
Dialogue: 0,0:27:19.56,0:27:22.09,Default,,0000,0000,0000,,Can we represent this again?
Dialogue: 0,0:27:22.09,0:27:22.65,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:27:22.65,0:27:24.30,Default,,0000,0000,0000,,You need to help me.
Dialogue: 0,0:27:24.30,0:27:29.34,Default,,0000,0000,0000,,Who is big F prime\Nwith respect to x?
Dialogue: 0,0:27:29.34,0:27:30.13,Default,,0000,0000,0000,,There is no x here.
Dialogue: 0,0:27:30.13,0:27:32.62,Default,,0000,0000,0000,,Thank God that's\Nlike a constant.
Dialogue: 0,0:27:32.62,0:27:36.17,Default,,0000,0000,0000,,I just have to take this\Nlittle one, f, and prime it
Dialogue: 0,0:27:36.17,0:27:36.96,Default,,0000,0000,0000,,with respect to x.
Dialogue: 0,0:27:36.96,0:27:41.05,Default,,0000,0000,0000,,And that's exactly what that's\Ngoing to be-- little f sub x.
Dialogue: 0,0:27:41.05,0:27:44.80,Default,,0000,0000,0000,,What is big F with respect to y?
Dialogue: 0,0:27:44.80,0:27:46.04,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:27:46.04,0:27:49.20,Default,,0000,0000,0000,,PROFESSOR: Little f sub\Ny prime with respect
Dialogue: 0,0:27:49.20,0:27:51.94,Default,,0000,0000,0000,,to y-- differentiated\Nwith respect to y.
Dialogue: 0,0:27:51.94,0:27:56.21,Default,,0000,0000,0000,,And finally, if I differentiated\Nwith respect to z,
Dialogue: 0,0:27:56.21,0:27:57.52,Default,,0000,0000,0000,,there is no z here, right?
Dialogue: 0,0:27:57.52,0:27:58.49,Default,,0000,0000,0000,,There is no z.
Dialogue: 0,0:27:58.49,0:28:00.00,Default,,0000,0000,0000,,So that's like a constant.
Dialogue: 0,0:28:00.00,0:28:04.54,Default,,0000,0000,0000,,Prime [INAUDIBLE] 0 and minus 1.
Dialogue: 0,0:28:04.54,0:28:05.77,Default,,0000,0000,0000,,So I know the gradient.
Dialogue: 0,0:28:05.77,0:28:06.88,Default,,0000,0000,0000,,I know the normal.
Dialogue: 0,0:28:06.88,0:28:09.14,Default,,0000,0000,0000,,This is the normal.
Dialogue: 0,0:28:09.14,0:28:14.18,Default,,0000,0000,0000,,Now, if somebody gives you\Nthe normal, there you are.
Dialogue: 0,0:28:14.18,0:28:20.23,Default,,0000,0000,0000,,You have the normal to the\Nsurface-- normal to surface.
Dialogue: 0,0:28:20.23,0:28:21.28,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:28:21.28,0:28:26.48,Default,,0000,0000,0000,,Equals normal to the tangent\Nplane to the surface.
Dialogue: 0,0:28:26.48,0:28:30.05,Default,,0000,0000,0000,,Normal or perpendicular\Nto the tangent plane-
Dialogue: 0,0:28:30.05,0:28:37.52,Default,,0000,0000,0000,,to the plane-- of the surface.
Dialogue: 0,0:28:37.52,0:28:41.47,Default,,0000,0000,0000,,At that point--\Npoint is the point p.
Dialogue: 0,0:28:41.47,0:28:44.50,Default,,0000,0000,0000,,
Dialogue: 0,0:28:44.50,0:28:52.53,Default,,0000,0000,0000,,All right, so if you were to\Nstudy a surface that's-- do you
Dialogue: 0,0:28:52.53,0:28:53.57,Default,,0000,0000,0000,,have a [INAUDIBLE]?
Dialogue: 0,0:28:53.57,0:28:54.70,Default,,0000,0000,0000,,STUDENT: Uh, no.
Dialogue: 0,0:28:54.70,0:28:55.52,Default,,0000,0000,0000,,Do you?
Dialogue: 0,0:28:55.52,0:28:56.10,Default,,0000,0000,0000,,PROFESSOR: OK.
Dialogue: 0,0:28:56.10,0:28:59.30,Default,,0000,0000,0000,,
Dialogue: 0,0:28:59.30,0:29:03.52,Default,,0000,0000,0000,,OK, I want to study\Nthe tangent plane
Dialogue: 0,0:29:03.52,0:29:05.27,Default,,0000,0000,0000,,at this point to the surface.
Dialogue: 0,0:29:05.27,0:29:06.67,Default,,0000,0000,0000,,Well, that's flat, Magdalena.
Dialogue: 0,0:29:06.67,0:29:08.46,Default,,0000,0000,0000,,You have no imagination.
Dialogue: 0,0:29:08.46,0:29:13.77,Default,,0000,0000,0000,,The tangent plane is this plane,\Nis the same as the surface.
Dialogue: 0,0:29:13.77,0:29:16.49,Default,,0000,0000,0000,,So, no fun-- no fun.
Dialogue: 0,0:29:16.49,0:29:19.88,Default,,0000,0000,0000,,How about I pick my\Nfavorite plane here
Dialogue: 0,0:29:19.88,0:29:24.69,Default,,0000,0000,0000,,and I take-- what is-- OK.
Dialogue: 0,0:29:24.69,0:29:26.76,Default,,0000,0000,0000,,I have-- this is\NChildren Internationals.
Dialogue: 0,0:29:26.76,0:29:30.30,Default,,0000,0000,0000,,I have a little girl\Nabroad that I'm sponsoring.
Dialogue: 0,0:29:30.30,0:29:34.50,Default,,0000,0000,0000,,So you have a point\Nhere and a plane
Dialogue: 0,0:29:34.50,0:29:38.78,Default,,0000,0000,0000,,that passes through that point.
Dialogue: 0,0:29:38.78,0:29:40.59,Default,,0000,0000,0000,,This is the tangent plane.
Dialogue: 0,0:29:40.59,0:29:43.61,Default,,0000,0000,0000,,And my finger is the normal.
Dialogue: 0,0:29:43.61,0:29:46.91,Default,,0000,0000,0000,,And the normal, we call\Nthat normal to the surface
Dialogue: 0,0:29:46.91,0:29:49.32,Default,,0000,0000,0000,,when it's normal to\Nthe tangent plane.
Dialogue: 0,0:29:49.32,0:29:52.97,Default,,0000,0000,0000,,At every point, this\Nis what the normal is.
Dialogue: 0,0:29:52.97,0:29:55.49,Default,,0000,0000,0000,,All right, can we write\Nthat based on chapter nine?
Dialogue: 0,0:29:55.49,0:29:59.18,Default,,0000,0000,0000,,Now I will see what you remember\Nfrom chapter nine if anything
Dialogue: 0,0:29:59.18,0:30:00.16,Default,,0000,0000,0000,,at all.
Dialogue: 0,0:30:00.16,0:30:03.58,Default,,0000,0000,0000,,
Dialogue: 0,0:30:03.58,0:30:09.43,Default,,0000,0000,0000,,All right, how do we\Nwrite the tangent plane
Dialogue: 0,0:30:09.43,0:30:11.83,Default,,0000,0000,0000,,if we know the normal?
Dialogue: 0,0:30:11.83,0:30:21.55,Default,,0000,0000,0000,,OK, review-- if the normal\Nvector is ai plus bj plus ck,
Dialogue: 0,0:30:21.55,0:30:28.06,Default,,0000,0000,0000,,that means the plane that\Nis perpendicular to it
Dialogue: 0,0:30:28.06,0:30:30.65,Default,,0000,0000,0000,,is of what form?
Dialogue: 0,0:30:30.65,0:30:37.55,Default,,0000,0000,0000,,Ax plus by plus cz\Nplus d equals 0, right?
Dialogue: 0,0:30:37.55,0:30:39.54,Default,,0000,0000,0000,,You've learned that\Nin chapter nine.
Dialogue: 0,0:30:39.54,0:30:43.37,Default,,0000,0000,0000,,Most of you learned that\Nlast semester in Calculus 2
Dialogue: 0,0:30:43.37,0:30:44.59,Default,,0000,0000,0000,,at the end.
Dialogue: 0,0:30:44.59,0:30:51.29,Default,,0000,0000,0000,,Now, if my normal is f sub\Nx, f sub y, and minus 1,
Dialogue: 0,0:30:51.29,0:30:52.81,Default,,0000,0000,0000,,those are ABC for God's sake.
Dialogue: 0,0:30:52.81,0:30:53.93,Default,,0000,0000,0000,,Well, good.
Dialogue: 0,0:30:53.93,0:30:59.40,Default,,0000,0000,0000,,Big A, big B, big C\Nat the given point.
Dialogue: 0,0:30:59.40,0:31:09.81,Default,,0000,0000,0000,,So I'm going to have f sub\Nx at the given point d times
Dialogue: 0,0:31:09.81,0:31:17.76,Default,,0000,0000,0000,,x plus f sub y at any\Ngiven point d times y.
Dialogue: 0,0:31:17.76,0:31:18.98,Default,,0000,0000,0000,,Who is c?
Dialogue: 0,0:31:18.98,0:31:20.42,Default,,0000,0000,0000,,C is minus 1.
Dialogue: 0,0:31:20.42,0:31:24.18,Default,,0000,0000,0000,,Minus 1 times z is--\Nsay you're being silly.
Dialogue: 0,0:31:24.18,0:31:26.55,Default,,0000,0000,0000,,Magdalena, why do\Nyou write minus 1?
Dialogue: 0,0:31:26.55,0:31:28.89,Default,,0000,0000,0000,,Just because I'm having fun.
Dialogue: 0,0:31:28.89,0:31:32.78,Default,,0000,0000,0000,,And plus, d equals 0.
Dialogue: 0,0:31:32.78,0:31:34.59,Default,,0000,0000,0000,,And you say, well,\Nwait, wait, wait.
Dialogue: 0,0:31:34.59,0:31:39.88,Default,,0000,0000,0000,,This starts looking like that\Nbut it's not the same thing.
Dialogue: 0,0:31:39.88,0:31:42.83,Default,,0000,0000,0000,,All right, what?
Dialogue: 0,0:31:42.83,0:31:44.24,Default,,0000,0000,0000,,How do you get to d?
Dialogue: 0,0:31:44.24,0:31:47.70,Default,,0000,0000,0000,,
Dialogue: 0,0:31:47.70,0:31:50.51,Default,,0000,0000,0000,,Now, actually, the\Nplane perpendicular
Dialogue: 0,0:31:50.51,0:31:55.18,Default,,0000,0000,0000,,to n that passes\Nthrough a given point
Dialogue: 0,0:31:55.18,0:31:58.92,Default,,0000,0000,0000,,can be written\Nmuch faster, right?
Dialogue: 0,0:31:58.92,0:32:06.02,Default,,0000,0000,0000,,So if a plane is perpendicular\Nto a certain line,
Dialogue: 0,0:32:06.02,0:32:09.54,Default,,0000,0000,0000,,how do we write if\Nwe know a point?
Dialogue: 0,0:32:09.54,0:32:15.19,Default,,0000,0000,0000,,If we know a point\Nin the normal ABC--
Dialogue: 0,0:32:15.19,0:32:18.92,Default,,0000,0000,0000,,I have to go backwards to\Nread it backwards-- then
Dialogue: 0,0:32:18.92,0:32:22.99,Default,,0000,0000,0000,,the plane is going\Nto be x minus x0
Dialogue: 0,0:32:22.99,0:32:29.53,Default,,0000,0000,0000,,plus b times y times y0 plus\Nc times z minus c0 equals 0.
Dialogue: 0,0:32:29.53,0:32:33.03,Default,,0000,0000,0000,,
Dialogue: 0,0:32:33.03,0:32:34.68,Default,,0000,0000,0000,,So who is the d?
Dialogue: 0,0:32:34.68,0:32:39.31,Default,,0000,0000,0000,,The d is all the constant\Nthat gets out of here.
Dialogue: 0,0:32:39.31,0:32:43.67,Default,,0000,0000,0000,,So the point x0, y0, z0\Nhas to verify the plane.
Dialogue: 0,0:32:43.67,0:32:47.08,Default,,0000,0000,0000,,And that's why when\Nyou plug in x0, y0, z0,
Dialogue: 0,0:32:47.08,0:32:50.42,Default,,0000,0000,0000,,you get 0 plus 0\Nplus 0 equals 0.
Dialogue: 0,0:32:50.42,0:32:53.49,Default,,0000,0000,0000,,That's what it means for a\Npoint to verify the plane.
Dialogue: 0,0:32:53.49,0:32:59.04,Default,,0000,0000,0000,,When you take the x0, y0, z0 and\Nyou plug it into the equation,
Dialogue: 0,0:32:59.04,0:33:02.77,Default,,0000,0000,0000,,you have to have an\Nidentity 0 equals 0.
Dialogue: 0,0:33:02.77,0:33:06.80,Default,,0000,0000,0000,,So this can be rewritten\Nzx plus by plus cz
Dialogue: 0,0:33:06.80,0:33:10.89,Default,,0000,0000,0000,,just like we did there plus a d.
Dialogue: 0,0:33:10.89,0:33:12.54,Default,,0000,0000,0000,,And who in the world is the d?
Dialogue: 0,0:33:12.54,0:33:18.73,Default,,0000,0000,0000,,The d will be exactly minus\Nax0 minus by0 minus cz0.
Dialogue: 0,0:33:18.73,0:33:22.94,Default,,0000,0000,0000,,If that makes you uncomfortable,\Nthis is in chapter nine.
Dialogue: 0,0:33:22.94,0:33:28.89,Default,,0000,0000,0000,,Look at the equation of a\Nplane and the normal to it.
Dialogue: 0,0:33:28.89,0:33:32.76,Default,,0000,0000,0000,,Now I know that I can do\Nbetter than that if I'm smart.
Dialogue: 0,0:33:32.76,0:33:35.85,Default,,0000,0000,0000,,So again, I collect the ABC.
Dialogue: 0,0:33:35.85,0:33:37.18,Default,,0000,0000,0000,,Now I know my ABC.
Dialogue: 0,0:33:37.18,0:33:40.32,Default,,0000,0000,0000,,
Dialogue: 0,0:33:40.32,0:33:42.22,Default,,0000,0000,0000,,I put them in here.
Dialogue: 0,0:33:42.22,0:33:46.17,Default,,0000,0000,0000,,So I have f sub x at\Nthe point in time.
Dialogue: 0,0:33:46.17,0:33:50.60,Default,,0000,0000,0000,,Oh, OK, x minus x0\Nplus, who is my b?
Dialogue: 0,0:33:50.60,0:33:57.12,Default,,0000,0000,0000,,F sub y computed at the\Npoint p times y minus y0.
Dialogue: 0,0:33:57.12,0:33:58.76,Default,,0000,0000,0000,,And, what?
Dialogue: 0,0:33:58.76,0:33:59.93,Default,,0000,0000,0000,,Minus, right?
Dialogue: 0,0:33:59.93,0:34:03.50,Default,,0000,0000,0000,,Minus-- minus 1.
Dialogue: 0,0:34:03.50,0:34:05.04,Default,,0000,0000,0000,,I'm not going to write minus 1.
Dialogue: 0,0:34:05.04,0:34:07.33,Default,,0000,0000,0000,,You're going to make fun of me.
Dialogue: 0,0:34:07.33,0:34:10.24,Default,,0000,0000,0000,,Minus z minus cz.
Dialogue: 0,0:34:10.24,0:34:12.06,Default,,0000,0000,0000,,And my proof is done.
Dialogue: 0,0:34:12.06,0:34:17.02,Default,,0000,0000,0000,,QED-- what does it mean, QED?
Dialogue: 0,0:34:17.02,0:34:19.62,Default,,0000,0000,0000,,In Latin.
Dialogue: 0,0:34:19.62,0:34:22.53,Default,,0000,0000,0000,,QED means I proved\Nwhat I wanted to prove.
Dialogue: 0,0:34:22.53,0:34:23.96,Default,,0000,0000,0000,,Do you know what it stands for?
Dialogue: 0,0:34:23.96,0:34:26.72,Default,,0000,0000,0000,,Did you take Latin, any of you?
Dialogue: 0,0:34:26.72,0:34:29.09,Default,,0000,0000,0000,,You took Latin?
Dialogue: 0,0:34:29.09,0:34:33.19,Default,,0000,0000,0000,,Quod erat demonstrandum.
Dialogue: 0,0:34:33.19,0:34:36.70,Default,,0000,0000,0000,,
Dialogue: 0,0:34:36.70,0:34:39.43,Default,,0000,0000,0000,,So this was to be proved.
Dialogue: 0,0:34:39.43,0:34:42.05,Default,,0000,0000,0000,,That's exactly what\Nit was to be proved.
Dialogue: 0,0:34:42.05,0:34:44.65,Default,,0000,0000,0000,,That, what, that\Nc minus z0, which
Dialogue: 0,0:34:44.65,0:34:50.18,Default,,0000,0000,0000,,was my fellow over here pretty\Nin pink, is going to be f sub x
Dialogue: 0,0:34:50.18,0:34:54.62,Default,,0000,0000,0000,,times x minus x0 plus yf\Nsub y times y minus y0.
Dialogue: 0,0:34:54.62,0:35:01.14,Default,,0000,0000,0000,,So now you know why the equation\Nof the tangent plane is that.
Dialogue: 0,0:35:01.14,0:35:04.52,Default,,0000,0000,0000,,I proved it more or less,\Nmaking some assumptions,
Dialogue: 0,0:35:04.52,0:35:06.83,Default,,0000,0000,0000,,some axioms as assumption.
Dialogue: 0,0:35:06.83,0:35:09.54,Default,,0000,0000,0000,,But you don't know\Nhow to use it.
Dialogue: 0,0:35:09.54,0:35:10.86,Default,,0000,0000,0000,,So let's use it.
Dialogue: 0,0:35:10.86,0:35:14.26,Default,,0000,0000,0000,,So for the same valley--\Nnot valley, hill--
Dialogue: 0,0:35:14.26,0:35:16.17,Default,,0000,0000,0000,,it was full of snow.
Dialogue: 0,0:35:16.17,0:35:19.38,Default,,0000,0000,0000,,Z equals 1 minus x\Nsquared-- what was you
Dialogue: 0,0:35:19.38,0:35:20.81,Default,,0000,0000,0000,,guys have forgotten?
Dialogue: 0,0:35:20.81,0:35:24.97,Default,,0000,0000,0000,,OK, 1 minus x squared\Nminus y squared.
Dialogue: 0,0:35:24.97,0:35:32.57,Default,,0000,0000,0000,,Find the tangent plane\Nat the following points.
Dialogue: 0,0:35:32.57,0:35:37.04,Default,,0000,0000,0000,,Ah, x0, y0 to be origin.
Dialogue: 0,0:35:37.04,0:35:39.47,Default,,0000,0000,0000,,And you say, did you\Nsay that that's trivial?
Dialogue: 0,0:35:39.47,0:35:40.48,Default,,0000,0000,0000,,Yes, it is trivial.
Dialogue: 0,0:35:40.48,0:35:42.95,Default,,0000,0000,0000,,But I'm going to do\Nit one more time.
Dialogue: 0,0:35:42.95,0:35:47.19,Default,,0000,0000,0000,,And what was my\N[INAUDIBLE] point before?
Dialogue: 0,0:35:47.19,0:35:48.63,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:35:48.63,0:35:53.43,Default,,0000,0000,0000,,PROFESSOR: 1 over\N2 and 1 over 2.
Dialogue: 0,0:35:53.43,0:35:57.79,Default,,0000,0000,0000,,OK, and what will be the\Ncorresponding point in 3D?
Dialogue: 0,0:35:57.79,0:36:01.50,Default,,0000,0000,0000,,1 over 2, 1 over 2, I plug in.
Dialogue: 0,0:36:01.50,0:36:03.54,Default,,0000,0000,0000,,Ah, yes.
Dialogue: 0,0:36:03.54,0:36:07.07,Default,,0000,0000,0000,,And with this, I hope\Nto finish the day so we
Dialogue: 0,0:36:07.07,0:36:10.24,Default,,0000,0000,0000,,can go to our other businesses.
Dialogue: 0,0:36:10.24,0:36:11.62,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:36:11.62,0:36:15.96,Default,,0000,0000,0000,,Now, I was not able-- I\Nhave to be honest with you.
Dialogue: 0,0:36:15.96,0:36:20.67,Default,,0000,0000,0000,,I was not able to memorize the\Nequation of a tangent plane
Dialogue: 0,0:36:20.67,0:36:27.01,Default,,0000,0000,0000,,when I was-- when I was young,\Nlike a freshman and sophomore.
Dialogue: 0,0:36:27.01,0:36:29.99,Default,,0000,0000,0000,,I wasn't ready to\Nunderstand that this
Dialogue: 0,0:36:29.99,0:36:33.22,Default,,0000,0000,0000,,is a linear approximation\Nof a curved something.
Dialogue: 0,0:36:33.22,0:36:35.53,Default,,0000,0000,0000,,This practically like\Nthe Taylor equation
Dialogue: 0,0:36:35.53,0:36:39.40,Default,,0000,0000,0000,,for functions of\Ntwo variables when
Dialogue: 0,0:36:39.40,0:36:42.71,Default,,0000,0000,0000,,you neglect the quadratic\Nthird term and so on.
Dialogue: 0,0:36:42.71,0:36:46.12,Default,,0000,0000,0000,,You just take the--\NI'll teach you
Dialogue: 0,0:36:46.12,0:36:52.15,Default,,0000,0000,0000,,next time when this is, a first\Norder linear approximation.
Dialogue: 0,0:36:52.15,0:36:54.01,Default,,0000,0000,0000,,All right, can we do\Nthis really quickly?
Dialogue: 0,0:36:54.01,0:36:55.80,Default,,0000,0000,0000,,It's going to be\Na piece of cake.
Dialogue: 0,0:36:55.80,0:36:56.63,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:36:56.63,0:36:58.36,Default,,0000,0000,0000,,Again, how do we do that?
Dialogue: 0,0:36:58.36,0:36:59.82,Default,,0000,0000,0000,,This is f of x and y.
Dialogue: 0,0:36:59.82,0:37:01.70,Default,,0000,0000,0000,,We computed that again.
Dialogue: 0,0:37:01.70,0:37:04.44,Default,,0000,0000,0000,,F of 0, 0 was this 0.
Dialogue: 0,0:37:04.44,0:37:08.49,Default,,0000,0000,0000,,Guys, if I say something\Nsilly, will you stop me?
Dialogue: 0,0:37:08.49,0:37:12.76,Default,,0000,0000,0000,,F of f sub x-- f\Nof y at 0, 0 is 0.
Dialogue: 0,0:37:12.76,0:37:14.33,Default,,0000,0000,0000,,So I have two slopes.
Dialogue: 0,0:37:14.33,0:37:15.52,Default,,0000,0000,0000,,Those are my hands.
Dialogue: 0,0:37:15.52,0:37:19.43,Default,,0000,0000,0000,,The slopes of my hands are 0.
Dialogue: 0,0:37:19.43,0:37:27.27,Default,,0000,0000,0000,,So the tangent plane will\Nbe z minus z0 equals 0.
Dialogue: 0,0:37:27.27,0:37:29.05,Default,,0000,0000,0000,,What is the 0?
Dialogue: 0,0:37:29.05,0:37:29.55,Default,,0000,0000,0000,,STUDENT: 1
Dialogue: 0,0:37:29.55,0:37:30.55,Default,,0000,0000,0000,,PROFESSOR: 1, excellent.
Dialogue: 0,0:37:30.55,0:37:31.75,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:37:31.75,0:37:32.80,Default,,0000,0000,0000,,PROFESSOR: Why is that 1?
Dialogue: 0,0:37:32.80,0:37:36.06,Default,,0000,0000,0000,,0 and 0 give me 1.
Dialogue: 0,0:37:36.06,0:37:39.88,Default,,0000,0000,0000,,So that was the picture\Nthat I had z equals 1
Dialogue: 0,0:37:39.88,0:37:42.84,Default,,0000,0000,0000,,as the tangent plane at\Nthe point corresponding
Dialogue: 0,0:37:42.84,0:37:44.82,Default,,0000,0000,0000,,to the origin.
Dialogue: 0,0:37:44.82,0:37:48.34,Default,,0000,0000,0000,,That look like the\Nnorth pole, 0, 0, 1.
Dialogue: 0,0:37:48.34,0:37:50.05,Default,,0000,0000,0000,,OK, no.
Dialogue: 0,0:37:50.05,0:37:52.55,Default,,0000,0000,0000,,It's the top of a hill.
Dialogue: 0,0:37:52.55,0:37:56.33,Default,,0000,0000,0000,,And finally, one last\Nthing [INAUDIBLE].
Dialogue: 0,0:37:56.33,0:37:58.20,Default,,0000,0000,0000,,Maybe you can do\Nthis by yourselves,
Dialogue: 0,0:37:58.20,0:38:01.06,Default,,0000,0000,0000,,but I will shut up if I can.
Dialogue: 0,0:38:01.06,0:38:03.39,Default,,0000,0000,0000,,I can't in general,\Nbut I'll shut up.
Dialogue: 0,0:38:03.39,0:38:09.08,Default,,0000,0000,0000,,Let's see-- f sub x at 1\Nover root 2, 1 over root 2.
Dialogue: 0,0:38:09.08,0:38:10.20,Default,,0000,0000,0000,,Why was that?
Dialogue: 0,0:38:10.20,0:38:11.90,Default,,0000,0000,0000,,What is f sub x?
Dialogue: 0,0:38:11.90,0:38:14.32,Default,,0000,0000,0000,,STUDENT: The square root of--\Nnegative square root of 2.
Dialogue: 0,0:38:14.32,0:38:17.37,Default,,0000,0000,0000,,PROFESSOR: Right,\Nwe've done that before.
Dialogue: 0,0:38:17.37,0:38:20.24,Default,,0000,0000,0000,,And you got exactly what\Nyou said-- [INAUDIBLE]
Dialogue: 0,0:38:20.24,0:38:24.76,Default,,0000,0000,0000,,2 f sub y at the same point.
Dialogue: 0,0:38:24.76,0:38:29.52,Default,,0000,0000,0000,,I am too lazy to write it\Ndown again-- minus root 2.
Dialogue: 0,0:38:29.52,0:38:32.94,Default,,0000,0000,0000,,And how do we actually\Nexpress the final answer
Dialogue: 0,0:38:32.94,0:38:37.26,Default,,0000,0000,0000,,so we can go home and\Nwhatever-- to the next class?
Dialogue: 0,0:38:37.26,0:38:39.26,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:38:39.26,0:38:39.76,Default,,0000,0000,0000,,No.
Dialogue: 0,0:38:39.76,0:38:40.97,Default,,0000,0000,0000,,What's the answer?
Dialogue: 0,0:38:40.97,0:38:44.14,Default,,0000,0000,0000,,Z minus-- now, attention.
Dialogue: 0,0:38:44.14,0:38:45.80,Default,,0000,0000,0000,,What is z0?
Dialogue: 0,0:38:45.80,0:38:46.73,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:38:46.73,0:38:48.82,Default,,0000,0000,0000,,PROFESSOR: 0, right.
Dialogue: 0,0:38:48.82,0:38:49.47,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:38:49.47,0:38:54.28,Default,,0000,0000,0000,,Because when I plug 1 over\Na 2, 1 over a 2, I got 0.
Dialogue: 0,0:38:54.28,0:38:56.78,Default,,0000,0000,0000,,0-- do I have to write it down?
Dialogue: 0,0:38:56.78,0:38:59.30,Default,,0000,0000,0000,,No, not unless I\Nwant to be silly.
Dialogue: 0,0:38:59.30,0:39:02.14,Default,,0000,0000,0000,,But if you do write\Ndown everything
Dialogue: 0,0:39:02.14,0:39:04.98,Default,,0000,0000,0000,,and you don't simplify\Nthe equation of the plane,
Dialogue: 0,0:39:04.98,0:39:08.65,Default,,0000,0000,0000,,we don't penalize you in\Nany way in the final, OK?
Dialogue: 0,0:39:08.65,0:39:14.14,Default,,0000,0000,0000,,So if you show your work like\Nthat, you're going to be fine.
Dialogue: 0,0:39:14.14,0:39:16.96,Default,,0000,0000,0000,,What is that 1 over 2?
Dialogue: 0,0:39:16.96,0:39:25.16,Default,,0000,0000,0000,,Plus minus root 2 times\Ny minus 1 over root 2.
Dialogue: 0,0:39:25.16,0:39:27.78,Default,,0000,0000,0000,,Is it elegant?
Dialogue: 0,0:39:27.78,0:39:30.82,Default,,0000,0000,0000,,No, it's not elegant at all.
Dialogue: 0,0:39:30.82,0:39:35.96,Default,,0000,0000,0000,,So as the last row for\Ntoday, one final line.
Dialogue: 0,0:39:35.96,0:39:40.06,Default,,0000,0000,0000,,Can we make it\Nlook more elegant?
Dialogue: 0,0:39:40.06,0:39:43.54,Default,,0000,0000,0000,,Do we care to make\Nit more elegant?
Dialogue: 0,0:39:43.54,0:39:47.61,Default,,0000,0000,0000,,Definitely some of you care.
Dialogue: 0,0:39:47.61,0:39:52.18,Default,,0000,0000,0000,,Z will be minus root 2x.
Dialogue: 0,0:39:52.18,0:39:56.66,Default,,0000,0000,0000,,I want to be consistent and\Nkeep the same style in y.
Dialogue: 0,0:39:56.66,0:39:58.93,Default,,0000,0000,0000,,And yet the constant\Ngoes wherever
Dialogue: 0,0:39:58.93,0:40:00.36,Default,,0000,0000,0000,,it wants to go at the end.
Dialogue: 0,0:40:00.36,0:40:01.99,Default,,0000,0000,0000,,What's that constant?
Dialogue: 0,0:40:01.99,0:40:03.38,Default,,0000,0000,0000,,STUDENT: 2 [INAUDIBLE].
Dialogue: 0,0:40:03.38,0:40:05.11,Default,,0000,0000,0000,,PROFESSOR: So you\Nsee what you have.
Dialogue: 0,0:40:05.11,0:40:06.15,Default,,0000,0000,0000,,You have this times that.
Dialogue: 0,0:40:06.15,0:40:07.91,Default,,0000,0000,0000,,It's a 1, this then that is a 1.
Dialogue: 0,0:40:07.91,0:40:10.10,Default,,0000,0000,0000,,1 plus 1 is 2.
Dialogue: 0,0:40:10.10,0:40:12.49,Default,,0000,0000,0000,,All right, are you\Nhappy with this?
Dialogue: 0,0:40:12.49,0:40:13.91,Default,,0000,0000,0000,,I'm not.
Dialogue: 0,0:40:13.91,0:40:15.93,Default,,0000,0000,0000,,I'm happy.
Dialogue: 0,0:40:15.93,0:40:18.26,Default,,0000,0000,0000,,You-- if this were\Na multiple choice,
Dialogue: 0,0:40:18.26,0:40:21.50,Default,,0000,0000,0000,,you would be able to\Nrecognize it right away.
Dialogue: 0,0:40:21.50,0:40:25.44,Default,,0000,0000,0000,,What's the standardized general\Nequation of a plane, though?
Dialogue: 0,0:40:25.44,0:40:29.14,Default,,0000,0000,0000,,Something x plus something y\Nplus something z plus something
Dialogue: 0,0:40:29.14,0:40:31.05,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,0:40:31.05,0:40:34.26,Default,,0000,0000,0000,,So if you wanted to\Nmake me very happy,
Dialogue: 0,0:40:34.26,0:40:38.56,Default,,0000,0000,0000,,you would still move everybody\Nto the left hand side.
Dialogue: 0,0:40:38.56,0:40:41.08,Default,,0000,0000,0000,,
Dialogue: 0,0:40:41.08,0:40:43.22,Default,,0000,0000,0000,,Do you want equal to or minus 3?
Dialogue: 0,0:40:43.22,0:40:45.10,Default,,0000,0000,0000,,Yes, it does.
Dialogue: 0,0:40:45.10,0:40:46.04,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:40:46.04,0:40:46.98,Default,,0000,0000,0000,,PROFESSOR: Huh?
Dialogue: 0,0:40:46.98,0:40:49.79,Default,,0000,0000,0000,,Negative 2-- is that OK?
Dialogue: 0,0:40:49.79,0:40:50.68,Default,,0000,0000,0000,,Is that fine?
Dialogue: 0,0:40:50.68,0:40:51.61,Default,,0000,0000,0000,,Are you guys done?
Dialogue: 0,0:40:51.61,0:40:52.40,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:40:52.40,0:40:53.56,Default,,0000,0000,0000,,Mm-mm.
Dialogue: 0,0:40:53.56,0:40:55.20,Default,,0000,0000,0000,,It's hard?
Dialogue: 0,0:40:55.20,0:40:56.28,Default,,0000,0000,0000,,No.
Dialogue: 0,0:40:56.28,0:40:58.77,Default,,0000,0000,0000,,Who said it's hard?
Dialogue: 0,0:40:58.77,0:41:05.18,Default,,0000,0000,0000,,So-- so I would work more\Ntangent planes next time.
Dialogue: 0,0:41:05.18,0:41:08.32,Default,,0000,0000,0000,,But I think it's something\Nthat we can practice on.
Dialogue: 0,0:41:08.32,0:41:12.60,Default,,0000,0000,0000,,And do expect one exercise\Nlike that from one
Dialogue: 0,0:41:12.60,0:41:16.41,Default,,0000,0000,0000,,of those, God knows,\N15, 16 on the final.
Dialogue: 0,0:41:16.41,0:41:18.01,Default,,0000,0000,0000,,I'm not sure about the midterm.
Dialogue: 0,0:41:18.01,0:41:19.86,Default,,0000,0000,0000,,I like this type of problem.
Dialogue: 0,0:41:19.86,0:41:23.23,Default,,0000,0000,0000,,So you might even see\Nsomething with tangent planes
Dialogue: 0,0:41:23.23,0:41:26.96,Default,,0000,0000,0000,,on the midterm-- normal to\Na surface tangent plane.
Dialogue: 0,0:41:26.96,0:41:28.07,Default,,0000,0000,0000,,It's a good topic.
Dialogue: 0,0:41:28.07,0:41:29.47,Default,,0000,0000,0000,,It's really pretty.
Dialogue: 0,0:41:29.47,0:41:33.20,Default,,0000,0000,0000,,For people who like to draw,\Nit's also nice to draw them.
Dialogue: 0,0:41:33.20,0:41:34.59,Default,,0000,0000,0000,,But do you have to?
Dialogue: 0,0:41:34.59,0:41:35.73,Default,,0000,0000,0000,,No.
Dialogue: 0,0:41:35.73,0:41:39.36,Default,,0000,0000,0000,,Some of you don't like to.
Dialogue: 0,0:41:39.36,0:41:43.16,Default,,0000,0000,0000,,OK, so now I say thank\Nyou for the attendance
Dialogue: 0,0:41:43.16,0:41:48.13,Default,,0000,0000,0000,,and I'll see you next time\Non Thursday-- on Tuesday.
Dialogue: 0,0:41:48.13,0:41:50.58,Default,,0000,0000,0000,,Happy Valentine's Day.
Dialogue: 0,0:41:50.58,0:41:52.19,Default,,0000,0000,0000,,