[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.97,Default,,0000,0000,0000,,PROFESSOR: Questions so far?
Dialogue: 0,0:00:01.97,0:00:03.03,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:00:03.03,0:00:03.94,Default,,0000,0000,0000,,PROFESSOR: Yes, ma'am.
Dialogue: 0,0:00:03.94,0:00:04.88,Default,,0000,0000,0000,,You are Megan?
Dialogue: 0,0:00:04.88,0:00:05.42,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,0:00:05.42,0:00:07.89,Default,,0000,0000,0000,,PROFESSOR: OK.
Dialogue: 0,0:00:07.89,0:00:11.09,Default,,0000,0000,0000,,STUDENT: I was just wondering\Nif we get like a form
Dialogue: 0,0:00:11.09,0:00:13.31,Default,,0000,0000,0000,,of [INAUDIBLE], note cards--
Dialogue: 0,0:00:13.31,0:00:15.71,Default,,0000,0000,0000,,PROFESSOR: No, you\N[INAUDIBLE] sheet whatsoever.
Dialogue: 0,0:00:15.71,0:00:22.19,Default,,0000,0000,0000,,So I think it's better that\NI review some of the formulas
Dialogue: 0,0:00:22.19,0:00:25.71,Default,,0000,0000,0000,,today that you are\Nexpected to know by heart,
Dialogue: 0,0:00:25.71,0:00:30.28,Default,,0000,0000,0000,,because they are also-- they\Nrequire you know to expect you
Dialogue: 0,0:00:30.28,0:00:34.44,Default,,0000,0000,0000,,to know the same formulas\Nby heart for the final
Dialogue: 0,0:00:34.44,0:00:36.37,Default,,0000,0000,0000,,with no cheat sheet.
Dialogue: 0,0:00:36.37,0:00:39.44,Default,,0000,0000,0000,,So the final will have\Nexactly the same policy,
Dialogue: 0,0:00:39.44,0:00:40.96,Default,,0000,0000,0000,,at the end of [INAUDIBLE].
Dialogue: 0,0:00:40.96,0:00:45.86,Default,,0000,0000,0000,,No calculator, no formula sheet,\Nno cheat sheet, no nothing,
Dialogue: 0,0:00:45.86,0:00:50.02,Default,,0000,0000,0000,,but you know I'm telling\Nyou guys this, except what
Dialogue: 0,0:00:50.02,0:00:50.76,Default,,0000,0000,0000,,you remember.
Dialogue: 0,0:00:50.76,0:00:58.11,Default,,0000,0000,0000,,
Dialogue: 0,0:00:58.11,0:01:01.91,Default,,0000,0000,0000,,Let me remind you\Nthat you are expected
Dialogue: 0,0:01:01.91,0:01:06.84,Default,,0000,0000,0000,,to know the equation\Nof the tangent plane.
Dialogue: 0,0:01:06.84,0:01:09.32,Default,,0000,0000,0000,,I'm not going to give them\Nin chronological order,
Dialogue: 0,0:01:09.32,0:01:23.01,Default,,0000,0000,0000,,but I think it's a good idea to\Nreview for midterm and final,
Dialogue: 0,0:01:23.01,0:01:30.99,Default,,0000,0000,0000,,some of the must-know formulas.
Dialogue: 0,0:01:30.99,0:01:37.46,Default,,0000,0000,0000,,
Dialogue: 0,0:01:37.46,0:01:42.30,Default,,0000,0000,0000,,One, well, I discussed this\Nbefore but I didn't remind you,
Dialogue: 0,0:01:42.30,0:01:51.02,Default,,0000,0000,0000,,differential of a function\Nof several variables.
Dialogue: 0,0:01:51.02,0:01:55.86,Default,,0000,0000,0000,,
Dialogue: 0,0:01:55.86,0:01:59.66,Default,,0000,0000,0000,,In particular, two\Nvariables most likely
Dialogue: 0,0:01:59.66,0:02:05.68,Default,,0000,0000,0000,,are the examples we've worked\Non a lot this semester.
Dialogue: 0,0:02:05.68,0:02:13.25,Default,,0000,0000,0000,,Number two, the definition\Nand especially formula,
Dialogue: 0,0:02:13.25,0:02:27.55,Default,,0000,0000,0000,,main formula for responding to\Ndirectional derivatives of F
Dialogue: 0,0:02:27.55,0:02:36.43,Default,,0000,0000,0000,,at P of coordinates X 0, Y 0,\Nin the direction U 1 and U 2,
Dialogue: 0,0:02:36.43,0:02:40.79,Default,,0000,0000,0000,,equals U.
Dialogue: 0,0:02:40.79,0:02:44.45,Default,,0000,0000,0000,,Just to test you,\NOK, well, I believe
Dialogue: 0,0:02:44.45,0:02:49.12,Default,,0000,0000,0000,,you know the formula\Nof the differential.
Dialogue: 0,0:02:49.12,0:02:56.21,Default,,0000,0000,0000,,But without me reminding you\Nwhat was that of two variables,
Dialogue: 0,0:02:56.21,0:02:59.79,Default,,0000,0000,0000,,I expect you to say d F equals--
Dialogue: 0,0:02:59.79,0:03:02.67,Default,,0000,0000,0000,,STUDENT: F [INAUDIBLE].
Dialogue: 0,0:03:02.67,0:03:06.78,Default,,0000,0000,0000,,D X plus F Y, D Y.
Dialogue: 0,0:03:06.78,0:03:08.20,Default,,0000,0000,0000,,PROFESSOR: How\Nabout-- thank you--
Dialogue: 0,0:03:08.20,0:03:11.12,Default,,0000,0000,0000,,how about the\Ndirectional derivative
Dialogue: 0,0:03:11.12,0:03:18.87,Default,,0000,0000,0000,,of F at P in the\Ndirection of the vector U?
Dialogue: 0,0:03:18.87,0:03:21.53,Default,,0000,0000,0000,,You will need the formula.
Dialogue: 0,0:03:21.53,0:03:22.42,Default,,0000,0000,0000,,Good, [INAUDIBLE].
Dialogue: 0,0:03:22.42,0:03:25.58,Default,,0000,0000,0000,,Yeah, that's the easiest\Nway to remember it,
Dialogue: 0,0:03:25.58,0:03:31.67,Default,,0000,0000,0000,,but that's not the first thing\NI want you to say, right?
Dialogue: 0,0:03:31.67,0:03:33.94,Default,,0000,0000,0000,,How did I write this?
Dialogue: 0,0:03:33.94,0:03:35.21,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:03:35.21,0:03:39.00,Default,,0000,0000,0000,,Of course, F of X-- thank\Nyou-- and [INAUDIBLE] 0,
Dialogue: 0,0:03:39.00,0:03:43.36,Default,,0000,0000,0000,,times Z 1 plus\Nderivative inspect Y,
Dialogue: 0,0:03:43.36,0:03:46.29,Default,,0000,0000,0000,,and X 0, Y 0 times U 2.
Dialogue: 0,0:03:46.29,0:03:51.38,Default,,0000,0000,0000,,What do we assume about it, F\NC 1 on the domain [INAUDIBLE]?
Dialogue: 0,0:03:51.38,0:03:54.25,Default,,0000,0000,0000,,
Dialogue: 0,0:03:54.25,0:04:00.61,Default,,0000,0000,0000,,Which means differential goal\Nwith continuous derivatives.
Dialogue: 0,0:04:00.61,0:04:04.48,Default,,0000,0000,0000,,This is what we assume\Nthrough chapter 11.
Dialogue: 0,0:04:04.48,0:04:13.92,Default,,0000,0000,0000,,Number three, I think I\Ntold you, but I'm not sure.
Dialogue: 0,0:04:13.92,0:04:16.69,Default,,0000,0000,0000,,But I think I did.
Dialogue: 0,0:04:16.69,0:04:24.81,Default,,0000,0000,0000,,Review the tangent\Nplane formula,
Dialogue: 0,0:04:24.81,0:04:27.18,Default,,0000,0000,0000,,formulas-- how about both?
Dialogue: 0,0:04:27.18,0:04:33.97,Default,,0000,0000,0000,,Well, only one is the one\NI consider relative for us.
Dialogue: 0,0:04:33.97,0:04:41.35,Default,,0000,0000,0000,,Which is Z equals F of X,\NY, will imply that at P
Dialogue: 0,0:04:41.35,0:04:46.20,Default,,0000,0000,0000,,with on the surface,\Neven as a reference, we
Dialogue: 0,0:04:46.20,0:04:51.04,Default,,0000,0000,0000,,have a tangent plane\Nof formula Z minus Z 0,
Dialogue: 0,0:04:51.04,0:04:52.61,Default,,0000,0000,0000,,equals-- who does it?
Dialogue: 0,0:04:52.61,0:04:55.80,Default,,0000,0000,0000,,OK, now you have\Nto remember this.
Dialogue: 0,0:04:55.80,0:04:57.91,Default,,0000,0000,0000,,Of course, it's your midterm.
Dialogue: 0,0:04:57.91,0:05:02.88,Default,,0000,0000,0000,,Review all of these things\Nby [INAUDIBLE] Thursday.
Dialogue: 0,0:05:02.88,0:05:09.51,Default,,0000,0000,0000,,Variable S of X,\N[INAUDIBLE] 0 at 0, times--
Dialogue: 0,0:05:09.51,0:05:11.59,Default,,0000,0000,0000,,STUDENT: X minus X 0.
Dialogue: 0,0:05:11.59,0:05:13.93,Default,,0000,0000,0000,,PROFESSOR: Thank you,\NRoberto, X 0 minus X 0,
Dialogue: 0,0:05:13.93,0:05:18.35,Default,,0000,0000,0000,,plus the same kind o thing in\Na different color, because I
Dialogue: 0,0:05:18.35,0:05:28.00,Default,,0000,0000,0000,,like to play [INAUDIBLE] orange,\NS Y, X 0, Y 0, Y minus Y 0.
Dialogue: 0,0:05:28.00,0:05:31.35,Default,,0000,0000,0000,,Don't come to the midterm-- you\Nbetter not come to the midterm,
Dialogue: 0,0:05:31.35,0:05:35.85,Default,,0000,0000,0000,,and you get a 0 for not\Nknowing the formulas, right?
Dialogue: 0,0:05:35.85,0:05:39.53,Default,,0000,0000,0000,,Now maybe you will see\Non this midterm, maybe
Dialogue: 0,0:05:39.53,0:05:43.35,Default,,0000,0000,0000,,not, maybe you'll see\Nit on the final-- what
Dialogue: 0,0:05:43.35,0:05:47.92,Default,,0000,0000,0000,,happens when you don't have\Nthe graph of a surface?
Dialogue: 0,0:05:47.92,0:05:52.29,Default,,0000,0000,0000,,
Dialogue: 0,0:05:52.29,0:06:03.74,Default,,0000,0000,0000,,Maybe you'll have an implicit\Nequation, an implicit equation
Dialogue: 0,0:06:03.74,0:06:15.56,Default,,0000,0000,0000,,where we write F of coordinates,\NX, Y and Z, equals a constant.
Dialogue: 0,0:06:15.56,0:06:18.59,Default,,0000,0000,0000,,Why is the tangent plane a P?
Dialogue: 0,0:06:18.59,0:06:25.48,Default,,0000,0000,0000,,Tangent plane, tangent plane\Nin both cases should be Y.
Dialogue: 0,0:06:25.48,0:06:31.50,Default,,0000,0000,0000,,Well, if you consider\Nthe first formula
Dialogue: 0,0:06:31.50,0:06:34.40,Default,,0000,0000,0000,,as a consequence\Nof the second one,
Dialogue: 0,0:06:34.40,0:06:37.75,Default,,0000,0000,0000,,that would be simply\Neasy, because you
Dialogue: 0,0:06:37.75,0:06:41.29,Default,,0000,0000,0000,,will have to write F of\NX Y minus Z equals 0.
Dialogue: 0,0:06:41.29,0:06:46.64,Default,,0000,0000,0000,,And there you are, the same\Nkind of formula in this.
Dialogue: 0,0:06:46.64,0:06:52.20,Default,,0000,0000,0000,,So what do you write--\Nremember the surface,
Dialogue: 0,0:06:52.20,0:06:54.45,Default,,0000,0000,0000,,the implicit formula.
Dialogue: 0,0:06:54.45,0:06:57.52,Default,,0000,0000,0000,,Who gave you the normal to\Nthe surface of a point P?
Dialogue: 0,0:06:57.52,0:07:00.31,Default,,0000,0000,0000,,
Dialogue: 0,0:07:00.31,0:07:01.32,Default,,0000,0000,0000,,No?
Dialogue: 0,0:07:01.32,0:07:04.07,Default,,0000,0000,0000,,The gradient of who?
Dialogue: 0,0:07:04.07,0:07:06.84,Default,,0000,0000,0000,,Not the gradient of the left,\Ndon't confuse-- the gradient
Dialogue: 0,0:07:06.84,0:07:09.12,Default,,0000,0000,0000,,of the big F, right?
Dialogue: 0,0:07:09.12,0:07:16.48,Default,,0000,0000,0000,,OK, at P. And the tangent\Nplane represents a what?
Dialogue: 0,0:07:16.48,0:07:20.17,Default,,0000,0000,0000,,The tangent plane represents\Nexactly the perpendicular plane
Dialogue: 0,0:07:20.17,0:07:26.49,Default,,0000,0000,0000,,that passes through the\Npoint P, and is [INAUDIBLE]
Dialogue: 0,0:07:26.49,0:07:28.72,Default,,0000,0000,0000,,to the normal.
Dialogue: 0,0:07:28.72,0:07:33.52,Default,,0000,0000,0000,,So you're going to have\Nyour surface, your normal,
Dialogue: 0,0:07:33.52,0:07:37.87,Default,,0000,0000,0000,,and the tangent plane, which\Nis perpendicular to the normal.
Dialogue: 0,0:07:37.87,0:07:40.58,Default,,0000,0000,0000,,Is this easy to remember,\Nmaybe for your final?
Dialogue: 0,0:07:40.58,0:07:45.66,Default,,0000,0000,0000,,I want to check if you know--\Nmake a list, this list,
Dialogue: 0,0:07:45.66,0:07:49.15,Default,,0000,0000,0000,,you have to post it in\Nthe bathroom or somewhere,
Dialogue: 0,0:07:49.15,0:07:52.14,Default,,0000,0000,0000,,on the wall or a closet.
Dialogue: 0,0:07:52.14,0:07:56.04,Default,,0000,0000,0000,,Because you need to know\Nthese things by the final.
Dialogue: 0,0:07:56.04,0:08:03.14,Default,,0000,0000,0000,,S of X at point P becomes\NS minus X 0 plus what?
Dialogue: 0,0:08:03.14,0:08:05.63,Default,,0000,0000,0000,,The same kind of thing, right?
Dialogue: 0,0:08:05.63,0:08:17.97,Default,,0000,0000,0000,,But it [INAUDIBLE] Y and Z.
Dialogue: 0,0:08:17.97,0:08:21.27,Default,,0000,0000,0000,,So if you have the\Ncuriosity to want
Dialogue: 0,0:08:21.27,0:08:25.53,Default,,0000,0000,0000,,to prove that the first colorful\Nformula for the tangent plane,
Dialogue: 0,0:08:25.53,0:08:31.55,Default,,0000,0000,0000,,using the red formula for the\Ntangent plane, it would come,
Dialogue: 0,0:08:31.55,0:08:33.27,Default,,0000,0000,0000,,is an immediate [INAUDIBLE].
Dialogue: 0,0:08:33.27,0:08:35.22,Default,,0000,0000,0000,,We've actually done that before.
Dialogue: 0,0:08:35.22,0:08:38.12,Default,,0000,0000,0000,,We even did the implicit\Nfunction theorem.
Dialogue: 0,0:08:38.12,0:08:39.95,Default,,0000,0000,0000,,There are some very\Nnice things you
Dialogue: 0,0:08:39.95,0:08:42.50,Default,,0000,0000,0000,,can do when you have a\Nfunction of several variables.
Dialogue: 0,0:08:42.50,0:08:48.13,Default,,0000,0000,0000,,And in particular, for a\Nfunction of two variables,
Dialogue: 0,0:08:48.13,0:08:50.46,Default,,0000,0000,0000,,makes it really easy.
Dialogue: 0,0:08:50.46,0:08:59.10,Default,,0000,0000,0000,,I'm gonna erase one,\Ntwo, three, and continue.
Dialogue: 0,0:08:59.10,0:09:02.37,Default,,0000,0000,0000,,
Dialogue: 0,0:09:02.37,0:09:04.92,Default,,0000,0000,0000,,So I guess when\NI leave the room,
Dialogue: 0,0:09:04.92,0:09:08.34,Default,,0000,0000,0000,,I have to be careful not\Nto leave the actual midterm
Dialogue: 0,0:09:08.34,0:09:10.46,Default,,0000,0000,0000,,in the room, although I\Nknow that you wouldn't even
Dialogue: 0,0:09:10.46,0:09:13.00,Default,,0000,0000,0000,,try to check my papers.
Dialogue: 0,0:09:13.00,0:09:16.30,Default,,0000,0000,0000,,
Dialogue: 0,0:09:16.30,0:09:20.93,Default,,0000,0000,0000,,I did also something\Nin this for finding
Dialogue: 0,0:09:20.93,0:09:23.88,Default,,0000,0000,0000,,a direction in which\Nthe function increases
Dialogue: 0,0:09:23.88,0:09:25.34,Default,,0000,0000,0000,,most rapidly.
Dialogue: 0,0:09:25.34,0:09:27.72,Default,,0000,0000,0000,,I don't have to\Nwrite it down, but I
Dialogue: 0,0:09:27.72,0:09:29.25,Default,,0000,0000,0000,,can remind you of the concept.
Dialogue: 0,0:09:29.25,0:09:33.41,Default,,0000,0000,0000,,
Dialogue: 0,0:09:33.41,0:09:39.08,Default,,0000,0000,0000,,So it's just the concept now,\Nno formula to actually memorize.
Dialogue: 0,0:09:39.08,0:09:43.14,Default,,0000,0000,0000,,But I'll still say number\Nfour, problem number four,
Dialogue: 0,0:09:43.14,0:09:52.04,Default,,0000,0000,0000,,because that's what I set\Nup on the actual exam.
Dialogue: 0,0:09:52.04,0:09:59.72,Default,,0000,0000,0000,,So what is the direction of\Nhighest ascend, deepest ascend?
Dialogue: 0,0:09:59.72,0:10:01.66,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:10:01.66,0:10:03.61,Default,,0000,0000,0000,,PROFESSOR: Is the\Ndirection of the plane.
Dialogue: 0,0:10:03.61,0:10:08.91,Default,,0000,0000,0000,,And what is the direction\Nof the steepest descent?
Dialogue: 0,0:10:08.91,0:10:14.48,Default,,0000,0000,0000,,The opposite of the\Ndirection of the grade.
Dialogue: 0,0:10:14.48,0:10:28.91,Default,,0000,0000,0000,,So the direction of\Nsteepest ascend and descend
Dialogue: 0,0:10:28.91,0:10:48.67,Default,,0000,0000,0000,,is the direction of for the\Ngraph Z equals F of X 1.
Dialogue: 0,0:10:48.67,0:10:53.01,Default,,0000,0000,0000,,This is the function [INAUDIBLE]\Nthat I'm talking about.
Dialogue: 0,0:10:53.01,0:11:00.55,Default,,0000,0000,0000,,Five, the direction\Nof U that you
Dialogue: 0,0:11:00.55,0:11:07.20,Default,,0000,0000,0000,,found at the previous problem,\NI didn't ask if it's unique, OK?
Dialogue: 0,0:11:07.20,0:11:09.68,Default,,0000,0000,0000,,Because that was one--\Nof course it's unique.
Dialogue: 0,0:11:09.68,0:11:12.14,Default,,0000,0000,0000,,Because we [INAUDIBLE] sizes.
Dialogue: 0,0:11:12.14,0:11:15.60,Default,,0000,0000,0000,,How do you say units\Nof sizing, [INAUDIBLE]?
Dialogue: 0,0:11:15.60,0:11:19.54,Default,,0000,0000,0000,,By deriving with it,\N[INAUDIBLE] a second,
Dialogue: 0,0:11:19.54,0:11:22.50,Default,,0000,0000,0000,,you have a U and a -U.
Dialogue: 0,0:11:22.50,0:11:24.96,Default,,0000,0000,0000,,STUDENT: So isn't the direction\Nthat the actual [INAUDIBLE]
Dialogue: 0,0:11:24.96,0:11:27.43,Default,,0000,0000,0000,,for is the gradient\Nof a normal vector?
Dialogue: 0,0:11:27.43,0:11:33.90,Default,,0000,0000,0000,,PROFESSOR: So yeah,\Nso the way I-- OK, you
Dialogue: 0,0:11:33.90,0:11:35.07,Default,,0000,0000,0000,,want me to read the problem?
Dialogue: 0,0:11:35.07,0:11:40.19,Default,,0000,0000,0000,,I'm going to read the actual\Nfunction So find the direction
Dialogue: 0,0:11:40.19,0:11:47.29,Default,,0000,0000,0000,,U, in which the function F of\NX Y, blah, blah, blah, blah,
Dialogue: 0,0:11:47.29,0:11:51.34,Default,,0000,0000,0000,,is here, it increases\Nmost rapidly.
Dialogue: 0,0:11:51.34,0:11:53.20,Default,,0000,0000,0000,,So what do you have to do?
Dialogue: 0,0:11:53.20,0:11:55.41,Default,,0000,0000,0000,,So the direction\Nof that is, what is
Dialogue: 0,0:11:55.41,0:11:58.37,Default,,0000,0000,0000,,the direction of that or this?
Dialogue: 0,0:11:58.37,0:12:11.86,Default,,0000,0000,0000,,U equals [INAUDIBLE]\Nrespectively minus U at P.
Dialogue: 0,0:12:11.86,0:12:21.86,Default,,0000,0000,0000,,Five, this direction U that you\Nfound at the previous problem,
Dialogue: 0,0:12:21.86,0:12:25.18,Default,,0000,0000,0000,,could be perpendicular\Nto a certain line, which
Dialogue: 0,0:12:25.18,0:12:26.18,Default,,0000,0000,0000,,of the following planes?
Dialogue: 0,0:12:26.18,0:12:32.26,Default,,0000,0000,0000,,
Dialogue: 0,0:12:32.26,0:12:33.95,Default,,0000,0000,0000,,I may give you multiple choice.
Dialogue: 0,0:12:33.95,0:12:37.51,Default,,0000,0000,0000,,Now what do you have\Nto do when you think
Dialogue: 0,0:12:37.51,0:12:44.84,Default,,0000,0000,0000,,the direction of-- the way\Nit's actually formulated
Dialogue: 0,0:12:44.84,0:12:47.98,Default,,0000,0000,0000,,is zero direction\Nis parallel to one
Dialogue: 0,0:12:47.98,0:12:50.88,Default,,0000,0000,0000,,of the following [INAUDIBLE]\Nplanes, which one?
Dialogue: 0,0:12:50.88,0:12:52.38,Default,,0000,0000,0000,,Let me give you an example.
Dialogue: 0,0:12:52.38,0:13:01.38,Default,,0000,0000,0000,,
Dialogue: 0,0:13:01.38,0:13:07.95,Default,,0000,0000,0000,,Z equals X [INAUDIBLE]\Nsquared, at P coordinates 1, 1.
Dialogue: 0,0:13:07.95,0:13:13.65,Default,,0000,0000,0000,,So that means X 0 is 1,\NY 0 is 1, and Z 0 is two.
Dialogue: 0,0:13:13.65,0:13:17.19,Default,,0000,0000,0000,,
Dialogue: 0,0:13:17.19,0:13:28.68,Default,,0000,0000,0000,,Find the direction\Nof the gradient of F.
Dialogue: 0,0:13:28.68,0:13:36.52,Default,,0000,0000,0000,,Let me put Z for alpha-- I'm\Nabusing my [INAUDIBLE]-- at P.
Dialogue: 0,0:13:36.52,0:13:50.09,Default,,0000,0000,0000,,And state which of the following\Nlines is parallel for this
Dialogue: 0,0:13:50.09,0:13:50.59,Default,,0000,0000,0000,,direction?
Dialogue: 0,0:13:50.59,0:13:58.35,Default,,0000,0000,0000,,
Dialogue: 0,0:13:58.35,0:14:02.78,Default,,0000,0000,0000,,A, lines in plane.
Dialogue: 0,0:14:02.78,0:14:07.13,Default,,0000,0000,0000,,
Dialogue: 0,0:14:07.13,0:14:08.85,Default,,0000,0000,0000,,X equals 2.
Dialogue: 0,0:14:08.85,0:14:11.66,Default,,0000,0000,0000,,B, Y equals 3.
Dialogue: 0,0:14:11.66,0:14:16.80,Default,,0000,0000,0000,,Or C, X plus 1 equals 0.
Dialogue: 0,0:14:16.80,0:14:20.16,Default,,0000,0000,0000,,D, these are lines in\Nplane in the plane,
Dialogue: 0,0:14:20.16,0:14:24.42,Default,,0000,0000,0000,,X Y. X plus Y. E,\Nnone of the above.
Dialogue: 0,0:14:24.42,0:14:29.67,Default,,0000,0000,0000,,
Dialogue: 0,0:14:29.67,0:14:31.69,Default,,0000,0000,0000,,So how are you going\Nto do that quickly?
Dialogue: 0,0:14:31.69,0:14:32.73,Default,,0000,0000,0000,,Well, it's easy, right?
Dialogue: 0,0:14:32.73,0:14:34.56,Default,,0000,0000,0000,,So what do I do?
Dialogue: 0,0:14:34.56,0:14:38.73,Default,,0000,0000,0000,,I say gradient 2 X\N2 Y, at the point
Dialogue: 0,0:14:38.73,0:14:41.55,Default,,0000,0000,0000,,1, 1-- you don't have to\Nwrite down everything.
Dialogue: 0,0:14:41.55,0:14:47.03,Default,,0000,0000,0000,,It's going to be the gradient\Nof F at P, will be 2, 2.
Dialogue: 0,0:14:47.03,0:14:53.18,Default,,0000,0000,0000,,That means U will\Nbe normalized 2, 2.
Dialogue: 0,0:14:53.18,0:14:54.74,Default,,0000,0000,0000,,What do you get?
Dialogue: 0,0:14:54.74,0:14:56.55,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:14:56.55,0:15:00.78,Default,,0000,0000,0000,,PROFESSOR: Well, what\Ndo people do normally
Dialogue: 0,0:15:00.78,0:15:04.02,Default,,0000,0000,0000,,if they want to do\Nit by the definition?
Dialogue: 0,0:15:04.02,0:15:06.45,Default,,0000,0000,0000,,They [INAUDIBLE]\Nthe vector 2, 2,
Dialogue: 0,0:15:06.45,0:15:08.84,Default,,0000,0000,0000,,by the square root [INAUDIBLE].
Dialogue: 0,0:15:08.84,0:15:11.40,Default,,0000,0000,0000,,Well, you could be a little\Nbit smarter than that,
Dialogue: 0,0:15:11.40,0:15:14.93,Default,,0000,0000,0000,,and say, F is the same\Nas the direction 1,
Dialogue: 0,0:15:14.93,0:15:18.36,Default,,0000,0000,0000,,1 divided by the square\Nroot of the sums,
Dialogue: 0,0:15:18.36,0:15:21.45,Default,,0000,0000,0000,,of a sum of the squares.
Dialogue: 0,0:15:21.45,0:15:23.04,Default,,0000,0000,0000,,It doesn't matter\Nwhich one you pick.
Dialogue: 0,0:15:23.04,0:15:28.57,Default,,0000,0000,0000,,All the co-linear ones\Nwill reveal the unique U.
Dialogue: 0,0:15:28.57,0:15:31.98,Default,,0000,0000,0000,,And that's exactly what\NI was trying to say,
Dialogue: 0,0:15:31.98,0:15:35.42,Default,,0000,0000,0000,,was this thinking by in just two\Nor three moves ahead of that.
Dialogue: 0,0:15:35.42,0:15:38.02,Default,,0000,0000,0000,,STUDENT: So that's the\Nsame as 2, 2 over 4?
Dialogue: 0,0:15:38.02,0:15:38.85,Default,,0000,0000,0000,,PROFESSOR: Yes, sir.
Dialogue: 0,0:15:38.85,0:15:43.32,Default,,0000,0000,0000,,It's the same as 2, 2 over\Nthe square root of 4 plus 4.
Dialogue: 0,0:15:43.32,0:15:46.88,Default,,0000,0000,0000,,But it's easier, why it's\Nsort of faster to do it.
Dialogue: 0,0:15:46.88,0:15:51.50,Default,,0000,0000,0000,,So why is that true\Nactually, Ryan is very right?
Dialogue: 0,0:15:51.50,0:15:53.86,Default,,0000,0000,0000,,Why is that true?
Dialogue: 0,0:15:53.86,0:15:56.16,Default,,0000,0000,0000,,Exactly because of that\Nuniqueness that I told you
Dialogue: 0,0:15:56.16,0:16:00.51,Default,,0000,0000,0000,,about last time, when you\Nsaid, well, [INAUDIBLE],
Dialogue: 0,0:16:00.51,0:16:01.72,Default,,0000,0000,0000,,what is that?
Dialogue: 0,0:16:01.72,0:16:06.59,Default,,0000,0000,0000,,So you get 1 over square root of\N2, and 1 over square root of 2,
Dialogue: 0,0:16:06.59,0:16:10.64,Default,,0000,0000,0000,,is that you [INAUDIBLE]\Nvector direction.
Dialogue: 0,0:16:10.64,0:16:14.29,Default,,0000,0000,0000,,Now without doing\Nfurther work, this
Dialogue: 0,0:16:14.29,0:16:16.86,Default,,0000,0000,0000,,is just a simple\Nmultiple question,
Dialogue: 0,0:16:16.86,0:16:18.35,Default,,0000,0000,0000,,of [INAUDIBLE] question.
Dialogue: 0,0:16:18.35,0:16:21.84,Default,,0000,0000,0000,,You are in front of your exam,\Nand you see lines in play.
Dialogue: 0,0:16:21.84,0:16:25.91,Default,,0000,0000,0000,,You close your eyes and see\Nall of-- I will see my what?
Dialogue: 0,0:16:25.91,0:16:29.38,Default,,0000,0000,0000,,You see all the lines in plane.
Dialogue: 0,0:16:29.38,0:16:33.40,Default,,0000,0000,0000,,Of all these lines,\Nyour favorite line
Dialogue: 0,0:16:33.40,0:16:38.92,Default,,0000,0000,0000,,has to have the same direction\Nas the vector U. Is X equals 2?
Dialogue: 0,0:16:38.92,0:16:40.53,Default,,0000,0000,0000,,No, that's nothing.
Dialogue: 0,0:16:40.53,0:16:41.35,Default,,0000,0000,0000,,Y equals 3?
Dialogue: 0,0:16:41.35,0:16:43.57,Default,,0000,0000,0000,,Those are horizontal, vertical.
Dialogue: 0,0:16:43.57,0:16:46.98,Default,,0000,0000,0000,,That's the direction of\Nthe first [INAUDIBLE].
Dialogue: 0,0:16:46.98,0:16:48.72,Default,,0000,0000,0000,,So is this true of C?
Dialogue: 0,0:16:48.72,0:16:49.39,Default,,0000,0000,0000,,No.
Dialogue: 0,0:16:49.39,0:16:51.01,Default,,0000,0000,0000,,STUDENT: No, that's\Nfor parallel lines.
Dialogue: 0,0:16:51.01,0:16:51.81,Default,,0000,0000,0000,,PROFESSOR: D?
Dialogue: 0,0:16:51.81,0:16:52.66,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,0:16:52.66,0:16:53.51,Default,,0000,0000,0000,,PROFESSOR: Right?
Dialogue: 0,0:16:53.51,0:16:58.25,Default,,0000,0000,0000,,So the incline X, Y--\Nthe first bisector
Dialogue: 0,0:16:58.25,0:16:59.74,Default,,0000,0000,0000,,is X equals Y [INAUDIBLE].
Dialogue: 0,0:16:59.74,0:17:05.49,Default,,0000,0000,0000,,Number C is Y minus X, which\Nis called the second bisector.
Dialogue: 0,0:17:05.49,0:17:08.88,Default,,0000,0000,0000,,You've seen that in college\Nalgebra-- high school algebra,
Dialogue: 0,0:17:08.88,0:17:10.09,Default,,0000,0000,0000,,more likely.
Dialogue: 0,0:17:10.09,0:17:12.98,Default,,0000,0000,0000,,So we call this first\Nbisector, second bisector.
Dialogue: 0,0:17:12.98,0:17:15.24,Default,,0000,0000,0000,,All right, so the\Nanswer is D. Do you have
Dialogue: 0,0:17:15.24,0:17:17.28,Default,,0000,0000,0000,,the same thing [INAUDIBLE]?
Dialogue: 0,0:17:17.28,0:17:19.77,Default,,0000,0000,0000,,On the two multiple\Nchoice things
Dialogue: 0,0:17:19.77,0:17:25.26,Default,,0000,0000,0000,,you have, you see very\Nwell, OK, I'm testing you.
Dialogue: 0,0:17:25.26,0:17:28.54,Default,,0000,0000,0000,,I didn't say anything.
Dialogue: 0,0:17:28.54,0:17:33.70,Default,,0000,0000,0000,,It was three feet away, OK?
Dialogue: 0,0:17:33.70,0:17:37.33,Default,,0000,0000,0000,,
Dialogue: 0,0:17:37.33,0:17:42.19,Default,,0000,0000,0000,,We have just a quick\Nanswer, and it's
Dialogue: 0,0:17:42.19,0:17:44.46,Default,,0000,0000,0000,,going to be easy,\Nwithout algebra,
Dialogue: 0,0:17:44.46,0:17:45.70,Default,,0000,0000,0000,,without computational stuff.
Dialogue: 0,0:17:45.70,0:17:48.55,Default,,0000,0000,0000,,
Dialogue: 0,0:17:48.55,0:17:50.90,Default,,0000,0000,0000,,Just from the first glance,\Nyou'll be able to answer.
Dialogue: 0,0:17:50.90,0:17:53.81,Default,,0000,0000,0000,,Number six, what\Nis the maximum rate
Dialogue: 0,0:17:53.81,0:17:58.80,Default,,0000,0000,0000,,of increase in the same\Ncase as in problem five?
Dialogue: 0,0:17:58.80,0:18:05.98,Default,,0000,0000,0000,,You say [INAUDIBLE], what\Nis the maximum, maximal rate
Dialogue: 0,0:18:05.98,0:18:12.19,Default,,0000,0000,0000,,of increase of a [INAUDIBLE]?
Dialogue: 0,0:18:12.19,0:18:14.67,Default,,0000,0000,0000,,And we all know what\NI'm talking about,
Dialogue: 0,0:18:14.67,0:18:16.66,Default,,0000,0000,0000,,although maybe not everybody.
Dialogue: 0,0:18:16.66,0:18:19.15,Default,,0000,0000,0000,,But this is the gradient.
Dialogue: 0,0:18:19.15,0:18:23.29,Default,,0000,0000,0000,,Who is giving you the\Nmaximum rate of increase?
Dialogue: 0,0:18:23.29,0:18:27.43,Default,,0000,0000,0000,,As I said last\Ntime in the review,
Dialogue: 0,0:18:27.43,0:18:29.50,Default,,0000,0000,0000,,that's actually the\Ndirectional derivative
Dialogue: 0,0:18:29.50,0:18:31.20,Default,,0000,0000,0000,,in the direction\Nof the gradient.
Dialogue: 0,0:18:31.20,0:18:35.24,Default,,0000,0000,0000,,But you are supposed to\Nknow without proving again
Dialogue: 0,0:18:35.24,0:18:37.49,Default,,0000,0000,0000,,that the directional\Nderivative and the direction
Dialogue: 0,0:18:37.49,0:18:41.14,Default,,0000,0000,0000,,of the gradient will\Ngive you that what?
Dialogue: 0,0:18:41.14,0:18:43.58,Default,,0000,0000,0000,,Gradient of norm of--
Dialogue: 0,0:18:43.58,0:18:44.47,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:18:44.47,0:18:50.84,Default,,0000,0000,0000,,PROFESSOR: Exactly, the\Nmagnitude of this F. So
Dialogue: 0,0:18:50.84,0:18:59.55,Default,,0000,0000,0000,,what does that-- what\Nwould that be in my case?
Dialogue: 0,0:18:59.55,0:19:02.01,Default,,0000,0000,0000,,[INAUDIBLE] pay\Nattention, please.
Dialogue: 0,0:19:02.01,0:19:04.62,Default,,0000,0000,0000,,Don't look at this,\Nif it's confusing you.
Dialogue: 0,0:19:04.62,0:19:06.42,Default,,0000,0000,0000,,Look at that, right?
Dialogue: 0,0:19:06.42,0:19:07.63,Default,,0000,0000,0000,,How much is that?
Dialogue: 0,0:19:07.63,0:19:11.52,Default,,0000,0000,0000,,
Dialogue: 0,0:19:11.52,0:19:14.12,Default,,0000,0000,0000,,All right, [INAUDIBLE].
Dialogue: 0,0:19:14.12,0:19:19.10,Default,,0000,0000,0000,,So you can put this\Nas [INAUDIBLE].
Dialogue: 0,0:19:19.10,0:19:23.82,Default,,0000,0000,0000,,So your multiple choice-- how\Nmany multiple choices do you
Dialogue: 0,0:19:23.82,0:19:24.32,Default,,0000,0000,0000,,have?
Dialogue: 0,0:19:24.32,0:19:25.32,Default,,0000,0000,0000,,Only two.
Dialogue: 0,0:19:25.32,0:19:28.50,Default,,0000,0000,0000,,It may seem like what\Nis the maximum rate?
Dialogue: 0,0:19:28.50,0:19:33.72,Default,,0000,0000,0000,,1, 0, 0, is telling-- that\Nmeans you have no increase.
Dialogue: 0,0:19:33.72,0:19:34.70,Default,,0000,0000,0000,,You're not moving.
Dialogue: 0,0:19:34.70,0:19:37.09,Default,,0000,0000,0000,,You're just lying\Nthere on the plane.
Dialogue: 0,0:19:37.09,0:19:37.90,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:19:37.90,0:19:40.05,Default,,0000,0000,0000,,What else?
Dialogue: 0,0:19:40.05,0:19:42.58,Default,,0000,0000,0000,,2 root 2, 2 [INAUDIBLE]\Nto infinity.
Dialogue: 0,0:19:42.58,0:19:43.32,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:19:43.32,0:19:45.43,Default,,0000,0000,0000,,I'm giving some\Nnonsensical choices.
Dialogue: 0,0:19:45.43,0:19:48.38,Default,,0000,0000,0000,,But one of them\Nwould be 2 root 2.
Dialogue: 0,0:19:48.38,0:19:53.75,Default,,0000,0000,0000,,So you would see, it would\Njump in front of your eyes.
Dialogue: 0,0:19:53.75,0:20:01.16,Default,,0000,0000,0000,,Number seven, I think I'm\Ngoing-- I thought about this,
Dialogue: 0,0:20:01.16,0:20:05.61,Default,,0000,0000,0000,,and I said one of\Nyou guys asked me,
Dialogue: 0,0:20:05.61,0:20:08.10,Default,,0000,0000,0000,,can you re-open any homework?
Dialogue: 0,0:20:08.10,0:20:10.26,Default,,0000,0000,0000,,And I said, nope.
Dialogue: 0,0:20:10.26,0:20:10.80,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:20:10.80,0:20:15.30,Default,,0000,0000,0000,,Because once the homework\Ncloses, automatically
Dialogue: 0,0:20:15.30,0:20:21.11,Default,,0000,0000,0000,,a few seconds later, all\Nthe answers are gonna be up.
Dialogue: 0,0:20:21.11,0:20:23.68,Default,,0000,0000,0000,,Do I have other\Nproblems handy to create
Dialogue: 0,0:20:23.68,0:20:29.47,Default,,0000,0000,0000,,a make-up for that individual\Nperson who had the problem?
Dialogue: 0,0:20:29.47,0:20:32.28,Default,,0000,0000,0000,,My cat almost died this\Nweek, but she said,
Dialogue: 0,0:20:32.28,0:20:35.45,Default,,0000,0000,0000,,but I have a treatment, and\Nhopefully she's going to live.
Dialogue: 0,0:20:35.45,0:20:40.04,Default,,0000,0000,0000,,So in situations of [INAUDIBLE],\Nlike an accident, a problem,
Dialogue: 0,0:20:40.04,0:20:44.22,Default,,0000,0000,0000,,[INAUDIBLE] hospitalization,\Nand so on, I'm sorry,
Dialogue: 0,0:20:44.22,0:20:46.48,Default,,0000,0000,0000,,I cannot re-open the homework.
Dialogue: 0,0:20:46.48,0:20:49.76,Default,,0000,0000,0000,,The homework is already up\Nthere with all the answers.
Dialogue: 0,0:20:49.76,0:20:54.43,Default,,0000,0000,0000,,When I extend homework, it's\Nstill doing that interval when
Dialogue: 0,0:20:54.43,0:20:56.20,Default,,0000,0000,0000,,you cannot see the answers.
Dialogue: 0,0:20:56.20,0:20:59.10,Default,,0000,0000,0000,,So I can extend it\Nby there [INAUDIBLE],
Dialogue: 0,0:20:59.10,0:21:00.17,Default,,0000,0000,0000,,that was an exception.
Dialogue: 0,0:21:00.17,0:21:03.44,Default,,0000,0000,0000,,So you have until the\Nfourth-- is the the fourth?
Dialogue: 0,0:21:03.44,0:21:05.11,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:21:05.11,0:21:07.99,Default,,0000,0000,0000,,But once that closes,\NI cannot re-open it.
Dialogue: 0,0:21:07.99,0:21:14.44,Default,,0000,0000,0000,,However, I thought of giving\Nyou a compensation midterm exam,
Dialogue: 0,0:21:14.44,0:21:16.95,Default,,0000,0000,0000,,contains an extra\Ncredit problem.
Dialogue: 0,0:21:16.95,0:21:20.81,Default,,0000,0000,0000,,Because once you told\Nme that, I started
Dialogue: 0,0:21:20.81,0:21:27.47,Default,,0000,0000,0000,,feeling bad for the two\Npeople who have problems.
Dialogue: 0,0:21:27.47,0:21:31.28,Default,,0000,0000,0000,,There were two or three people\Nwho had very serious problems
Dialogue: 0,0:21:31.28,0:21:33.50,Default,,0000,0000,0000,,this past weekend.
Dialogue: 0,0:21:33.50,0:21:36.82,Default,,0000,0000,0000,,So in the midterm, you have\Nthat extra credit problem,
Dialogue: 0,0:21:36.82,0:21:41.31,Default,,0000,0000,0000,,that is meant to touch up\Na little bit of let's say
Dialogue: 0,0:21:41.31,0:21:44.72,Default,,0000,0000,0000,,if you missed a few\Nproblems from the homework,
Dialogue: 0,0:21:44.72,0:21:48.44,Default,,0000,0000,0000,,you had some bad day, whatever.
Dialogue: 0,0:21:48.44,0:21:51.36,Default,,0000,0000,0000,,So you have ten\Nproblems plus one.
Dialogue: 0,0:21:51.36,0:21:56.88,Default,,0000,0000,0000,,Seven, you've seen that\Nbefore I told you about it.
Dialogue: 0,0:21:56.88,0:21:59.77,Default,,0000,0000,0000,,It's an easy problem.
Dialogue: 0,0:21:59.77,0:22:06.72,Default,,0000,0000,0000,,You have Z equals\NF over X Y. And I'm
Dialogue: 0,0:22:06.72,0:22:28.14,Default,,0000,0000,0000,,saying compute the volume of the\Nbody that lies below the graph
Dialogue: 0,0:22:28.14,0:22:40.26,Default,,0000,0000,0000,,and above the unit\N[INAUDIBLE] D. Fine.
Dialogue: 0,0:22:40.26,0:22:45.68,Default,,0000,0000,0000,,Eight, unfortunately\Neight have [INAUDIBLE]
Dialogue: 0,0:22:45.68,0:22:49.12,Default,,0000,0000,0000,,was disclosed because\NRyan was dreaming
Dialogue: 0,0:22:49.12,0:22:50.94,Default,,0000,0000,0000,,of the problems in the midterm.
Dialogue: 0,0:22:50.94,0:22:55.63,Default,,0000,0000,0000,,But it was something like\Nthat, very good information.
Dialogue: 0,0:22:55.63,0:23:01.86,Default,,0000,0000,0000,,So I would say a problem\Nlike that, maybe a plane that
Dialogue: 0,0:23:01.86,0:23:04.64,Default,,0000,0000,0000,,is cut in what?
Dialogue: 0,0:23:04.64,0:23:08.73,Default,,0000,0000,0000,,The plane's coordinates form\Nsomething like a tetrahedron,
Dialogue: 0,0:23:08.73,0:23:13.66,Default,,0000,0000,0000,,find the volume,\Nsomething like that.
Dialogue: 0,0:23:13.66,0:23:21.34,Default,,0000,0000,0000,,Nine, again without giving\Nyou the exact values,
Dialogue: 0,0:23:21.34,0:23:25.69,Default,,0000,0000,0000,,you will have a function\NF of X, and U of X,
Dialogue: 0,0:23:25.69,0:23:29.78,Default,,0000,0000,0000,,I'd say positive over\Na certain interval.
Dialogue: 0,0:23:29.78,0:23:35.01,Default,,0000,0000,0000,,
Dialogue: 0,0:23:35.01,0:23:44.61,Default,,0000,0000,0000,,Set up the double integral,\Nset up a double integral
Dialogue: 0,0:23:44.61,0:23:51.90,Default,,0000,0000,0000,,for the area of the domain\Nbetween F and G contained.
Dialogue: 0,0:23:51.90,0:24:01.36,Default,,0000,0000,0000,,
Dialogue: 0,0:24:01.36,0:24:15.40,Default,,0000,0000,0000,,Compute that, and also reverse\Nthe order of integration
Dialogue: 0,0:24:15.40,0:24:16.51,Default,,0000,0000,0000,,to check your work.
Dialogue: 0,0:24:16.51,0:24:19.38,Default,,0000,0000,0000,,
Dialogue: 0,0:24:19.38,0:24:24.99,Default,,0000,0000,0000,,Your answer, because here,\Na multiple choice answer,
Dialogue: 0,0:24:24.99,0:24:27.84,Default,,0000,0000,0000,,[INAUDIBLE] answer, no guesses.
Dialogue: 0,0:24:27.84,0:24:30.27,Default,,0000,0000,0000,,It's going to not be\Nhard at all-- very
Dialogue: 0,0:24:30.27,0:24:32.87,Default,,0000,0000,0000,,nice, friendly functions, very\Nnice, friendly [INAUDIBLE]
Dialogue: 0,0:24:32.87,0:24:34.33,Default,,0000,0000,0000,,functions.
Dialogue: 0,0:24:34.33,0:24:37.93,Default,,0000,0000,0000,,Just I have done this\Nbefore, but I'm not
Dialogue: 0,0:24:37.93,0:24:39.53,Default,,0000,0000,0000,,going to repeat what it was.
Dialogue: 0,0:24:39.53,0:24:45.02,Default,,0000,0000,0000,,I did it in the-- it's like\Nthe one I did last week.
Dialogue: 0,0:24:45.02,0:24:48.87,Default,,0000,0000,0000,,
Dialogue: 0,0:24:48.87,0:24:53.46,Default,,0000,0000,0000,,All right, so remember you\Nwrite the vertical strip thing,
Dialogue: 0,0:24:53.46,0:24:56.40,Default,,0000,0000,0000,,integration with respect to Y\Nfirst, and then with respect
Dialogue: 0,0:24:56.40,0:24:58.57,Default,,0000,0000,0000,,to X. You switch to\Nthe horizontal strip
Dialogue: 0,0:24:58.57,0:25:01.58,Default,,0000,0000,0000,,method of integration\Nwith respect
Dialogue: 0,0:25:01.58,0:25:07.60,Default,,0000,0000,0000,,to X first, and then with\Nrespect to Y. Okie-doke?
Dialogue: 0,0:25:07.60,0:25:13.03,Default,,0000,0000,0000,,And the actual algebra\Nhere will be [INAUDIBLE]
Dialogue: 0,0:25:13.03,0:25:16.48,Default,,0000,0000,0000,,expect to be done in one line.
Dialogue: 0,0:25:16.48,0:25:19.76,Default,,0000,0000,0000,,So you will have something\Nextremely simple.
Dialogue: 0,0:25:19.76,0:25:24.29,Default,,0000,0000,0000,,Ten-- it's another long exam.
Dialogue: 0,0:25:24.29,0:25:27.61,Default,,0000,0000,0000,,So I have to try\Nto test everything
Dialogue: 0,0:25:27.61,0:25:31.06,Default,,0000,0000,0000,,you know without you\Nspending more than one minute
Dialogue: 0,0:25:31.06,0:25:36.77,Default,,0000,0000,0000,,per problem, just to conceive\Nthe result. Formally,
Dialogue: 0,0:25:36.77,0:25:38.99,Default,,0000,0000,0000,,hold on-- now nine, I split it.
Dialogue: 0,0:25:38.99,0:25:41.49,Default,,0000,0000,0000,,Because I felt pity for you.
Dialogue: 0,0:25:41.49,0:25:45.75,Default,,0000,0000,0000,,So I put [INAUDIBLE], I put\Njust set up the level integral,
Dialogue: 0,0:25:45.75,0:25:50.07,Default,,0000,0000,0000,,and reverse the\Norder of integration.
Dialogue: 0,0:25:50.07,0:25:53.55,Default,,0000,0000,0000,,So you have to write integral\Nintegral equals integral
Dialogue: 0,0:25:53.55,0:25:57.21,Default,,0000,0000,0000,,integral, nothing else,\Nno answer, no number.
Dialogue: 0,0:25:57.21,0:26:07.03,Default,,0000,0000,0000,,And ten, actually compute\Nany of the two integrals
Dialogue: 0,0:26:07.03,0:26:14.88,Default,,0000,0000,0000,,at number nine to find\Nthe area of the domain.
Dialogue: 0,0:26:14.88,0:26:18.34,Default,,0000,0000,0000,,
Dialogue: 0,0:26:18.34,0:26:22.16,Default,,0000,0000,0000,,Just like we did\Nlast time, and you
Dialogue: 0,0:26:22.16,0:26:24.46,Default,,0000,0000,0000,,don't have a calculator, OK?
Dialogue: 0,0:26:24.46,0:26:27.78,Default,,0000,0000,0000,,Suppose your answer will\Nbe-- what was last time?
Dialogue: 0,0:26:27.78,0:26:30.44,Default,,0000,0000,0000,,One over six, I don't know.
Dialogue: 0,0:26:30.44,0:26:33.25,Default,,0000,0000,0000,,If you give me decimals,\NI will be very upset.
Dialogue: 0,0:26:33.25,0:26:36.76,Default,,0000,0000,0000,,You have to give me the precise\Nanswer for that problem,
Dialogue: 0,0:26:36.76,0:26:40.65,Default,,0000,0000,0000,,because it's so easy to\Ncompute that you would have
Dialogue: 0,0:26:40.65,0:26:47.28,Default,,0000,0000,0000,,no need for using a calculator\Nor software, or any kind
Dialogue: 0,0:26:47.28,0:26:50.66,Default,,0000,0000,0000,,of electronic device.
Dialogue: 0,0:26:50.66,0:26:56.69,Default,,0000,0000,0000,,And finally number\N11-- and number 11, I
Dialogue: 0,0:26:56.69,0:27:01.14,Default,,0000,0000,0000,,shouldn't say what it is,\Nbecause it's extra credit.
Dialogue: 0,0:27:01.14,0:27:03.37,Default,,0000,0000,0000,,But I'll still say what it is.
Dialogue: 0,0:27:03.37,0:27:05.48,Default,,0000,0000,0000,,It's some simple\Nintegral where you
Dialogue: 0,0:27:05.48,0:27:08.30,Default,,0000,0000,0000,,are going to have to use\Nspherical coordinates.
Dialogue: 0,0:27:08.30,0:27:09.84,Default,,0000,0000,0000,,And shut up,\N[INAUDIBLE], because you
Dialogue: 0,0:27:09.84,0:27:11.74,Default,,0000,0000,0000,,are talking too much.
Dialogue: 0,0:27:11.74,0:27:16.35,Default,,0000,0000,0000,,So again, number 11 will\Nbe a triple integral
Dialogue: 0,0:27:16.35,0:27:17.86,Default,,0000,0000,0000,,that is easy to compute.
Dialogue: 0,0:27:17.86,0:27:22.61,Default,,0000,0000,0000,,And when you're going-- well,\Nyou don't have to use vehicle.
Dialogue: 0,0:27:22.61,0:27:26.11,Default,,0000,0000,0000,,You can still do it with\Ncylindrical coordinates,
Dialogue: 0,0:27:26.11,0:27:27.35,Default,,0000,0000,0000,,for example.
Dialogue: 0,0:27:27.35,0:27:33.30,Default,,0000,0000,0000,,But it's a big pain doing\Nthe cylindrical coordinates
Dialogue: 0,0:27:33.30,0:27:35.12,Default,,0000,0000,0000,,for that kind of problem.
Dialogue: 0,0:27:35.12,0:27:41.08,Default,,0000,0000,0000,,So imagine maybe I'm looking\Nat the domain to be a sphere.
Dialogue: 0,0:27:41.08,0:27:43.98,Default,,0000,0000,0000,,
Dialogue: 0,0:27:43.98,0:27:49.18,Default,,0000,0000,0000,,The problems we worked\Nas a training in class,
Dialogue: 0,0:27:49.18,0:27:53.33,Default,,0000,0000,0000,,are actually harder than\Nthe ones I put on the exam.
Dialogue: 0,0:27:53.33,0:27:58.10,Default,,0000,0000,0000,,I have a professor who's a grad\Nstudent, and he used to say,
Dialogue: 0,0:27:58.10,0:28:00.52,Default,,0000,0000,0000,,the easy problems\Nare the professor
Dialogue: 0,0:28:00.52,0:28:02.93,Default,,0000,0000,0000,,to work in classes examples.
Dialogue: 0,0:28:02.93,0:28:07.96,Default,,0000,0000,0000,,The hard problems are the\Nstudents who have on the exam.
Dialogue: 0,0:28:07.96,0:28:10.39,Default,,0000,0000,0000,,I think exactly the opposite.
Dialogue: 0,0:28:10.39,0:28:17.56,Default,,0000,0000,0000,,Because when you will in\Ntraining for any kind of sports
Dialogue: 0,0:28:17.56,0:28:21.31,Default,,0000,0000,0000,,or a skill or music, you\Nhave to train yourself
Dialogue: 0,0:28:21.31,0:28:26.32,Default,,0000,0000,0000,,above the level of\Nyour competition.
Dialogue: 0,0:28:26.32,0:28:29.04,Default,,0000,0000,0000,,Otherwise your\Ncompetition will be bored.
Dialogue: 0,0:28:29.04,0:28:30.62,Default,,0000,0000,0000,,So what you're\Ndoing with training
Dialogue: 0,0:28:30.62,0:28:32.62,Default,,0000,0000,0000,,should not always\Nbe how whether you
Dialogue: 0,0:28:32.62,0:28:37.03,Default,,0000,0000,0000,,are an athlete or a\Nmathematician or a violinist
Dialogue: 0,0:28:37.03,0:28:39.42,Default,,0000,0000,0000,,or whatever.
Dialogue: 0,0:28:39.42,0:28:44.15,Default,,0000,0000,0000,,So you're not going to see\Nsomething like intersector
Dialogue: 0,0:28:44.15,0:28:48.70,Default,,0000,0000,0000,,cylinders, passing one through\Nthe other, one the cone,
Dialogue: 0,0:28:48.70,0:28:51.24,Default,,0000,0000,0000,,ice cream cone will\Nbe doing a parabola,
Dialogue: 0,0:28:51.24,0:28:55.28,Default,,0000,0000,0000,,then the cone is full of ice\Ncream-- nothing like that.
Dialogue: 0,0:28:55.28,0:28:57.73,Default,,0000,0000,0000,,Something simpler.
Dialogue: 0,0:28:57.73,0:29:02.20,Default,,0000,0000,0000,,And you may guess what it is,\Nbut keep in mind that force
Dialogue: 0,0:29:02.20,0:29:06.29,Default,,0000,0000,0000,,of speed components, you\Nhave to know the Jacobian,
Dialogue: 0,0:29:06.29,0:29:07.47,Default,,0000,0000,0000,,don't hesitate.
Dialogue: 0,0:29:07.47,0:29:09.58,Default,,0000,0000,0000,,That's assumed to be memorized.
Dialogue: 0,0:29:09.58,0:29:11.56,Default,,0000,0000,0000,,Don't ask me in the\Nmiddle of the exam.
Dialogue: 0,0:29:11.56,0:29:14.10,Default,,0000,0000,0000,,Why was the Jacobian\N[INAUDIBLE] components?
Dialogue: 0,0:29:14.10,0:29:17.17,Default,,0000,0000,0000,,You are supposed to\Nknow that as being what?
Dialogue: 0,0:29:17.17,0:29:17.88,Default,,0000,0000,0000,,What was that?
Dialogue: 0,0:29:17.88,0:29:18.87,Default,,0000,0000,0000,,Roberto knows.
Dialogue: 0,0:29:18.87,0:29:19.70,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:29:19.70,0:29:21.77,Default,,0000,0000,0000,,PROFESSOR: [INAUDIBLE]\Nassign by, and by
Dialogue: 0,0:29:21.77,0:29:24.86,Default,,0000,0000,0000,,was what, for a friend of yours?
Dialogue: 0,0:29:24.86,0:29:28.82,Default,,0000,0000,0000,,The latitude from Santa Claus,\Nmeasured down all the way
Dialogue: 0,0:29:28.82,0:29:31.09,Default,,0000,0000,0000,,to [INAUDIBLE].
Dialogue: 0,0:29:31.09,0:29:35.21,Default,,0000,0000,0000,,Theta is the longitude\Nfrom 0 to 2 pi.
Dialogue: 0,0:29:35.21,0:29:40.45,Default,,0000,0000,0000,,You are going to have\Nsome very nice domain.
Dialogue: 0,0:29:40.45,0:29:42.03,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:29:42.03,0:29:42.70,Default,,0000,0000,0000,,That's it, guys.
Dialogue: 0,0:29:42.70,0:29:45.39,Default,,0000,0000,0000,,That's what the exam will say.
Dialogue: 0,0:29:45.39,0:29:47.57,Default,,0000,0000,0000,,I'm asking you for a few things.
Dialogue: 0,0:29:47.57,0:29:51.61,Default,,0000,0000,0000,,First of all, you are already\Nprepared, I guarantee it.
Dialogue: 0,0:29:51.61,0:29:55.01,Default,,0000,0000,0000,,Do not stay up late at night.
Dialogue: 0,0:29:55.01,0:29:57.90,Default,,0000,0000,0000,,The biggest mistake\Nstudents make
Dialogue: 0,0:29:57.90,0:30:02.40,Default,,0000,0000,0000,,is staying up the night before a\Nmidterm or a final because they
Dialogue: 0,0:30:02.40,0:30:04.45,Default,,0000,0000,0000,,want to study everything.
Dialogue: 0,0:30:04.45,0:30:06.20,Default,,0000,0000,0000,,That's bad.
Dialogue: 0,0:30:06.20,0:30:10.41,Default,,0000,0000,0000,,The next day you will be tired\Nand you won't perform as well.
Dialogue: 0,0:30:10.41,0:30:15.70,Default,,0000,0000,0000,,Second of all, do not\Nbe nervous at all.
Dialogue: 0,0:30:15.70,0:30:17.15,Default,,0000,0000,0000,,You have no reason\Nto be nervous.
Dialogue: 0,0:30:17.15,0:30:19.45,Default,,0000,0000,0000,,You have plenty of time.
Dialogue: 0,0:30:19.45,0:30:24.36,Default,,0000,0000,0000,,You have plenty of\Nthings to write down.
Dialogue: 0,0:30:24.36,0:30:28.75,Default,,0000,0000,0000,,OK, about the way I grade.
Dialogue: 0,0:30:28.75,0:30:31.18,Default,,0000,0000,0000,,If you leave the problem\Ncompletely blank,
Dialogue: 0,0:30:31.18,0:30:33.30,Default,,0000,0000,0000,,yes, that's a zero.
Dialogue: 0,0:30:33.30,0:30:39.24,Default,,0000,0000,0000,,But if you provide me with at\Nleast a hint, a formula that
Dialogue: 0,0:30:39.24,0:30:41.58,Default,,0000,0000,0000,,serves you-- not just\Nan arbitrary formula
Dialogue: 0,0:30:41.58,0:30:43.02,Default,,0000,0000,0000,,that has nothing to do with it.
Dialogue: 0,0:30:43.02,0:30:46.38,Default,,0000,0000,0000,,But a formula that's\Nin the regulation,
Dialogue: 0,0:30:46.38,0:30:49.74,Default,,0000,0000,0000,,I give you partial\Ncredit for everything.
Dialogue: 0,0:30:49.74,0:30:52.96,Default,,0000,0000,0000,,So you have no\Nreason to freak out.
Dialogue: 0,0:30:52.96,0:30:57.32,Default,,0000,0000,0000,,Even if you mess up, let's\Nsay, one or two problems,
Dialogue: 0,0:30:57.32,0:31:00.84,Default,,0000,0000,0000,,your algebra at the\Nend, you should still
Dialogue: 0,0:31:00.84,0:31:04.15,Default,,0000,0000,0000,,gather together lots of credit.
Dialogue: 0,0:31:04.15,0:31:11.54,Default,,0000,0000,0000,,I wrote the exam especially\Nbecause I, myself,
Dialogue: 0,0:31:11.54,0:31:14.53,Default,,0000,0000,0000,,hate some medical\Nanswers from web work.
Dialogue: 0,0:31:14.53,0:31:18.40,Default,,0000,0000,0000,,I made the answers to\Nbe fat and sassy, not
Dialogue: 0,0:31:18.40,0:31:20.61,Default,,0000,0000,0000,,like the ones in the web works.
Dialogue: 0,0:31:20.61,0:31:24.94,Default,,0000,0000,0000,,So something that you\Ncan do even mentally,
Dialogue: 0,0:31:24.94,0:31:28.60,Default,,0000,0000,0000,,not have to struggle\Nwith the answer.
Dialogue: 0,0:31:28.60,0:31:31.34,Default,,0000,0000,0000,,Several people have\Nasked me to go over
Dialogue: 0,0:31:31.34,0:31:38.58,Default,,0000,0000,0000,,the last two problems of the\Nhomework, and I'll go forward.
Dialogue: 0,0:31:38.58,0:31:40.89,Default,,0000,0000,0000,,I'll give you an example\Nof a problem that bothered
Dialogue: 0,0:31:40.89,0:31:45.23,Default,,0000,0000,0000,,a few bothered a few people.
Dialogue: 0,0:31:45.23,0:31:51.85,Default,,0000,0000,0000,,And it's somewhat interesting\Nbecause, of course,
Dialogue: 0,0:31:51.85,0:31:57.58,Default,,0000,0000,0000,,I make the algebra\Neasier than it is.
Dialogue: 0,0:31:57.58,0:32:01.17,Default,,0000,0000,0000,,You have 2, 3, 7,\N9, I don't know.
Dialogue: 0,0:32:01.17,0:32:06.96,Default,,0000,0000,0000,,You have a function x and y that\Nare both functions of u and v.
Dialogue: 0,0:32:06.96,0:32:10.62,Default,,0000,0000,0000,,And instead of asking you\Nfor the determinant-- well,
Dialogue: 0,0:32:10.62,0:32:14.56,Default,,0000,0000,0000,,one of them may have asked\Nyou for the determinant,
Dialogue: 0,0:32:14.56,0:32:19.33,Default,,0000,0000,0000,,the functions x, derivative\Nof x, y with [INAUDIBLE] u,
Dialogue: 0,0:32:19.33,0:32:20.36,Default,,0000,0000,0000,,v as Jacobian.
Dialogue: 0,0:32:20.36,0:32:25.40,Default,,0000,0000,0000,,But the other one was asking\Nyou for just the opposite.
Dialogue: 0,0:32:25.40,0:32:27.39,Default,,0000,0000,0000,,Well, several people\Ndidn't see that.
Dialogue: 0,0:32:27.39,0:32:30.72,Default,,0000,0000,0000,,And they kept asking me, so\NI answered some 20 questions
Dialogue: 0,0:32:30.72,0:32:34.37,Default,,0000,0000,0000,,during the weekend exactly\Nabout problems like that.
Dialogue: 0,0:32:34.37,0:32:36.23,Default,,0000,0000,0000,,Which doesn't bother me.
Dialogue: 0,0:32:36.23,0:32:40.27,Default,,0000,0000,0000,,I think I would have had\Nthe same problem when
Dialogue: 0,0:32:40.27,0:32:41.73,Default,,0000,0000,0000,,I was like you.
Dialogue: 0,0:32:41.73,0:32:44.69,Default,,0000,0000,0000,,The easy part on\Nthe first Jacobian
Dialogue: 0,0:32:44.69,0:32:49.85,Default,,0000,0000,0000,,is that you have 1, 1, 1\Nminus 1, whatever that is.
Dialogue: 0,0:32:49.85,0:32:54.48,Default,,0000,0000,0000,,The definition is\Nx sub u, x sub v,
Dialogue: 0,0:32:54.48,0:32:57.85,Default,,0000,0000,0000,,y sub u, y sub v. These\Nare the partial derivatives
Dialogue: 0,0:32:57.85,0:32:59.67,Default,,0000,0000,0000,,and that's called Jacobian.
Dialogue: 0,0:32:59.67,0:33:06.15,Default,,0000,0000,0000,,And what you have, you\Nhave an easy answer.
Dialogue: 0,0:33:06.15,0:33:09.38,Default,,0000,0000,0000,,In this case, you have\Nthe answer negative 2.
Dialogue: 0,0:33:09.38,0:33:15.71,Default,,0000,0000,0000,,And if you, however, are asked\Nby the author of the problem,
Dialogue: 0,0:33:15.71,0:33:17.90,Default,,0000,0000,0000,,whoever created the\Nproblem of this.
Dialogue: 0,0:33:17.90,0:33:19.62,Default,,0000,0000,0000,,And you put negative\N2, it's going
Dialogue: 0,0:33:19.62,0:33:21.82,Default,,0000,0000,0000,,to say no, this is not correct.
Dialogue: 0,0:33:21.82,0:33:24.09,Default,,0000,0000,0000,,And this is what happened\Nto several people.
Dialogue: 0,0:33:24.09,0:33:26.83,Default,,0000,0000,0000,,Now, there are two\Nways around it.
Dialogue: 0,0:33:26.83,0:33:30.99,Default,,0000,0000,0000,,There are two ways\Nyou can solve that.
Dialogue: 0,0:33:30.99,0:33:34.33,Default,,0000,0000,0000,,STUDENT: On the work, it\Nsays the reverse [INAUDIBLE].
Dialogue: 0,0:33:34.33,0:33:37.19,Default,,0000,0000,0000,,That Jacobian times the\Nreverse Jacobian [INAUDIBLE].
Dialogue: 0,0:33:37.19,0:33:40.60,Default,,0000,0000,0000,,
Dialogue: 0,0:33:40.60,0:33:45.72,Default,,0000,0000,0000,,PROFESSOR: I want\Nto say why that is.
Dialogue: 0,0:33:45.72,0:33:48.92,Default,,0000,0000,0000,,For a student who doesn't know\Nwhy this Jacobian is exactly
Dialogue: 0,0:33:48.92,0:33:53.58,Default,,0000,0000,0000,,j inverse, there are still\Nchances the student can say,
Dialogue: 0,0:33:53.58,0:33:56.87,Default,,0000,0000,0000,,well, here's how smart I am.
Dialogue: 0,0:33:56.87,0:34:01.34,Default,,0000,0000,0000,,I'm going to say u out, v\Nout in terms of x and y.
Dialogue: 0,0:34:01.34,0:34:03.56,Default,,0000,0000,0000,,I inverse the\Nfunctions because they
Dialogue: 0,0:34:03.56,0:34:06.51,Default,,0000,0000,0000,,are linear functions\N[INAUDIBLE] linear system.
Dialogue: 0,0:34:06.51,0:34:08.85,Default,,0000,0000,0000,,So I say x plus y.
Dialogue: 0,0:34:08.85,0:34:12.06,Default,,0000,0000,0000,,This is elimination called--\Nwhen we were little,
Dialogue: 0,0:34:12.06,0:34:15.89,Default,,0000,0000,0000,,this was called elimination 2u.
Dialogue: 0,0:34:15.89,0:34:22.28,Default,,0000,0000,0000,,x minus y equals 2v.
Dialogue: 0,0:34:22.28,0:34:27.99,Default,,0000,0000,0000,,So u is x plus y over 2.
Dialogue: 0,0:34:27.99,0:34:30.07,Default,,0000,0000,0000,,That means 1/2 x, 1/2 y.
Dialogue: 0,0:34:30.07,0:34:30.78,Default,,0000,0000,0000,,Right, guys?
Dialogue: 0,0:34:30.78,0:34:31.28,Default,,0000,0000,0000,,I'm right?
Dialogue: 0,0:34:31.28,0:34:32.32,Default,,0000,0000,0000,,STUDENT: Mm-hmm.
Dialogue: 0,0:34:32.32,0:34:33.42,Default,,0000,0000,0000,,PROFESSOR: OK.
Dialogue: 0,0:34:33.42,0:34:38.32,Default,,0000,0000,0000,,And 1/2 of x and minus 1/2 of y.
Dialogue: 0,0:34:38.32,0:34:40.16,Default,,0000,0000,0000,,And then, what does\Nthe student say?
Dialogue: 0,0:34:40.16,0:34:42.53,Default,,0000,0000,0000,,I know what I'm going to do.
Dialogue: 0,0:34:42.53,0:34:48.02,Default,,0000,0000,0000,,Just by the same definition,\NI say the du, v dx,
Dialogue: 0,0:34:48.02,0:34:51.30,Default,,0000,0000,0000,,y as I have an inverse function.
Dialogue: 0,0:34:51.30,0:34:53.74,Default,,0000,0000,0000,,And I knew how to\Ninvert the system.
Dialogue: 0,0:34:53.74,0:34:56.15,Default,,0000,0000,0000,,I get 1/2.
Dialogue: 0,0:34:56.15,0:34:59.71,Default,,0000,0000,0000,,Not matrix, Magdalena,\Nnow, determinant.
Dialogue: 0,0:34:59.71,0:35:04.22,Default,,0000,0000,0000,,1/2, 1/2 and 1/2, minus 1/2.
Dialogue: 0,0:35:04.22,0:35:07.16,Default,,0000,0000,0000,,
Dialogue: 0,0:35:07.16,0:35:09.13,Default,,0000,0000,0000,,And guess what?
Dialogue: 0,0:35:09.13,0:35:09.90,Default,,0000,0000,0000,,What do I get?
Dialogue: 0,0:35:09.90,0:35:14.77,Default,,0000,0000,0000,,Exactly what it was saying,\Nbut I did it the long way.
Dialogue: 0,0:35:14.77,0:35:21.17,Default,,0000,0000,0000,,I got minus 1 over 4 minus 1\Nover 4, which is minus 1/2.
Dialogue: 0,0:35:21.17,0:35:21.67,Default,,0000,0000,0000,,Which is--
Dialogue: 0,0:35:21.67,0:35:22.38,Default,,0000,0000,0000,,STUDENT: Inverse.
Dialogue: 0,0:35:22.38,0:35:23.99,Default,,0000,0000,0000,,PROFESSOR: The inverse of that.
Dialogue: 0,0:35:23.99,0:35:26.70,Default,,0000,0000,0000,,
Dialogue: 0,0:35:26.70,0:35:33.08,Default,,0000,0000,0000,,And you are going to ask me,\NOK, I don't understand why.
Dialogue: 0,0:35:33.08,0:35:35.65,Default,,0000,0000,0000,,That's why I want to\Ntell you a story that I
Dialogue: 0,0:35:35.65,0:35:37.39,Default,,0000,0000,0000,,think is beautiful.
Dialogue: 0,0:35:37.39,0:35:40.72,Default,,0000,0000,0000,,The book doesn't start like\Nthat, because the book doesn't
Dialogue: 0,0:35:40.72,0:35:45.37,Default,,0000,0000,0000,,necessarily have enough\Nspace to remind you
Dialogue: 0,0:35:45.37,0:35:50.18,Default,,0000,0000,0000,,everything you learned in\NCalc 1 when you are in Calc 3.
Dialogue: 0,0:35:50.18,0:35:54.63,Default,,0000,0000,0000,,But if you think of what you\Nlearned in Calc 1, in Calc 1
Dialogue: 0,0:35:54.63,0:35:58.50,Default,,0000,0000,0000,,your professor-- I'm sure that\Nhe or she showed you this.
Dialogue: 0,0:35:58.50,0:36:01.82,Default,,0000,0000,0000,,If you have a function\Ny equals f of x,
Dialogue: 0,0:36:01.82,0:36:06.10,Default,,0000,0000,0000,,assume this is a c1 function\Nand everything is nice.
Dialogue: 0,0:36:06.10,0:36:14.04,Default,,0000,0000,0000,,And then you have that\Nf prime of x exists
Dialogue: 0,0:36:14.04,0:36:15.33,Default,,0000,0000,0000,,and it's continuous everywhere.
Dialogue: 0,0:36:15.33,0:36:17.57,Default,,0000,0000,0000,,That's what it means c1.
Dialogue: 0,0:36:17.57,0:36:20.97,Default,,0000,0000,0000,,And you want to\Ninvert this function.
Dialogue: 0,0:36:20.97,0:36:25.03,Default,,0000,0000,0000,,You want to invert this\Nfunction around the point x0.
Dialogue: 0,0:36:25.03,0:36:38.03,Default,,0000,0000,0000,,So you know that at\Nleast for some interval
Dialogue: 0,0:36:38.03,0:36:41.03,Default,,0000,0000,0000,,that f is one-to-one.
Dialogue: 0,0:36:41.03,0:36:42.49,Default,,0000,0000,0000,,So it's invertible.
Dialogue: 0,0:36:42.49,0:36:48.96,Default,,0000,0000,0000,,
Dialogue: 0,0:36:48.96,0:36:53.59,Default,,0000,0000,0000,,What is the derivative\Nof [INAUDIBLE]?
Dialogue: 0,0:36:53.59,0:36:56.42,Default,,0000,0000,0000,,
Dialogue: 0,0:36:56.42,0:37:00.27,Default,,0000,0000,0000,,Somebody asks you, so what\Nis the-- the derivative
Dialogue: 0,0:37:00.27,0:37:02.03,Default,,0000,0000,0000,,of the inverse\Nfunction is a function
Dialogue: 0,0:37:02.03,0:37:06.40,Default,,0000,0000,0000,,of x with respect to x.
Dialogue: 0,0:37:06.40,0:37:08.56,Default,,0000,0000,0000,,[INAUDIBLE]?
Dialogue: 0,0:37:08.56,0:37:10.00,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:37:10.00,0:37:15.68,Default,,0000,0000,0000,,
Dialogue: 0,0:37:15.68,0:37:17.77,Default,,0000,0000,0000,,Remind yourself\Nhow you did that.
Dialogue: 0,0:37:17.77,0:37:18.97,Default,,0000,0000,0000,,Was this hard?
Dialogue: 0,0:37:18.97,0:37:23.47,Default,,0000,0000,0000,,
Dialogue: 0,0:37:23.47,0:37:31.45,Default,,0000,0000,0000,,Anybody remembers the formula\Nfor the inverse function?
Dialogue: 0,0:37:31.45,0:37:32.35,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:37:32.35,0:37:35.21,Default,,0000,0000,0000,,PROFESSOR: 1 over f prime of x.
Dialogue: 0,0:37:35.21,0:37:38.65,Default,,0000,0000,0000,,
Dialogue: 0,0:37:38.65,0:37:42.09,Default,,0000,0000,0000,,So assume that you do\Nthat at the next 0,
Dialogue: 0,0:37:42.09,0:37:45.52,Default,,0000,0000,0000,,assume that f prime of\Nx0 is different from 0.
Dialogue: 0,0:37:45.52,0:37:49.25,Default,,0000,0000,0000,,Now, how would you\Nprove that and how--
Dialogue: 0,0:37:49.25,0:37:51.52,Default,,0000,0000,0000,,well, too much memorization.
Dialogue: 0,0:37:51.52,0:38:03.12,Default,,0000,0000,0000,,This is what we are doing\Nin-- the derivative of e
Dialogue: 0,0:38:03.12,0:38:06.68,Default,,0000,0000,0000,,to the x was what?
Dialogue: 0,0:38:06.68,0:38:10.35,Default,,0000,0000,0000,,What was the derivative\Nof natural log of this?
Dialogue: 0,0:38:10.35,0:38:11.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:38:11.74,0:38:12.36,Default,,0000,0000,0000,,PROFESSOR: 1/x.
Dialogue: 0,0:38:12.36,0:38:15.12,Default,,0000,0000,0000,,
Dialogue: 0,0:38:15.12,0:38:18.84,Default,,0000,0000,0000,,Now, when you have an\Narbitrary function f
Dialogue: 0,0:38:18.84,0:38:24.02,Default,,0000,0000,0000,,and you compose with inverse,\Nwhat is it by definition?
Dialogue: 0,0:38:24.02,0:38:25.16,Default,,0000,0000,0000,,X equals x.
Dialogue: 0,0:38:25.16,0:38:26.53,Default,,0000,0000,0000,,So this is the\Nidentity function.
Dialogue: 0,0:38:26.53,0:38:32.20,Default,,0000,0000,0000,,
Dialogue: 0,0:38:32.20,0:38:35.07,Default,,0000,0000,0000,,Chain rule tells\Nyou, wait a minute.
Dialogue: 0,0:38:35.07,0:38:41.20,Default,,0000,0000,0000,,Chain rule tell you how\Nto prime the whole thing.
Dialogue: 0,0:38:41.20,0:38:45.58,Default,,0000,0000,0000,,So I prime-- what\Nis this animal?
Dialogue: 0,0:38:45.58,0:38:49.63,Default,,0000,0000,0000,,F of f inverse of x is the\Ncomposition of functions,
Dialogue: 0,0:38:49.63,0:38:50.26,Default,,0000,0000,0000,,right?
Dialogue: 0,0:38:50.26,0:38:56.42,Default,,0000,0000,0000,,So apply chain rule to f of\Nf inverse of x, all prime,
Dialogue: 0,0:38:56.42,0:38:57.67,Default,,0000,0000,0000,,with respect to x.
Dialogue: 0,0:38:57.67,0:39:00.60,Default,,0000,0000,0000,,What is x prime\Nwith respect to x?
Dialogue: 0,0:39:00.60,0:39:03.36,Default,,0000,0000,0000,,x prime with respect to x is 1.
Dialogue: 0,0:39:03.36,0:39:05.16,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:39:05.16,0:39:05.82,Default,,0000,0000,0000,,Chain rule.
Dialogue: 0,0:39:05.82,0:39:07.22,Default,,0000,0000,0000,,What does the chain rule say?
Dialogue: 0,0:39:07.22,0:39:11.48,Default,,0000,0000,0000,,Chain rule says f\Nprime of f inverse
Dialogue: 0,0:39:11.48,0:39:16.85,Default,,0000,0000,0000,,of x times a function on\Nthe outside prime first.
Dialogue: 0,0:39:16.85,0:39:21.18,Default,,0000,0000,0000,,We go from the outside to the\Ninside one step at a time.
Dialogue: 0,0:39:21.18,0:39:24.48,Default,,0000,0000,0000,,Derivative of the guy--\Nyou cover f with your hand.
Dialogue: 0,0:39:24.48,0:39:27.14,Default,,0000,0000,0000,,Derivative of the guy inside,\Nthe core function inside,
Dialogue: 0,0:39:27.14,0:39:33.02,Default,,0000,0000,0000,,that will simply be f inverse\Nof x prime with respect to x
Dialogue: 0,0:39:33.02,0:39:35.06,Default,,0000,0000,0000,,equals 1.
Dialogue: 0,0:39:35.06,0:39:37.02,Default,,0000,0000,0000,,That was x prime.
Dialogue: 0,0:39:37.02,0:39:41.97,Default,,0000,0000,0000,,So the derivative of the\Ninverse function-- all right.
Dialogue: 0,0:39:41.97,0:39:51.80,Default,,0000,0000,0000,,f inverse prime is 1 over\Nf prime of f inverse of x.
Dialogue: 0,0:39:51.80,0:39:56.18,Default,,0000,0000,0000,,
Dialogue: 0,0:39:56.18,0:39:59.52,Default,,0000,0000,0000,,So if you think of\Nthis being the y,
Dialogue: 0,0:39:59.52,0:40:08.98,Default,,0000,0000,0000,,you have f inverse prime at\Ny equals 1 over-- well, yeah.
Dialogue: 0,0:40:08.98,0:40:16.23,Default,,0000,0000,0000,,If you put it at x, it's\Nf prime of f inverse of x.
Dialogue: 0,0:40:16.23,0:40:20.42,Default,,0000,0000,0000,,Because the f\Ninverse-- this is x.
Dialogue: 0,0:40:20.42,0:40:22.30,Default,,0000,0000,0000,,This is f of x.
Dialogue: 0,0:40:22.30,0:40:24.65,Default,,0000,0000,0000,,This is the y [INAUDIBLE].
Dialogue: 0,0:40:24.65,0:40:28.62,Default,,0000,0000,0000,,When you have f inverse,\Nx is the image of y.
Dialogue: 0,0:40:28.62,0:40:30.53,Default,,0000,0000,0000,,So f inverse has an input.
Dialogue: 0,0:40:30.53,0:40:33.27,Default,,0000,0000,0000,,
Dialogue: 0,0:40:33.27,0:40:35.15,Default,,0000,0000,0000,,How is this called?
Dialogue: 0,0:40:35.15,0:40:37.24,Default,,0000,0000,0000,,In the domain of f inverse.
Dialogue: 0,0:40:37.24,0:40:40.04,Default,,0000,0000,0000,,That means, who is\Nin the domain of f?
Dialogue: 0,0:40:40.04,0:40:41.23,Default,,0000,0000,0000,,f inverse of x.
Dialogue: 0,0:40:41.23,0:40:43.44,Default,,0000,0000,0000,,So one is x, one is y.
Dialogue: 0,0:40:43.44,0:40:47.12,Default,,0000,0000,0000,,
Dialogue: 0,0:40:47.12,0:40:52.25,Default,,0000,0000,0000,,So keep that in mind that when\Nyou have to invert a function,
Dialogue: 0,0:40:52.25,0:40:53.06,Default,,0000,0000,0000,,what do you do?
Dialogue: 0,0:40:53.06,0:41:02.07,Default,,0000,0000,0000,,You say 1 over the\Nderivative of the initial--
Dialogue: 0,0:41:02.07,0:41:05.50,Default,,0000,0000,0000,,so the derivative of\Nthe inverse function
Dialogue: 0,0:41:05.50,0:41:09.92,Default,,0000,0000,0000,,is 1 over derivative\Nof our initial function
Dialogue: 0,0:41:09.92,0:41:13.64,Default,,0000,0000,0000,,at the corresponding point.
Dialogue: 0,0:41:13.64,0:41:19.06,Default,,0000,0000,0000,,This is how you did the\Nderivative for the Calc 1
Dialogue: 0,0:41:19.06,0:41:20.89,Default,,0000,0000,0000,,people.
Dialogue: 0,0:41:20.89,0:41:21.64,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:41:21.64,0:41:34.49,Default,,0000,0000,0000,,
Dialogue: 0,0:41:34.49,0:41:38.07,Default,,0000,0000,0000,,So how do I apply that formula?
Dialogue: 0,0:41:38.07,0:41:40.95,Default,,0000,0000,0000,,Well, I have two functions here.
Dialogue: 0,0:41:40.95,0:41:48.58,Default,,0000,0000,0000,,One is e to the x and\None is natural log of x.
Dialogue: 0,0:41:48.58,0:41:50.50,Default,,0000,0000,0000,,How do I know they are\Ninverse to one another?
Dialogue: 0,0:41:50.50,0:41:54.64,Default,,0000,0000,0000,,Their graphs should be\Nsymmetric with respect to the?
Dialogue: 0,0:41:54.64,0:41:56.31,Default,,0000,0000,0000,,STUDENT: y equals x.
Dialogue: 0,0:41:56.31,0:41:59.15,Default,,0000,0000,0000,,PROFESSOR: With respect\Nto the first [INAUDIBLE].
Dialogue: 0,0:41:59.15,0:42:02.94,Default,,0000,0000,0000,,
Dialogue: 0,0:42:02.94,0:42:09.37,Default,,0000,0000,0000,,Assume that f of\Nx is e to the x.
Dialogue: 0,0:42:09.37,0:42:11.19,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:42:11.19,0:42:18.55,Default,,0000,0000,0000,,f inverse of-- well,\Nlet's say f inverse of y.
Dialogue: 0,0:42:18.55,0:42:22.53,Default,,0000,0000,0000,,That would be natural\Nlog of y, right?
Dialogue: 0,0:42:22.53,0:42:28.51,Default,,0000,0000,0000,,
Dialogue: 0,0:42:28.51,0:42:35.24,Default,,0000,0000,0000,,So what if you put\Nhere, what is f inverse?
Dialogue: 0,0:42:35.24,0:42:38.08,Default,,0000,0000,0000,,
Dialogue: 0,0:42:38.08,0:42:39.83,Default,,0000,0000,0000,,Natural log.
Dialogue: 0,0:42:39.83,0:42:43.86,Default,,0000,0000,0000,,Let's say a simple way to\Nwrite this, simple division.
Dialogue: 0,0:42:43.86,0:42:48.24,Default,,0000,0000,0000,,
Dialogue: 0,0:42:48.24,0:42:51.96,Default,,0000,0000,0000,,According to that formula,\Nhow would you do the math?
Dialogue: 0,0:42:51.96,0:43:01.98,Default,,0000,0000,0000,,You go f inverse prime of x must\Nbe 1 over the derivative of f
Dialogue: 0,0:43:01.98,0:43:06.88,Default,,0000,0000,0000,,with respect of f inverse x.
Dialogue: 0,0:43:06.88,0:43:09.52,Default,,0000,0000,0000,,So you go, wait a minute.
Dialogue: 0,0:43:09.52,0:43:12.24,Default,,0000,0000,0000,,OK, who is f inverse of x?
Dialogue: 0,0:43:12.24,0:43:20.69,Default,,0000,0000,0000,,
Dialogue: 0,0:43:20.69,0:43:24.70,Default,,0000,0000,0000,,Sorry, if f of x is e to the\Nx, who is f inverse of x?
Dialogue: 0,0:43:24.70,0:43:31.50,Default,,0000,0000,0000,,
Dialogue: 0,0:43:31.50,0:43:34.11,Default,,0000,0000,0000,,You want me to change a letter?
Dialogue: 0,0:43:34.11,0:43:36.24,Default,,0000,0000,0000,,I can put a y here.
Dialogue: 0,0:43:36.24,0:43:39.03,Default,,0000,0000,0000,,But in any case, I want to\Nconvince you that this is 1/x.
Dialogue: 0,0:43:39.03,0:43:42.18,Default,,0000,0000,0000,,
Dialogue: 0,0:43:42.18,0:43:43.82,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:43:43.82,0:43:47.49,Default,,0000,0000,0000,,Because f prime is e to the x.
Dialogue: 0,0:43:47.49,0:43:53.06,Default,,0000,0000,0000,,This is going to be e\Nto the f inverse of x,
Dialogue: 0,0:43:53.06,0:43:58.20,Default,,0000,0000,0000,,which is e to the natural\Nlog of x, which is x.
Dialogue: 0,0:43:58.20,0:44:01.09,Default,,0000,0000,0000,,That's why I have x here.
Dialogue: 0,0:44:01.09,0:44:05.46,Default,,0000,0000,0000,,So again, if f of x\Nequals e to the x, then f
Dialogue: 0,0:44:05.46,0:44:11.03,Default,,0000,0000,0000,,inverse of x is the\Nnational log of x.
Dialogue: 0,0:44:11.03,0:44:13.21,Default,,0000,0000,0000,,By this formula,\Nyou know that you
Dialogue: 0,0:44:13.21,0:44:19.15,Default,,0000,0000,0000,,have to compute natural\Nlog-- this is f inverse.
Dialogue: 0,0:44:19.15,0:44:22.54,Default,,0000,0000,0000,,Natural log of x prime, right?
Dialogue: 0,0:44:22.54,0:44:24.78,Default,,0000,0000,0000,,What is this by that formula?
Dialogue: 0,0:44:24.78,0:44:31.24,Default,,0000,0000,0000,,1 over the derivative of f prime\Ncomputed at f inverse of x.
Dialogue: 0,0:44:31.24,0:44:33.52,Default,,0000,0000,0000,,Now, who is f inverse of x?
Dialogue: 0,0:44:33.52,0:44:35.06,Default,,0000,0000,0000,,f inverse of x is\Nnatural log of x.
Dialogue: 0,0:44:35.06,0:44:36.65,Default,,0000,0000,0000,,So again, let me write.
Dialogue: 0,0:44:36.65,0:44:41.40,Default,,0000,0000,0000,,All this guy here in the orange\Nthing, this [INAUDIBLE] f
Dialogue: 0,0:44:41.40,0:44:44.92,Default,,0000,0000,0000,,inverse of x is ln x.
Dialogue: 0,0:44:44.92,0:44:47.74,Default,,0000,0000,0000,,Who is f prime?
Dialogue: 0,0:44:47.74,0:44:50.63,Default,,0000,0000,0000,,f prime of x is e to the x.
Dialogue: 0,0:44:50.63,0:44:53.39,Default,,0000,0000,0000,,So f prime of\Nnatural log of x will
Dialogue: 0,0:44:53.39,0:44:56.03,Default,,0000,0000,0000,,be e to the natural log of x.
Dialogue: 0,0:44:56.03,0:44:59.50,Default,,0000,0000,0000,,Applied to natural\Nlog of x, which is x.
Dialogue: 0,0:44:59.50,0:45:02.74,Default,,0000,0000,0000,,So you got it.
Dialogue: 0,0:45:02.74,0:45:04.04,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:45:04.04,0:45:10.28,Default,,0000,0000,0000,,So remember, this formula,\Nprofessors actually avoid.
Dialogue: 0,0:45:10.28,0:45:11.30,Default,,0000,0000,0000,,They say, oh my god.
Dialogue: 0,0:45:11.30,0:45:13.74,Default,,0000,0000,0000,,My students will never\Nunderstand this composition
Dialogue: 0,0:45:13.74,0:45:15.61,Default,,0000,0000,0000,,thing, derivative.
Dialogue: 0,0:45:15.61,0:45:17.49,Default,,0000,0000,0000,,So Magdalena, I don't care.
Dialogue: 0,0:45:17.49,0:45:18.95,Default,,0000,0000,0000,,You are the\Nundergraduate director.
Dialogue: 0,0:45:18.95,0:45:21.31,Default,,0000,0000,0000,,I'll never give it like that.
Dialogue: 0,0:45:21.31,0:45:23.19,Default,,0000,0000,0000,,That's a mistake.
Dialogue: 0,0:45:23.19,0:45:25.66,Default,,0000,0000,0000,,They should show it\Nto you like that.
Dialogue: 0,0:45:25.66,0:45:31.23,Default,,0000,0000,0000,,f inverse prime of x equals\N1 over f prime of what?
Dialogue: 0,0:45:31.23,0:45:33.75,Default,,0000,0000,0000,,Of the inverse image of x.
Dialogue: 0,0:45:33.75,0:45:37.98,Default,,0000,0000,0000,,Because you act on x.
Dialogue: 0,0:45:37.98,0:45:43.59,Default,,0000,0000,0000,,So if x is acting on--\Nthis is f of x, then
Dialogue: 0,0:45:43.59,0:45:47.35,Default,,0000,0000,0000,,you have to invert by\Nacting on f of x, like this
Dialogue: 0,0:45:47.35,0:45:48.53,Default,,0000,0000,0000,,and like that.
Dialogue: 0,0:45:48.53,0:45:54.79,Default,,0000,0000,0000,,If x is in the domain of f\Ninverse, that means what?
Dialogue: 0,0:45:54.79,0:46:00.95,Default,,0000,0000,0000,,That in the domain of f, you\Nhave f inverse of x as input.
Dialogue: 0,0:46:00.95,0:46:04.43,Default,,0000,0000,0000,,So instead of giving\Nyou the formula,
Dialogue: 0,0:46:04.43,0:46:08.04,Default,,0000,0000,0000,,they just make you\Nmemorize the formulas
Dialogue: 0,0:46:08.04,0:46:12.42,Default,,0000,0000,0000,,for the inverse functions,\Nlike-- believe me,
Dialogue: 0,0:46:12.42,0:46:16.40,Default,,0000,0000,0000,,you take e to the x\N[INAUDIBLE] derivative.
Dialogue: 0,0:46:16.40,0:46:19.80,Default,,0000,0000,0000,,You take natural log,\Nit's 1/x derivative.
Dialogue: 0,0:46:19.80,0:46:23.89,Default,,0000,0000,0000,,Don't worry about the fact that\Nthey are inverse to one another
Dialogue: 0,0:46:23.89,0:46:26.64,Default,,0000,0000,0000,,and you an relate the\Nderivatives of two
Dialogue: 0,0:46:26.64,0:46:28.14,Default,,0000,0000,0000,,inverse functions.
Dialogue: 0,0:46:28.14,0:46:33.01,Default,,0000,0000,0000,,They try to stay out of trouble\Nbecause this is hard to follow.
Dialogue: 0,0:46:33.01,0:46:36.49,Default,,0000,0000,0000,,You could see that you had\N[INAUDIBLE] a little bit
Dialogue: 0,0:46:36.49,0:46:37.16,Default,,0000,0000,0000,,and concentrate.
Dialogue: 0,0:46:37.16,0:46:38.71,Default,,0000,0000,0000,,What is this woman saying?
Dialogue: 0,0:46:38.71,0:46:41.52,Default,,0000,0000,0000,,This looks hard.
Dialogue: 0,0:46:41.52,0:46:46.14,Default,,0000,0000,0000,,But it's the same process\Nthat happens in the Jacobian.
Dialogue: 0,0:46:46.14,0:46:52.62,Default,,0000,0000,0000,,So in the Jacobian of a\Nfunction of two variables.
Dialogue: 0,0:46:52.62,0:46:57.03,Default,,0000,0000,0000,,
Dialogue: 0,0:46:57.03,0:47:05.45,Default,,0000,0000,0000,,Now, remember the signed\Narea that I told you about.
Dialogue: 0,0:47:05.45,0:47:08.61,Default,,0000,0000,0000,,Signed area notion.
Dialogue: 0,0:47:08.61,0:47:09.65,Default,,0000,0000,0000,,What did we say?
Dialogue: 0,0:47:09.65,0:47:13.83,Default,,0000,0000,0000,,We said that dA is\Ndx dy, but it's not
Dialogue: 0,0:47:13.83,0:47:17.42,Default,,0000,0000,0000,,the way they explain in\Nthe book because it's
Dialogue: 0,0:47:17.42,0:47:20.65,Default,,0000,0000,0000,,more like a wedge thing.
Dialogue: 0,0:47:20.65,0:47:25.64,Default,,0000,0000,0000,,And that wedge thingy had\Na meaning in the sense
Dialogue: 0,0:47:25.64,0:47:30.86,Default,,0000,0000,0000,,that if you were to not take\Nthe exterior derivative dx dy,
Dialogue: 0,0:47:30.86,0:47:35.61,Default,,0000,0000,0000,,but take dy wedge dx,\Nit would change sign.
Dialogue: 0,0:47:35.61,0:47:37.85,Default,,0000,0000,0000,,So we thought of\Nsigned area before.
Dialogue: 0,0:47:37.85,0:47:42.16,Default,,0000,0000,0000,,When we did dx wedge\Ndy, what did we
Dialogue: 0,0:47:42.16,0:47:44.72,Default,,0000,0000,0000,,get in terms of Jacobian?
Dialogue: 0,0:47:44.72,0:47:49.28,Default,,0000,0000,0000,,We get j d r\N[INAUDIBLE] coordinates.
Dialogue: 0,0:47:49.28,0:47:52.65,Default,,0000,0000,0000,,Do you remember what this j was?
Dialogue: 0,0:47:52.65,0:47:53.54,Default,,0000,0000,0000,,STUDENT: r.
Dialogue: 0,0:47:53.54,0:47:56.14,Default,,0000,0000,0000,,PROFESSOR: Very good, r.
Dialogue: 0,0:47:56.14,0:48:00.76,Default,,0000,0000,0000,,In the case, in the simple\Ncase of Cartesian versus polar,
Dialogue: 0,0:48:00.76,0:48:06.54,Default,,0000,0000,0000,,Cartesian going to polar,\Nyou have a function f.
Dialogue: 0,0:48:06.54,0:48:09.64,Default,,0000,0000,0000,,Coming back it's called\Nthe inverse function.
Dialogue: 0,0:48:09.64,0:48:15.24,Default,,0000,0000,0000,,So I'm asking, this is the\NJacobian of which function?
Dialogue: 0,0:48:15.24,0:48:18.16,Default,,0000,0000,0000,,This is the Jacobian of\Nthe function that goes
Dialogue: 0,0:48:18.16,0:48:20.50,Default,,0000,0000,0000,,from [INAUDIBLE] theta to x, y.
Dialogue: 0,0:48:20.50,0:48:23.09,Default,,0000,0000,0000,,If I want the Jacobian\Nof the function that
Dialogue: 0,0:48:23.09,0:48:25.41,Default,,0000,0000,0000,,goes from x, y into\N[INAUDIBLE] theta,
Dialogue: 0,0:48:25.41,0:48:30.12,Default,,0000,0000,0000,,I should write--\Nwell, d r d theta
Dialogue: 0,0:48:30.12,0:48:33.89,Default,,0000,0000,0000,,will be something times dx dy.
Dialogue: 0,0:48:33.89,0:48:36.76,Default,,0000,0000,0000,,And now you understand\Nbetter what's going on.
Dialogue: 0,0:48:36.76,0:48:39.18,Default,,0000,0000,0000,,1/j.
Dialogue: 0,0:48:39.18,0:48:40.16,Default,,0000,0000,0000,,1/j.
Dialogue: 0,0:48:40.16,0:48:42.71,Default,,0000,0000,0000,,So Matthew was\Nright, in the sense
Dialogue: 0,0:48:42.71,0:48:48.39,Default,,0000,0000,0000,,that he said why are you so\Nclumsy and go ahead and compute
Dialogue: 0,0:48:48.39,0:48:50.15,Default,,0000,0000,0000,,again u, v?
Dialogue: 0,0:48:50.15,0:48:52.36,Default,,0000,0000,0000,,Express u, v in terms of x, y.
Dialogue: 0,0:48:52.36,0:48:54.71,Default,,0000,0000,0000,,You waste your time\Nand get minus a 1/2.
Dialogue: 0,0:48:54.71,0:48:55.66,Default,,0000,0000,0000,,What was that, guys?
Dialogue: 0,0:48:55.66,0:48:56.85,Default,,0000,0000,0000,,Minus 1/2.
Dialogue: 0,0:48:56.85,0:49:01.75,Default,,0000,0000,0000,,When I'm telling you that\Nfor the inverse mapping,
Dialogue: 0,0:49:01.75,0:49:05.38,Default,,0000,0000,0000,,the Jacobian you get is\Nthe inverse of a Jacobian.
Dialogue: 0,0:49:05.38,0:49:06.76,Default,,0000,0000,0000,,It's very simple.
Dialogue: 0,0:49:06.76,0:49:08.60,Default,,0000,0000,0000,,It's a very simple relationship.
Dialogue: 0,0:49:08.60,0:49:10.53,Default,,0000,0000,0000,,I could observe that.
Dialogue: 0,0:49:10.53,0:49:11.75,Default,,0000,0000,0000,,And he was right.
Dialogue: 0,0:49:11.75,0:49:19.15,Default,,0000,0000,0000,,So keep in mind that when you\Nhave Jacobian of the map where
Dialogue: 0,0:49:19.15,0:49:24.90,Default,,0000,0000,0000,,x, y are functions of u, v,\Nthis is 1 over the Jacobian
Dialogue: 0,0:49:24.90,0:49:30.04,Default,,0000,0000,0000,,where you have u, v\Nas functions of x, y.
Dialogue: 0,0:49:30.04,0:49:32.98,Default,,0000,0000,0000,,So you have inverse mapping.
Dialogue: 0,0:49:32.98,0:49:36.44,Default,,0000,0000,0000,,In Advanced Calculus, you\Nmay learn a little bit more
Dialogue: 0,0:49:36.44,0:49:37.86,Default,,0000,0000,0000,,about the inverse\Nmapping theorem.
Dialogue: 0,0:49:37.86,0:49:40.82,Default,,0000,0000,0000,,This is what I'm talking about.
Dialogue: 0,0:49:40.82,0:49:42.42,Default,,0000,0000,0000,,For the inverse\Nmapping theorem, you
Dialogue: 0,0:49:42.42,0:49:47.94,Default,,0000,0000,0000,,go, well, if the derivative\Nof these two with respect
Dialogue: 0,0:49:47.94,0:49:50.43,Default,,0000,0000,0000,,to these two are done as\Nj Jacobian, the derivative
Dialogue: 0,0:49:50.43,0:49:53.80,Default,,0000,0000,0000,,of these two with\Nrespect to these two
Dialogue: 0,0:49:53.80,0:49:58.61,Default,,0000,0000,0000,,in a Jacobian [INAUDIBLE]\Nexactly j inverse, or 1/j.
Dialogue: 0,0:49:58.61,0:50:01.70,Default,,0000,0000,0000,,j is a real number.
Dialogue: 0,0:50:01.70,0:50:05.68,Default,,0000,0000,0000,,So for a real number, whether\NI write 1/j or j inverse,
Dialogue: 0,0:50:05.68,0:50:09.05,Default,,0000,0000,0000,,it's the same.
Dialogue: 0,0:50:09.05,0:50:12.19,Default,,0000,0000,0000,,So as an application, do\Nyou have to know all this?
Dialogue: 0,0:50:12.19,0:50:13.94,Default,,0000,0000,0000,,No, you don't.
Dialogue: 0,0:50:13.94,0:50:21.48,Default,,0000,0000,0000,,But as an application, let\Nme ask you the following.
Dialogue: 0,0:50:21.48,0:50:33.71,Default,,0000,0000,0000,,
Dialogue: 0,0:50:33.71,0:50:37.69,Default,,0000,0000,0000,,Something harder than\N[INAUDIBLE] in the book.
Dialogue: 0,0:50:37.69,0:50:41.10,Default,,0000,0000,0000,,In the book, you have\Nsimple transformations.
Dialogue: 0,0:50:41.10,0:50:45.73,Default,,0000,0000,0000,,What is the Jacobian\Nof r theta-- theta, phi
Dialogue: 0,0:50:45.73,0:50:46.32,Default,,0000,0000,0000,,or phi, theta.
Dialogue: 0,0:50:46.32,0:50:47.47,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:50:47.47,0:50:49.89,Default,,0000,0000,0000,,If I swap the two, I\Nstill have the same thing.
Dialogue: 0,0:50:49.89,0:50:53.77,Default,,0000,0000,0000,,
Dialogue: 0,0:50:53.77,0:50:56.64,Default,,0000,0000,0000,,If a determinant swaps\Ntwo rows or two columns,
Dialogue: 0,0:50:56.64,0:50:59.50,Default,,0000,0000,0000,,do you guys know what happens?
Dialogue: 0,0:50:59.50,0:51:01.88,Default,,0000,0000,0000,,You took linear algebra.
Dialogue: 0,0:51:01.88,0:51:02.64,Default,,0000,0000,0000,,STUDENT: Swap.
Dialogue: 0,0:51:02.64,0:51:04.72,Default,,0000,0000,0000,,PROFESSOR: You swap two\Nrows or two columns.
Dialogue: 0,0:51:04.72,0:51:05.59,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:51:05.59,0:51:07.99,Default,,0000,0000,0000,,PROFESSOR: It's going to pick\Nup a minus sign, very good.
Dialogue: 0,0:51:07.99,0:51:10.39,Default,,0000,0000,0000,,But only three people in\Nthis class figured it out.
Dialogue: 0,0:51:10.39,0:51:13.27,Default,,0000,0000,0000,,
Dialogue: 0,0:51:13.27,0:51:14.06,Default,,0000,0000,0000,,How shall I denote?
Dialogue: 0,0:51:14.06,0:51:16.89,Default,,0000,0000,0000,,Not j, but the notation\Nwas [INAUDIBLE].
Dialogue: 0,0:51:16.89,0:51:19.74,Default,,0000,0000,0000,,And this is j.
Dialogue: 0,0:51:19.74,0:51:23.83,Default,,0000,0000,0000,,So [INAUDIBLE] phi theta\Nover [INAUDIBLE] x, y, z.
Dialogue: 0,0:51:23.83,0:51:25.66,Default,,0000,0000,0000,,How do you compute them?
Dialogue: 0,0:51:25.66,0:51:28.67,Default,,0000,0000,0000,,You say, no, I'm not\Ngoing to compute it
Dialogue: 0,0:51:28.67,0:51:32.53,Default,,0000,0000,0000,,by hand because until tomorrow\NI'm not going to finish it.
Dialogue: 0,0:51:32.53,0:51:34.81,Default,,0000,0000,0000,,STUDENT: Does that\Nneed a 3 by 3 matrix?
Dialogue: 0,0:51:34.81,0:51:36.64,Default,,0000,0000,0000,,PROFESSOR: It's a determinant.
Dialogue: 0,0:51:36.64,0:51:39.75,Default,,0000,0000,0000,,So when you were to\Nwrite this, you're
Dialogue: 0,0:51:39.75,0:51:45.80,Default,,0000,0000,0000,,not going to do it because it's\Na killer for somebody to work
Dialogue: 0,0:51:45.80,0:51:49.31,Default,,0000,0000,0000,,like that in spherical\Ncoordinates with only
Dialogue: 0,0:51:49.31,0:51:50.93,Default,,0000,0000,0000,,those inverse functions.
Dialogue: 0,0:51:50.93,0:51:56.76,Default,,0000,0000,0000,,Do you remember as a review\Nwhat spherical coordinates were?
Dialogue: 0,0:51:56.76,0:52:00.89,Default,,0000,0000,0000,,x, y, z versus r, theta, phi.
Dialogue: 0,0:52:00.89,0:52:02.08,Default,,0000,0000,0000,,We reviewed that.
Dialogue: 0,0:52:02.08,0:52:03.29,Default,,0000,0000,0000,,Theta was the longitude.
Dialogue: 0,0:52:03.29,0:52:05.71,Default,,0000,0000,0000,,Phi was the latitude\Nfrom the North Pole.
Dialogue: 0,0:52:05.71,0:52:08.15,Default,,0000,0000,0000,,So x was-- who remembers that?
Dialogue: 0,0:52:08.15,0:52:08.98,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:52:08.98,0:52:13.96,Default,,0000,0000,0000,,
Dialogue: 0,0:52:13.96,0:52:18.96,Default,,0000,0000,0000,,PROFESSOR: Cosine theta\Nr sine phi sine theta.
Dialogue: 0,0:52:18.96,0:52:21.13,Default,,0000,0000,0000,,And z was the adjacent guy.
Dialogue: 0,0:52:21.13,0:52:24.12,Default,,0000,0000,0000,,Remember, this was the thingy?
Dialogue: 0,0:52:24.12,0:52:27.22,Default,,0000,0000,0000,,And this was the phi.
Dialogue: 0,0:52:27.22,0:52:31.62,Default,,0000,0000,0000,,And to express x, the\Nphi was adjacent to it.
Dialogue: 0,0:52:31.62,0:52:33.08,Default,,0000,0000,0000,,And that's why you\Nhave cosine phi.
Dialogue: 0,0:52:33.08,0:52:36.14,Default,,0000,0000,0000,,
Dialogue: 0,0:52:36.14,0:52:41.31,Default,,0000,0000,0000,,It's a killer if somebody wants\Nto pull out the r, phi, theta.
Dialogue: 0,0:52:41.31,0:52:43.14,Default,,0000,0000,0000,,First of all, r will be easy.
Dialogue: 0,0:52:43.14,0:52:46.16,Default,,0000,0000,0000,,But the other ones are a\Nlittle bit of a headache.
Dialogue: 0,0:52:46.16,0:52:47.68,Default,,0000,0000,0000,,And with all those\Nbig functions,
Dialogue: 0,0:52:47.68,0:52:51.19,Default,,0000,0000,0000,,you would waste a lot of time\Nto compute the determinant.
Dialogue: 0,0:52:51.19,0:52:52.15,Default,,0000,0000,0000,,What do you do?
Dialogue: 0,0:52:52.15,0:52:55.07,Default,,0000,0000,0000,,You say, well, didn't\Nyou say that if I
Dialogue: 0,0:52:55.07,0:52:58.45,Default,,0000,0000,0000,,take the inverse mapping,\Nthe Jacobian would be
Dialogue: 0,0:52:58.45,0:53:01.17,Default,,0000,0000,0000,,1 over the original Jacobian?
Dialogue: 0,0:53:01.17,0:53:03.96,Default,,0000,0000,0000,,Yes, I just said that.
Dialogue: 0,0:53:03.96,0:53:09.01,Default,,0000,0000,0000,,So go ahead and remember what\Nthe original Jacobian was
Dialogue: 0,0:53:09.01,0:53:13.43,Default,,0000,0000,0000,,and leave us alone\Nyou're going to say.
Dialogue: 0,0:53:13.43,0:53:15.94,Default,,0000,0000,0000,,And you're right.
Dialogue: 0,0:53:15.94,0:53:21.66,Default,,0000,0000,0000,,What was that I just said\Nthe other Jacobian was?
Dialogue: 0,0:53:21.66,0:53:22.65,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:53:22.65,0:53:24.63,Default,,0000,0000,0000,,PROFESSOR: You told me.
Dialogue: 0,0:53:24.63,0:53:26.83,Default,,0000,0000,0000,,I forgot it already.
Dialogue: 0,0:53:26.83,0:53:29.33,Default,,0000,0000,0000,,r squared sine phi, right?
Dialogue: 0,0:53:29.33,0:53:33.32,Default,,0000,0000,0000,,So if somebody's asking\Nyou to solve this problem,
Dialogue: 0,0:53:33.32,0:53:36.60,Default,,0000,0000,0000,,you don't need to\Nwrite out anything.
Dialogue: 0,0:53:36.60,0:53:38.08,Default,,0000,0000,0000,,Just 1 over [INAUDIBLE].
Dialogue: 0,0:53:38.08,0:53:39.96,Default,,0000,0000,0000,,I'm done.
Dialogue: 0,0:53:39.96,0:53:41.55,Default,,0000,0000,0000,,But I'm not going to ask you.
Dialogue: 0,0:53:41.55,0:53:44.93,Default,,0000,0000,0000,,Of course, I saw\Nthis problem exactly.
Dialogue: 0,0:53:44.93,0:53:47.90,Default,,0000,0000,0000,,Find the Jacobian of\Nthe inverse mapping
Dialogue: 0,0:53:47.90,0:53:49.83,Default,,0000,0000,0000,,for the spherical coordinates.
Dialogue: 0,0:53:49.83,0:53:54.22,Default,,0000,0000,0000,,That was given at Princeton\Nin Advanced Calculus.
Dialogue: 0,0:53:54.22,0:53:59.28,Default,,0000,0000,0000,,There were three variables, and\Nthen there was a generalization
Dialogue: 0,0:53:59.28,0:54:00.88,Default,,0000,0000,0000,,to [INAUDIBLE] variables.
Dialogue: 0,0:54:00.88,0:54:02.66,Default,,0000,0000,0000,,But based on this\Nat Princeton, I'm
Dialogue: 0,0:54:02.66,0:54:04.35,Default,,0000,0000,0000,,not going to give you\Nanything like that
Dialogue: 0,0:54:04.35,0:54:05.42,Default,,0000,0000,0000,,to compute in the exam.
Dialogue: 0,0:54:05.42,0:54:08.34,Default,,0000,0000,0000,,
Dialogue: 0,0:54:08.34,0:54:12.39,Default,,0000,0000,0000,,And I just expect that you\Nknow your basics about how
Dialogue: 0,0:54:12.39,0:54:15.74,Default,,0000,0000,0000,,to compute triple integrals.
Dialogue: 0,0:54:15.74,0:54:20.42,Default,,0000,0000,0000,,Use the Jacobians and\Nbe successful with it.
Dialogue: 0,0:54:20.42,0:54:25.32,Default,,0000,0000,0000,,
Dialogue: 0,0:54:25.32,0:54:28.63,Default,,0000,0000,0000,,Let's do one last\Nproblem about the review.
Dialogue: 0,0:54:28.63,0:54:30.38,Default,,0000,0000,0000,,Although, it's not\Nin the midterm,
Dialogue: 0,0:54:30.38,0:54:36.53,Default,,0000,0000,0000,,but I would like to-- I'd\Nlike to see how you solve it.
Dialogue: 0,0:54:36.53,0:55:01.27,Default,,0000,0000,0000,,
Dialogue: 0,0:55:01.27,0:55:05.76,Default,,0000,0000,0000,,A student from another\Nclass, Calc 3, came to me.
Dialogue: 0,0:55:05.76,0:55:10.24,Default,,0000,0000,0000,,And I was hesitant about even\Nhelping him on the homework
Dialogue: 0,0:55:10.24,0:55:16.07,Default,,0000,0000,0000,,because we're not supposed\Nto help our college students.
Dialogue: 0,0:55:16.07,0:55:19.04,Default,,0000,0000,0000,,So I told him, did you go\Nto the tutoring center?
Dialogue: 0,0:55:19.04,0:55:22.49,Default,,0000,0000,0000,,And he said yes, but they\Ncouldn't help him much.
Dialogue: 0,0:55:22.49,0:55:24.47,Default,,0000,0000,0000,,So I said, OK.
Dialogue: 0,0:55:24.47,0:55:27.40,Default,,0000,0000,0000,,So let me see the problem.
Dialogue: 0,0:55:27.40,0:55:28.94,Default,,0000,0000,0000,,He showed me the\Nproblem and I wanted
Dialogue: 0,0:55:28.94,0:55:32.59,Default,,0000,0000,0000,,to talk about this\Nproblem with you.
Dialogue: 0,0:55:32.59,0:55:36.91,Default,,0000,0000,0000,,This is not a hard problem, OK?
Dialogue: 0,0:55:36.91,0:55:41.57,Default,,0000,0000,0000,,You just have to see\Nwhat this is about.
Dialogue: 0,0:55:41.57,0:55:43.05,Default,,0000,0000,0000,,Understand what this is about.
Dialogue: 0,0:55:43.05,0:55:49.50,Default,,0000,0000,0000,,
Dialogue: 0,0:55:49.50,0:55:56.20,Default,,0000,0000,0000,,So you have the z equals x\Nsquared plus y squared, which
Dialogue: 0,0:55:56.20,0:55:59.92,Default,,0000,0000,0000,,is the [INAUDIBLE].
Dialogue: 0,0:55:59.92,0:56:01.41,Default,,0000,0000,0000,,Sorry about my typos.
Dialogue: 0,0:56:01.41,0:56:07.85,Default,,0000,0000,0000,,
Dialogue: 0,0:56:07.85,0:56:10.68,Default,,0000,0000,0000,,We didn't write this\Nproblem in the book.
Dialogue: 0,0:56:10.68,0:56:16.33,Default,,0000,0000,0000,,So I suspect that his instructor\Ncame up with this problem.
Dialogue: 0,0:56:16.33,0:56:19.58,Default,,0000,0000,0000,,This is a cone.
Dialogue: 0,0:56:19.58,0:56:21.44,Default,,0000,0000,0000,,We only look at\Nthe upper halves.
Dialogue: 0,0:56:21.44,0:56:24.03,Default,,0000,0000,0000,,
Dialogue: 0,0:56:24.03,0:56:25.56,Default,,0000,0000,0000,,Do these surfaces intersect?
Dialogue: 0,0:56:25.56,0:56:34.43,Default,,0000,0000,0000,,
Dialogue: 0,0:56:34.43,0:56:43.48,Default,,0000,0000,0000,,Draw the body between\Nthem if the case.
Dialogue: 0,0:56:43.48,0:56:48.98,Default,,0000,0000,0000,,
Dialogue: 0,0:56:48.98,0:56:51.39,Default,,0000,0000,0000,,And compute the\Nvolume of that body.
Dialogue: 0,0:56:51.39,0:57:01.53,Default,,0000,0000,0000,,
Dialogue: 0,0:57:01.53,0:57:03.48,Default,,0000,0000,0000,,And what do you think\Nmy reaction was?
Dialogue: 0,0:57:03.48,0:57:05.17,Default,,0000,0000,0000,,Oh, this is a piece of cake.
Dialogue: 0,0:57:05.17,0:57:07.29,Default,,0000,0000,0000,,And it is a piece of cake.
Dialogue: 0,0:57:07.29,0:57:10.53,Default,,0000,0000,0000,,But you need to\Nlearn Calc 3 first
Dialogue: 0,0:57:10.53,0:57:14.98,Default,,0000,0000,0000,,in order to help other\Npeople do Calc 3 problems.
Dialogue: 0,0:57:14.98,0:57:16.61,Default,,0000,0000,0000,,Especially if they\Nare not in the book.
Dialogue: 0,0:57:16.61,0:57:21.91,Default,,0000,0000,0000,,So one has to have a very good\Nunderstanding of the theory
Dialogue: 0,0:57:21.91,0:57:26.31,Default,,0000,0000,0000,,and of geometry, analytic\Ngeometry, and conics
Dialogue: 0,0:57:26.31,0:57:32.25,Default,,0000,0000,0000,,before they move onto\Ntriple integrals and so on.
Dialogue: 0,0:57:32.25,0:57:36.47,Default,,0000,0000,0000,,Can you imagine these with\Nthe eyes of your imagination?
Dialogue: 0,0:57:36.47,0:57:37.74,Default,,0000,0000,0000,,Can we draw them?
Dialogue: 0,0:57:37.74,0:57:38.79,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:57:38.79,0:57:42.65,Default,,0000,0000,0000,,We better draw them because\Nthey are not nasty to draw.
Dialogue: 0,0:57:42.65,0:57:46.86,Default,,0000,0000,0000,,Of course this looks\Nlike the Tower of Pisa.
Dialogue: 0,0:57:46.86,0:57:49.59,Default,,0000,0000,0000,,Let me do it again.
Dialogue: 0,0:57:49.59,0:57:50.09,Default,,0000,0000,0000,,Better.
Dialogue: 0,0:57:50.09,0:57:53.41,Default,,0000,0000,0000,,x, y, and z.
Dialogue: 0,0:57:53.41,0:57:56.03,Default,,0000,0000,0000,,And then I'll take the cone.
Dialogue: 0,0:57:56.03,0:58:01.69,Default,,0000,0000,0000,,Well, let me draw\Nthe paraboloid first.
Dialogue: 0,0:58:01.69,0:58:04.18,Default,,0000,0000,0000,,Kind of sort of.
Dialogue: 0,0:58:04.18,0:58:07.08,Default,,0000,0000,0000,,And then the cone.
Dialogue: 0,0:58:07.08,0:58:10.07,Default,,0000,0000,0000,,I hate myself when\NI cannot draw.
Dialogue: 0,0:58:10.07,0:58:14.06,Default,,0000,0000,0000,,
Dialogue: 0,0:58:14.06,0:58:20.38,Default,,0000,0000,0000,,If you were to cut, slice\Nup, it could be this.
Dialogue: 0,0:58:20.38,0:58:24.52,Default,,0000,0000,0000,,And who asked me last\Ntime, was it Alex, or Ryan,
Dialogue: 0,0:58:24.52,0:58:29.53,Default,,0000,0000,0000,,or maybe somebody else, who\Nsaid maybe we could do that even
Dialogue: 0,0:58:29.53,0:58:31.01,Default,,0000,0000,0000,,in Calc 2 by--
Dialogue: 0,0:58:31.01,0:58:31.59,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:58:31.59,0:58:32.27,Default,,0000,0000,0000,,PROFESSOR: You asked me.
Dialogue: 0,0:58:32.27,0:58:32.93,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:58:32.93,0:58:34.68,Default,,0000,0000,0000,,PROFESSOR: If you\Ntake a leaf like that
Dialogue: 0,0:58:34.68,0:58:37.34,Default,,0000,0000,0000,,and you rotate it\Naround the body,
Dialogue: 0,0:58:37.34,0:58:41.13,Default,,0000,0000,0000,,like in-- using one\Nof the two methods
Dialogue: 0,0:58:41.13,0:58:44.40,Default,,0000,0000,0000,,that you learned in Calc 2.
Dialogue: 0,0:58:44.40,0:58:45.63,Default,,0000,0000,0000,,Well, we can do that.
Dialogue: 0,0:58:45.63,0:58:49.34,Default,,0000,0000,0000,,But you see we have in Calc 3.
Dialogue: 0,0:58:49.34,0:58:52.82,Default,,0000,0000,0000,,So I would like to\Nwrite that in terms
Dialogue: 0,0:58:52.82,0:58:58.60,Default,,0000,0000,0000,,of the volume of the body\Nfaster with knowledge I have.
Dialogue: 0,0:58:58.60,0:58:59.43,Default,,0000,0000,0000,,Do they intersect?
Dialogue: 0,0:58:59.43,0:59:00.60,Default,,0000,0000,0000,,And where do they intersect?
Dialogue: 0,0:59:00.60,0:59:02.53,Default,,0000,0000,0000,,And how do I find this out?
Dialogue: 0,0:59:02.53,0:59:04.35,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:59:04.35,0:59:06.08,Default,,0000,0000,0000,,PROFESSOR: Yes.
Dialogue: 0,0:59:06.08,0:59:16.74,Default,,0000,0000,0000,,I have to make them equal and\Nsolve for z, and then the rest.
Dialogue: 0,0:59:16.74,0:59:18.59,Default,,0000,0000,0000,,How do I solve for z?
Dialogue: 0,0:59:18.59,0:59:23.73,Default,,0000,0000,0000,,Well, z equals z0 gives\Nme two possibilities.
Dialogue: 0,0:59:23.73,0:59:28.25,Default,,0000,0000,0000,,One is z equals 0\Nand 1 is z equals 1
Dialogue: 0,0:59:28.25,0:59:33.71,Default,,0000,0000,0000,,because this is the same as\Nwriting z times z minus 1
Dialogue: 0,0:59:33.71,0:59:35.06,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,0:59:35.06,0:59:36.31,Default,,0000,0000,0000,,So where do they intersect?
Dialogue: 0,0:59:36.31,0:59:37.72,Default,,0000,0000,0000,,They intersect\Nhere at the origin
Dialogue: 0,0:59:37.72,0:59:42.06,Default,,0000,0000,0000,,and they intersect\Nwhere z equals 1.
Dialogue: 0,0:59:42.06,0:59:46.98,Default,,0000,0000,0000,,And where z equals 1, I'm\Ngoing to have what circle?
Dialogue: 0,0:59:46.98,0:59:49.19,Default,,0000,0000,0000,,The unit circle.
Dialogue: 0,0:59:49.19,0:59:52.60,Default,,0000,0000,0000,,I'll draw over--\NI'll make it in red.
Dialogue: 0,0:59:52.60,0:59:58.19,Default,,0000,0000,0000,,This is x squared plus y squared\Nequals 1 at the altitude 1,
Dialogue: 0,0:59:58.19,0:59:59.75,Default,,0000,0000,0000,,z equals 1.
Dialogue: 0,0:59:59.75,1:00:02.41,Default,,0000,0000,0000,,This is the plane z equals 1.
Dialogue: 0,1:00:02.41,1:00:07.56,Default,,0000,0000,0000,,
Dialogue: 0,1:00:07.56,1:00:15.07,Default,,0000,0000,0000,,OK, so how many ways\Nto do this are there?
Dialogue: 0,1:00:15.07,1:00:21.97,Default,,0000,0000,0000,,When we were in Chapter 12,\Nwe said the triple integral
Dialogue: 0,1:00:21.97,1:00:23.56,Default,,0000,0000,0000,,will give me the volume.
Dialogue: 0,1:00:23.56,1:00:26.05,Default,,0000,0000,0000,,So the volume will\Nbe triple integral
Dialogue: 0,1:00:26.05,1:00:31.28,Default,,0000,0000,0000,,of a certain body-- of\N1 over a certain body
Dialogue: 0,1:00:31.28,1:00:40.95,Default,,0000,0000,0000,,dv, where the body is\Nthe body of revolution
Dialogue: 0,1:00:40.95,1:00:50.48,Default,,0000,0000,0000,,created by the motion\Nof-- what is this thing?
Dialogue: 0,1:00:50.48,1:00:51.48,Default,,0000,0000,0000,,What shall we call it?
Dialogue: 0,1:00:51.48,1:00:53.48,Default,,0000,0000,0000,,A wing.
Dialogue: 0,1:00:53.48,1:00:53.98,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:00:53.98,1:00:57.48,Default,,0000,0000,0000,,
Dialogue: 0,1:00:57.48,1:01:03.48,Default,,0000,0000,0000,,Domain D. No, domain D is\Nusually what's on [INAUDIBLE].
Dialogue: 0,1:01:03.48,1:01:09.48,Default,,0000,0000,0000,,
Dialogue: 0,1:01:09.48,1:01:10.48,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:01:10.48,1:01:13.98,Default,,0000,0000,0000,,STUDENT: L for leaf?
Dialogue: 0,1:01:13.98,1:01:14.98,Default,,0000,0000,0000,,PROFESSOR: L for leaf.
Dialogue: 0,1:01:14.98,1:01:15.48,Default,,0000,0000,0000,,Wonderful.
Dialogue: 0,1:01:15.48,1:01:15.98,Default,,0000,0000,0000,,I like that.
Dialogue: 0,1:01:15.98,1:01:21.48,Default,,0000,0000,0000,,L. OK.
Dialogue: 0,1:01:21.48,1:01:26.73,Default,,0000,0000,0000,,So I can write it up as\Na triple integral how?
Dialogue: 0,1:01:26.73,1:01:29.58,Default,,0000,0000,0000,,Is it easy to use it in\Nspherical coordinates?
Dialogue: 0,1:01:29.58,1:01:30.08,Default,,0000,0000,0000,,No.
Dialogue: 0,1:01:30.08,1:01:32.82,Default,,0000,0000,0000,,That's not a spherical\Ncoordinate problem.
Dialogue: 0,1:01:32.82,1:01:35.41,Default,,0000,0000,0000,,That's a cylindrical\Ncoordinate problem.
Dialogue: 0,1:01:35.41,1:01:36.18,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,1:01:36.18,1:01:38.54,Default,,0000,0000,0000,,I'm going to have to\Nthink where I live.
Dialogue: 0,1:01:38.54,1:01:44.82,Default,,0000,0000,0000,,I live above a beautiful disk,\Nwhich is the shadowy plane.
Dialogue: 0,1:01:44.82,1:01:48.07,Default,,0000,0000,0000,,And that beautiful disk\Nhas exactly radius 1.
Dialogue: 0,1:01:48.07,1:01:48.74,Default,,0000,0000,0000,,So we are lucky.
Dialogue: 0,1:01:48.74,1:01:53.33,Default,,0000,0000,0000,,That's the unit disk, x squared\Nplus y squared less than 1
Dialogue: 0,1:01:53.33,1:01:55.61,Default,,0000,0000,0000,,and greater than 0.
Dialogue: 0,1:01:55.61,1:02:00.23,Default,,0000,0000,0000,,So when I revolve, I'm\Nusing polar coordinates.
Dialogue: 0,1:02:00.23,1:02:02.83,Default,,0000,0000,0000,,And that means I'm using\Ncylindrical coordinates, which
Dialogue: 0,1:02:02.83,1:02:05.17,Default,,0000,0000,0000,,is practically the same thing.
Dialogue: 0,1:02:05.17,1:02:09.72,Default,,0000,0000,0000,,r will be between what and what?
Dialogue: 0,1:02:09.72,1:02:10.59,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:02:10.59,1:02:13.41,Default,,0000,0000,0000,,PROFESSOR: 0 to 1, very good.
Dialogue: 0,1:02:13.41,1:02:14.30,Default,,0000,0000,0000,,Theta?
Dialogue: 0,1:02:14.30,1:02:15.18,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:02:15.18,1:02:17.45,Default,,0000,0000,0000,,PROFESSOR: 0 to 2 pi.
Dialogue: 0,1:02:17.45,1:02:19.21,Default,,0000,0000,0000,,How about z?
Dialogue: 0,1:02:19.21,1:02:21.32,Default,,0000,0000,0000,,z is the z from\Ncylindrical coordinates.
Dialogue: 0,1:02:21.32,1:02:24.68,Default,,0000,0000,0000,,STUDENT: Square root x\Nsquared plus y squared--
Dialogue: 0,1:02:24.68,1:02:27.08,Default,,0000,0000,0000,,PROFESSOR: Who is on the bottom?
Dialogue: 0,1:02:27.08,1:02:29.49,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:02:29.49,1:02:34.82,Default,,0000,0000,0000,,PROFESSOR: So the z is between--\Nlet me write it in x first,
Dialogue: 0,1:02:34.82,1:02:36.71,Default,,0000,0000,0000,,and then switch to polar.
Dialogue: 0,1:02:36.71,1:02:37.61,Default,,0000,0000,0000,,Is that OK?
Dialogue: 0,1:02:37.61,1:02:38.15,Default,,0000,0000,0000,,STUDENT: Yeah
Dialogue: 0,1:02:38.15,1:02:39.02,Default,,0000,0000,0000,,PROFESSOR: All right.
Dialogue: 0,1:02:39.02,1:02:42.18,Default,,0000,0000,0000,,So what do I write on\Nthe left-hand side?
Dialogue: 0,1:02:42.18,1:02:43.23,Default,,0000,0000,0000,,I need water.
Dialogue: 0,1:02:43.23,1:02:44.10,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:02:44.10,1:02:50.82,Default,,0000,0000,0000,,
Dialogue: 0,1:02:50.82,1:02:52.05,Default,,0000,0000,0000,,PROFESSOR: Who is smaller?
Dialogue: 0,1:02:52.05,1:02:53.97,Default,,0000,0000,0000,,Who is smaller?
Dialogue: 0,1:02:53.97,1:02:56.87,Default,,0000,0000,0000,,Square root of x\Nsquared y squared or x
Dialogue: 0,1:02:56.87,1:02:59.08,Default,,0000,0000,0000,,squared plus y squared?
Dialogue: 0,1:02:59.08,1:03:00.78,Default,,0000,0000,0000,,STUDENT: Square root over.
Dialogue: 0,1:03:00.78,1:03:01.90,Default,,0000,0000,0000,,PROFESSOR: This is smaller.
Dialogue: 0,1:03:01.90,1:03:02.78,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:03:02.78,1:03:04.70,Default,,0000,0000,0000,,STUDENT: Because [INAUDIBLE].
Dialogue: 0,1:03:04.70,1:03:07.11,Default,,0000,0000,0000,,PROFESSOR: So it's less than 1.
Dialogue: 0,1:03:07.11,1:03:08.25,Default,,0000,0000,0000,,I mean, less than 1.
Dialogue: 0,1:03:08.25,1:03:09.64,Default,,0000,0000,0000,,This is less than 1.
Dialogue: 0,1:03:09.64,1:03:11.11,Default,,0000,0000,0000,,It's between 0 and 1.
Dialogue: 0,1:03:11.11,1:03:14.58,Default,,0000,0000,0000,,
Dialogue: 0,1:03:14.58,1:03:18.75,Default,,0000,0000,0000,,So I was trying to explain\Nthis to my son, but I couldn't.
Dialogue: 0,1:03:18.75,1:03:20.21,Default,,0000,0000,0000,,But he's 10.
Dialogue: 0,1:03:20.21,1:03:21.68,Default,,0000,0000,0000,,It's so hard.
Dialogue: 0,1:03:21.68,1:03:32.69,Default,,0000,0000,0000,,So I said compare square\Nroot of 0.04 with 0.04.
Dialogue: 0,1:03:32.69,1:03:35.20,Default,,0000,0000,0000,,This is smaller, obviously.
Dialogue: 0,1:03:35.20,1:03:36.85,Default,,0000,0000,0000,,This is 0.2.
Dialogue: 0,1:03:36.85,1:03:37.71,Default,,0000,0000,0000,,He can understand.
Dialogue: 0,1:03:37.71,1:03:40.72,Default,,0000,0000,0000,,
Dialogue: 0,1:03:40.72,1:03:43.29,Default,,0000,0000,0000,,So this is what we're doing.
Dialogue: 0,1:03:43.29,1:03:47.34,Default,,0000,0000,0000,,We are saying that\Nthis is x squared
Dialogue: 0,1:03:47.34,1:03:52.24,Default,,0000,0000,0000,,plus y squared, the round\Nthing on the bottom.
Dialogue: 0,1:03:52.24,1:04:01.27,Default,,0000,0000,0000,,And this is going to be on the\Ntop, square root of x squared
Dialogue: 0,1:04:01.27,1:04:03.38,Default,,0000,0000,0000,,plus y squared\Nfrom the cylinder,
Dialogue: 0,1:04:03.38,1:04:07.12,Default,,0000,0000,0000,,from the cone-- sorry\Nguys, the upper half.
Dialogue: 0,1:04:07.12,1:04:08.79,Default,,0000,0000,0000,,Because I only work\Nwith the upper half.
Dialogue: 0,1:04:08.79,1:04:12.32,Default,,0000,0000,0000,,Everything is about\Nthe sea level.
Dialogue: 0,1:04:12.32,1:04:15.42,Default,,0000,0000,0000,,Good, now let's write\Nout the whole thing.
Dialogue: 0,1:04:15.42,1:04:18.90,Default,,0000,0000,0000,,So I have integral from\Nthe polar coordinates,
Dialogue: 0,1:04:18.90,1:04:19.97,Default,,0000,0000,0000,,from what to what?
Dialogue: 0,1:04:19.97,1:04:23.32,Default,,0000,0000,0000,,
Dialogue: 0,1:04:23.32,1:04:24.46,Default,,0000,0000,0000,,STUDENT: r squared to r.
Dialogue: 0,1:04:24.46,1:04:30.63,Default,,0000,0000,0000,,PROFESSOR: r squared to\Nr, 0 to 1, 0 to 2 pi.
Dialogue: 0,1:04:30.63,1:04:35.12,Default,,0000,0000,0000,,So the order of integration\Nwould be dz dr d theta.
Dialogue: 0,1:04:35.12,1:04:38.19,Default,,0000,0000,0000,,
Dialogue: 0,1:04:38.19,1:04:40.38,Default,,0000,0000,0000,,And what's inside here?
Dialogue: 0,1:04:40.38,1:04:42.98,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:04:42.98,1:04:43.80,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,1:04:43.80,1:04:44.43,Default,,0000,0000,0000,,STUDENT: r.
Dialogue: 0,1:04:44.43,1:04:47.55,Default,,0000,0000,0000,,PROFESSOR: r,\Nexcellent, r-- why r?
Dialogue: 0,1:04:47.55,1:04:49.90,Default,,0000,0000,0000,,Because 1 was 1.
Dialogue: 0,1:04:49.90,1:04:58.65,Default,,0000,0000,0000,,But dv is Jacobian times dr\Nd theta dz-- dz, dr, d theta.
Dialogue: 0,1:04:58.65,1:05:02.10,Default,,0000,0000,0000,,So this is going to be the r\Nfrom the change of coordinates,
Dialogue: 0,1:05:02.10,1:05:03.92,Default,,0000,0000,0000,,the Jacobian.
Dialogue: 0,1:05:03.92,1:05:05.40,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,1:05:05.40,1:05:06.39,Default,,0000,0000,0000,,Well, let's do it.
Dialogue: 0,1:05:06.39,1:05:08.36,Default,,0000,0000,0000,,Come on, this shouldn't be hard.
Dialogue: 0,1:05:08.36,1:05:13.85,Default,,0000,0000,0000,,We can even separate\Nthe functions.
Dialogue: 0,1:05:13.85,1:05:19.59,Default,,0000,0000,0000,,And I got you some tricks.
Dialogue: 0,1:05:19.59,1:05:21.89,Default,,0000,0000,0000,,The first one we\Nhave to work it out.
Dialogue: 0,1:05:21.89,1:05:22.89,Default,,0000,0000,0000,,We have no other choice.
Dialogue: 0,1:05:22.89,1:05:27.91,Default,,0000,0000,0000,,So I'm going to have\Nthe integral from 0
Dialogue: 0,1:05:27.91,1:05:32.06,Default,,0000,0000,0000,,to 2 pi, integral from 0 to 1.
Dialogue: 0,1:05:32.06,1:05:33.31,Default,,0000,0000,0000,,And then I go what?
Dialogue: 0,1:05:33.31,1:05:39.73,Default,,0000,0000,0000,,I go integral of what you see\Nwith z, the z between r and r
Dialogue: 0,1:05:39.73,1:05:46.64,Default,,0000,0000,0000,,squared times r dr d theta.
Dialogue: 0,1:05:46.64,1:05:48.38,Default,,0000,0000,0000,,Who is going on my nerves?
Dialogue: 0,1:05:48.38,1:05:49.52,Default,,0000,0000,0000,,Not you guys.
Dialogue: 0,1:05:49.52,1:05:53.84,Default,,0000,0000,0000,,Here, there is a guy that\Ngoes on my nerves-- the theta.
Dialogue: 0,1:05:53.84,1:05:56.70,Default,,0000,0000,0000,,I can get rid of him, and I\Nsay, I don't need the theta.
Dialogue: 0,1:05:56.70,1:05:58.66,Default,,0000,0000,0000,,I've got things\Nto do with the r.
Dialogue: 0,1:05:58.66,1:06:05.81,Default,,0000,0000,0000,,So I go 2 pi, which is the\Nintegral from 0 to 2 pi of 1
Dialogue: 0,1:06:05.81,1:06:06.52,Default,,0000,0000,0000,,d theta.
Dialogue: 0,1:06:06.52,1:06:08.71,Default,,0000,0000,0000,,2 pi goes out.
Dialogue: 0,1:06:08.71,1:06:14.79,Default,,0000,0000,0000,,Now 2 pi times integral\Nfrom 0 to 1 of what?
Dialogue: 0,1:06:14.79,1:06:17.49,Default,,0000,0000,0000,,What's the simplest\Nway to write it?
Dialogue: 0,1:06:17.49,1:06:20.35,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:06:20.35,1:06:22.30,Default,,0000,0000,0000,,PROFESSOR: r squared in the end.
Dialogue: 0,1:06:22.30,1:06:23.98,Default,,0000,0000,0000,,I mean, I do the whole\Nthing in the end.
Dialogue: 0,1:06:23.98,1:06:31.34,Default,,0000,0000,0000,,I have r squared minus\Nr cubed, right guys?
Dialogue: 0,1:06:31.34,1:06:33.11,Default,,0000,0000,0000,,Are you with me?
Dialogue: 0,1:06:33.11,1:06:36.18,Default,,0000,0000,0000,,dr, so this is when I did it.
Dialogue: 0,1:06:36.18,1:06:40.32,Default,,0000,0000,0000,,But I didn't do the\Nanti-derivative, not yet.
Dialogue: 0,1:06:40.32,1:06:42.21,Default,,0000,0000,0000,,I did not apply the fundamental.
Dialogue: 0,1:06:42.21,1:06:46.67,Default,,0000,0000,0000,,Now you apply the\Nfundamental [INAUDIBLE]
Dialogue: 0,1:06:46.67,1:06:48.20,Default,,0000,0000,0000,,and tell me what you get.
Dialogue: 0,1:06:48.20,1:06:50.52,Default,,0000,0000,0000,,What is this?
Dialogue: 0,1:06:50.52,1:06:53.78,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] 1/12.
Dialogue: 0,1:06:53.78,1:06:58.56,Default,,0000,0000,0000,,PROFESSOR: 1/12, that's very\Ngood-- r cubed over 3 minus r
Dialogue: 0,1:06:58.56,1:07:04.53,Default,,0000,0000,0000,,to the 4 over 4, 1/3 minus\N1/4, 1/12, very good.
Dialogue: 0,1:07:04.53,1:07:09.43,Default,,0000,0000,0000,,So you have 2 pi times\N1/12 equals pi over 6.
Dialogue: 0,1:07:09.43,1:07:11.92,Default,,0000,0000,0000,,Thank god, we got it.
Dialogue: 0,1:07:11.92,1:07:12.69,Default,,0000,0000,0000,,Was it hard?
Dialogue: 0,1:07:12.69,1:07:15.46,Default,,0000,0000,0000,,
Dialogue: 0,1:07:15.46,1:07:17.98,Default,,0000,0000,0000,,Would you have spent two\Ndays without doing this?
Dialogue: 0,1:07:17.98,1:07:22.22,Default,,0000,0000,0000,,I think you would have\Ngotten it by yourselves.
Dialogue: 0,1:07:22.22,1:07:25.42,Default,,0000,0000,0000,,Am I right, with no problem?
Dialogue: 0,1:07:25.42,1:07:26.44,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,1:07:26.44,1:07:30.09,Default,,0000,0000,0000,,Because I think you\Nworked enough problems
Dialogue: 0,1:07:30.09,1:07:32.90,Default,,0000,0000,0000,,to master the material,\Nand you are prepared.
Dialogue: 0,1:07:32.90,1:07:37.17,Default,,0000,0000,0000,,And this is not a\Nsurprise for you
Dialogue: 0,1:07:37.17,1:07:40.59,Default,,0000,0000,0000,,like it is for many\Nstudents in other classes.
Dialogue: 0,1:07:40.59,1:07:41.57,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:07:41.57,1:07:44.52,Default,,0000,0000,0000,,STUDENT: Can you put that\None in spherical coordinates?
Dialogue: 0,1:07:44.52,1:07:45.84,Default,,0000,0000,0000,,PROFESSOR: You can.
Dialogue: 0,1:07:45.84,1:07:50.48,Default,,0000,0000,0000,,That is going to be a hassle.
Dialogue: 0,1:07:50.48,1:07:55.07,Default,,0000,0000,0000,,I would do one more\Nproblem that is not
Dialogue: 0,1:07:55.07,1:07:57.07,Default,,0000,0000,0000,,quite appropriate\Nfor spherical work,
Dialogue: 0,1:07:57.07,1:07:59.64,Default,,0000,0000,0000,,but I want to do it [INAUDIBLE].
Dialogue: 0,1:07:59.64,1:08:03.10,Default,,0000,0000,0000,,Because it looks like the\Nones I gave you as a homework,
Dialogue: 0,1:08:03.10,1:08:05.68,Default,,0000,0000,0000,,and several people\Nstruggled with that.
Dialogue: 0,1:08:05.68,1:08:13.23,Default,,0000,0000,0000,,And I want to see how it's done\Nsince not everybody finished
Dialogue: 0,1:08:13.23,1:08:13.73,Default,,0000,0000,0000,,it.
Dialogue: 0,1:08:13.73,1:08:16.51,Default,,0000,0000,0000,,
Dialogue: 0,1:08:16.51,1:08:21.35,Default,,0000,0000,0000,,Given [INAUDIBLE] numbers,\Nyou have a flat plane
Dialogue: 0,1:08:21.35,1:08:26.85,Default,,0000,0000,0000,,z equals a at some\Naltitude a and a cone
Dialogue: 0,1:08:26.85,1:08:29.18,Default,,0000,0000,0000,,exactly like the cone\NI gave you before.
Dialogue: 0,1:08:29.18,1:08:32.60,Default,,0000,0000,0000,,And of course this is\Nnot just like you asked.
Dialogue: 0,1:08:32.60,1:08:36.15,Default,,0000,0000,0000,,This is not very appropriate\Nfor spherical coordinates.
Dialogue: 0,1:08:36.15,1:08:38.95,Default,,0000,0000,0000,,It's appropriate\Nfor cylindrical.
Dialogue: 0,1:08:38.95,1:08:41.60,Default,,0000,0000,0000,,But they ask you\Nto do it in both.
Dialogue: 0,1:08:41.60,1:08:44.07,Default,,0000,0000,0000,,Remember that problem, guys?
Dialogue: 0,1:08:44.07,1:08:49.30,Default,,0000,0000,0000,,So you have the volume of, or\Nsome function, or something.
Dialogue: 0,1:08:49.30,1:08:52.61,Default,,0000,0000,0000,,And they say, put it in\Nboth spherical coordinates
Dialogue: 0,1:08:52.61,1:08:57.26,Default,,0000,0000,0000,,and cylindrical coordinates.
Dialogue: 0,1:08:57.26,1:08:59.39,Default,,0000,0000,0000,,And let's assume that\Nyou don't know what
Dialogue: 0,1:08:59.39,1:09:00.71,Default,,0000,0000,0000,,function you are integrating.
Dialogue: 0,1:09:00.71,1:09:02.82,Default,,0000,0000,0000,,I'm working too\Nmuch with volumes.
Dialogue: 0,1:09:02.82,1:09:04.73,Default,,0000,0000,0000,,Let's suppose that\Nyou are simply
Dialogue: 0,1:09:04.73,1:09:10.13,Default,,0000,0000,0000,,integrating in function\NF of x, y, z dV, which
Dialogue: 0,1:09:10.13,1:09:17.10,Default,,0000,0000,0000,,is dx dy dx over the body of\Nthe [INAUDIBLE], of the-- this
Dialogue: 0,1:09:17.10,1:09:23.21,Default,,0000,0000,0000,,is the flat cone, the\Nflat ice cream cone.
Dialogue: 0,1:09:23.21,1:09:28.37,Default,,0000,0000,0000,,Then somebody licked your\Nice cream up to this point.
Dialogue: 0,1:09:28.37,1:09:31.30,Default,,0000,0000,0000,,And you are left\Nwith the ice cream
Dialogue: 0,1:09:31.30,1:09:36.72,Default,,0000,0000,0000,,only under this at the level\Nof the rim of the waffle.
Dialogue: 0,1:09:36.72,1:09:39.84,Default,,0000,0000,0000,,
Dialogue: 0,1:09:39.84,1:09:43.78,Default,,0000,0000,0000,,Let's break this into two-- they\Ndon't ask you to compute it.
Dialogue: 0,1:09:43.78,1:09:51.21,Default,,0000,0000,0000,,They ask you to set up\Ncylindrical coordinates
Dialogue: 0,1:09:51.21,1:09:55.12,Default,,0000,0000,0000,,and set up the\Nspherical coordinates.
Dialogue: 0,1:09:55.12,1:09:56.60,Default,,0000,0000,0000,,But thank you for the idea.
Dialogue: 0,1:09:56.60,1:09:58.79,Default,,0000,0000,0000,,That was great.
Dialogue: 0,1:09:58.79,1:10:01.59,Default,,0000,0000,0000,,So let's see, how hard is it?
Dialogue: 0,1:10:01.59,1:10:06.89,Default,,0000,0000,0000,,I think it's very easy in\Ncylindrical coordinates.
Dialogue: 0,1:10:06.89,1:10:08.78,Default,,0000,0000,0000,,What do you do in\Ncylindrical coordinates?
Dialogue: 0,1:10:08.78,1:10:10.01,Default,,0000,0000,0000,,You say, well, wait a minute.
Dialogue: 0,1:10:10.01,1:10:13.94,Default,,0000,0000,0000,,If z equals a has\Nto be intersected
Dialogue: 0,1:10:13.94,1:10:17.81,Default,,0000,0000,0000,,with z squared\Nequals x squared, I
Dialogue: 0,1:10:17.81,1:10:19.64,Default,,0000,0000,0000,,know the circle that\NI'm going to get
Dialogue: 0,1:10:19.64,1:10:21.40,Default,,0000,0000,0000,,is going to be a piece of cake.
Dialogue: 0,1:10:21.40,1:10:25.87,Default,,0000,0000,0000,,x squared plus y squared\Nequals a squared.
Dialogue: 0,1:10:25.87,1:10:33.88,Default,,0000,0000,0000,,So really my ice cream\Ncone has the radius a.
Dialogue: 0,1:10:33.88,1:10:35.15,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:10:35.15,1:10:36.07,Default,,0000,0000,0000,,Is it true?
Dialogue: 0,1:10:36.07,1:10:40.06,Default,,0000,0000,0000,,Is it true that the radius\Nof this licked ice cream cone
Dialogue: 0,1:10:40.06,1:10:40.56,Default,,0000,0000,0000,,is a?
Dialogue: 0,1:10:40.56,1:10:41.26,Default,,0000,0000,0000,,STUDENT: Mhmm.
Dialogue: 0,1:10:41.26,1:10:44.25,Default,,0000,0000,0000,,PROFESSOR: It is true.
Dialogue: 0,1:10:44.25,1:10:49.26,Default,,0000,0000,0000,,Whatever that a was-- yours\Nwas 43, 34, 37, god knows what,
Dialogue: 0,1:10:49.26,1:10:52.06,Default,,0000,0000,0000,,doesn't matter.
Dialogue: 0,1:10:52.06,1:10:58.10,Default,,0000,0000,0000,,I would foresee-- I'm not\Na prophet or even a witch.
Dialogue: 0,1:10:58.10,1:10:58.66,Default,,0000,0000,0000,,I am a witch.
Dialogue: 0,1:10:58.66,1:11:03.40,Default,,0000,0000,0000,,But anyway, I would\Nnot foresee somebody
Dialogue: 0,1:11:03.40,1:11:04.82,Default,,0000,0000,0000,,giving you a hard\Nproblem to solve
Dialogue: 0,1:11:04.82,1:11:07.20,Default,,0000,0000,0000,,like that computationally.
Dialogue: 0,1:11:07.20,1:11:10.60,Default,,0000,0000,0000,,But on the final, they can\Nmake you set up the limits
Dialogue: 0,1:11:10.60,1:11:13.02,Default,,0000,0000,0000,,and leave it like that.
Dialogue: 0,1:11:13.02,1:11:16.90,Default,,0000,0000,0000,,So how do we do cylindrical?
Dialogue: 0,1:11:16.90,1:11:18.73,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,1:11:18.73,1:11:21.87,Default,,0000,0000,0000,,So r will be from 0 to a.
Dialogue: 0,1:11:21.87,1:11:24.42,Default,,0000,0000,0000,,And god, that's easy.
Dialogue: 0,1:11:24.42,1:11:27.24,Default,,0000,0000,0000,,0 to 2 pi is going\Nto be for the theta.
Dialogue: 0,1:11:27.24,1:11:28.69,Default,,0000,0000,0000,,First I write dz.
Dialogue: 0,1:11:28.69,1:11:32.28,Default,,0000,0000,0000,,Then I do dr and d theta.
Dialogue: 0,1:11:32.28,1:11:35.54,Default,,0000,0000,0000,,
Dialogue: 0,1:11:35.54,1:11:40.04,Default,,0000,0000,0000,,Theta will be between 0 and\N2 pi, r between 0 and a.
Dialogue: 0,1:11:40.04,1:11:44.34,Default,,0000,0000,0000,,z-- you guys have to\Ntell me, because it's
Dialogue: 0,1:11:44.34,1:11:47.82,Default,,0000,0000,0000,,between a bottom and a top.
Dialogue: 0,1:11:47.82,1:11:51.34,Default,,0000,0000,0000,,And I was about to\Ntake this to drink.
Dialogue: 0,1:11:51.34,1:11:52.21,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:11:52.21,1:12:02.45,Default,,0000,0000,0000,,
Dialogue: 0,1:12:02.45,1:12:04.88,Default,,0000,0000,0000,,PROFESSOR: r is the\None on the bottom,
Dialogue: 0,1:12:04.88,1:12:08.77,Default,,0000,0000,0000,,and a is the one on the top.
Dialogue: 0,1:12:08.77,1:12:10.72,Default,,0000,0000,0000,,And I think that's clear\Nto everybody, right?
Dialogue: 0,1:12:10.72,1:12:15.79,Default,,0000,0000,0000,,Is there anything\Nmissing obviously?
Dialogue: 0,1:12:15.79,1:12:19.70,Default,,0000,0000,0000,,So what do I do when they\Nask me on the final--
Dialogue: 0,1:12:19.70,1:12:22.87,Default,,0000,0000,0000,,when I say this is a\Nmysterious function, what
Dialogue: 0,1:12:22.87,1:12:24.42,Default,,0000,0000,0000,,do you put in here?
Dialogue: 0,1:12:24.42,1:12:28.98,Default,,0000,0000,0000,,F of x, y, z, yes,\Nbut yes and no.
Dialogue: 0,1:12:28.98,1:12:40.34,Default,,0000,0000,0000,,Because you say F of x of r z\Ntheta, y of r z theta, z of god
Dialogue: 0,1:12:40.34,1:12:41.62,Default,,0000,0000,0000,,knows z, z.
Dialogue: 0,1:12:41.62,1:12:43.67,Default,,0000,0000,0000,,z is the same, do\Nyou understand?
Dialogue: 0,1:12:43.67,1:12:45.74,Default,,0000,0000,0000,,So you indicate\Nto the poor people
Dialogue: 0,1:12:45.74,1:12:51.11,Default,,0000,0000,0000,,that I'm not going to stay in\Nx, y, z, because I'm not stupid.
Dialogue: 0,1:12:51.11,1:12:53.51,Default,,0000,0000,0000,,I'm going to transform\Nthe whole thing
Dialogue: 0,1:12:53.51,1:12:56.37,Default,,0000,0000,0000,,so it's going to be expressed\Nin terms of these letters-- r
Dialogue: 0,1:12:56.37,1:12:59.89,Default,,0000,0000,0000,,theta and z.
Dialogue: 0,1:12:59.89,1:13:02.10,Default,,0000,0000,0000,,Do you have to write all this?
Dialogue: 0,1:13:02.10,1:13:05.10,Default,,0000,0000,0000,,If you were a professional\Nwriting the math paper, yes,
Dialogue: 0,1:13:05.10,1:13:08.80,Default,,0000,0000,0000,,you have to, or a math book\Nor whatever, you have to.
Dialogue: 0,1:13:08.80,1:13:11.73,Default,,0000,0000,0000,,But you can also skip it\Nand put the F. I'm not
Dialogue: 0,1:13:11.73,1:13:13.04,Default,,0000,0000,0000,,going to take off points.
Dialogue: 0,1:13:13.04,1:13:15.53,Default,,0000,0000,0000,,I will understand.
Dialogue: 0,1:13:15.53,1:13:19.12,Default,,0000,0000,0000,,Times r-- very good.
Dialogue: 0,1:13:19.12,1:13:21.32,Default,,0000,0000,0000,,Never forget about\Nyour nice Jacobian.
Dialogue: 0,1:13:21.32,1:13:26.35,Default,,0000,0000,0000,,If you forget the r, this\Nis no good, 0 points,
Dialogue: 0,1:13:26.35,1:13:30.48,Default,,0000,0000,0000,,even with all the setup you\Ntried to do going into it.
Dialogue: 0,1:13:30.48,1:13:34.41,Default,,0000,0000,0000,,OK, finally let's see.
Dialogue: 0,1:13:34.41,1:13:37.59,Default,,0000,0000,0000,,How you do this in\Nspherical is not-- yes, sir.
Dialogue: 0,1:13:37.59,1:13:40.12,Default,,0000,0000,0000,,STUDENT: When you're\Nfinding the volume,
Dialogue: 0,1:13:40.12,1:13:43.66,Default,,0000,0000,0000,,isn't it with a triple integral,\Ndon't you just put a 1?
Dialogue: 0,1:13:43.66,1:13:44.42,Default,,0000,0000,0000,,PROFESSOR: Hm?
Dialogue: 0,1:13:44.42,1:13:46.00,Default,,0000,0000,0000,,STUDENT: When you're\Nfinding a volume?
Dialogue: 0,1:13:46.00,1:13:48.54,Default,,0000,0000,0000,,PROFESSOR: No, I didn't\Nsay-- I just said,
Dialogue: 0,1:13:48.54,1:13:51.47,Default,,0000,0000,0000,,but you probably were\Nthinking of [INAUDIBLE].
Dialogue: 0,1:13:51.47,1:13:53.76,Default,,0000,0000,0000,,I said, I gave you\Ntoo many volumes.
Dialogue: 0,1:13:53.76,1:13:56.71,Default,,0000,0000,0000,,I just said, and\NI'm tired of saying
Dialogue: 0,1:13:56.71,1:13:59.14,Default,,0000,0000,0000,,volume of this, volume of that.
Dialogue: 0,1:13:59.14,1:14:01.02,Default,,0000,0000,0000,,And in the actual\Nproblem, they may
Dialogue: 0,1:14:01.02,1:14:04.86,Default,,0000,0000,0000,,ask you to do triple\Nintegral of any function,
Dialogue: 0,1:14:04.86,1:14:07.91,Default,,0000,0000,0000,,differentiable function\Nor continuous function,
Dialogue: 0,1:14:07.91,1:14:12.70,Default,,0000,0000,0000,,over a volume, over a body.
Dialogue: 0,1:14:12.70,1:14:15.63,Default,,0000,0000,0000,,So this could be-- in\Nthe next chapter we're
Dialogue: 0,1:14:15.63,1:14:16.92,Default,,0000,0000,0000,,going to see some applications.
Dialogue: 0,1:14:16.92,1:14:20.01,Default,,0000,0000,0000,,
Dialogue: 0,1:14:20.01,1:14:24.69,Default,,0000,0000,0000,,I maybe saw some in 12.6 like\Nmass moment, those things.
Dialogue: 0,1:14:24.69,1:14:28.15,Default,,0000,0000,0000,,But in three coordinates,\Nyou have other functions
Dialogue: 0,1:14:28.15,1:14:29.89,Default,,0000,0000,0000,,that are these functions.
Dialogue: 0,1:14:29.89,1:14:35.15,Default,,0000,0000,0000,,You'll have that included,\Nrow z, x, y, z, and so on.
Dialogue: 0,1:14:35.15,1:14:37.28,Default,,0000,0000,0000,,OK, good.
Dialogue: 0,1:14:37.28,1:14:39.65,Default,,0000,0000,0000,,When you would integrate a\Ndensity function in that case,
Dialogue: 0,1:14:39.65,1:14:42.10,Default,,0000,0000,0000,,you will have a mass.
Dialogue: 0,1:14:42.10,1:14:45.37,Default,,0000,0000,0000,,Because you integrate this, d\Nover volume, you'd have a mass.
Dialogue: 0,1:14:45.37,1:14:49.85,Default,,0000,0000,0000,,OK, in this case,\Nwe have to be smart,
Dialogue: 0,1:14:49.85,1:14:56.19,Default,,0000,0000,0000,,say F times r squared\Nsine phi is the Jacobian.
Dialogue: 0,1:14:56.19,1:15:02.50,Default,,0000,0000,0000,,This is a function in r\Nphi and theta, right guys?
Dialogue: 0,1:15:02.50,1:15:04.23,Default,,0000,0000,0000,,We don't care what it is.
Dialogue: 0,1:15:04.23,1:15:05.96,Default,,0000,0000,0000,,We are going to have\Nthe d something,
Dialogue: 0,1:15:05.96,1:15:07.10,Default,,0000,0000,0000,,d something, d something.
Dialogue: 0,1:15:07.10,1:15:08.74,Default,,0000,0000,0000,,The question is, which ones?
Dialogue: 0,1:15:08.74,1:15:13.76,Default,,0000,0000,0000,,Because it's not obvious\Nat all, except for theta.
Dialogue: 0,1:15:13.76,1:15:15.18,Default,,0000,0000,0000,,Theta is nice.
Dialogue: 0,1:15:15.18,1:15:16.16,Default,,0000,0000,0000,,He's so nice.
Dialogue: 0,1:15:16.16,1:15:20.14,Default,,0000,0000,0000,,And we say, OK theta,\Nwe are grateful to you.
Dialogue: 0,1:15:20.14,1:15:25.41,Default,,0000,0000,0000,,We put you at the end, because\Nit's a complete rotation.
Dialogue: 0,1:15:25.41,1:15:31.13,Default,,0000,0000,0000,,And we know you are between 0\Nand 2 pi, very reliable guy.
Dialogue: 0,1:15:31.13,1:15:33.100,Default,,0000,0000,0000,,Phi is not so reliable.
Dialogue: 0,1:15:33.100,1:15:35.77,Default,,0000,0000,0000,,Well, phi is a nice guy.
Dialogue: 0,1:15:35.77,1:15:39.97,Default,,0000,0000,0000,,But he puts us through\Na little bit of work.
Dialogue: 0,1:15:39.97,1:15:41.39,Default,,0000,0000,0000,,Do we like to work?
Dialogue: 0,1:15:41.39,1:15:45.97,Default,,0000,0000,0000,,Well, not so much,\Nbut we'll try.
Dialogue: 0,1:15:45.97,1:15:52.63,Default,,0000,0000,0000,,So we need to know a little\Nbit more about this triangle.
Dialogue: 0,1:15:52.63,1:15:56.10,Default,,0000,0000,0000,,
Dialogue: 0,1:15:56.10,1:16:01.06,Default,,0000,0000,0000,,We need to understand a little\Nbit more about this triangle.
Dialogue: 0,1:16:01.06,1:16:05.28,Default,,0000,0000,0000,,STUDENT: Well, the angle\Nbetween the angle at the bottom
Dialogue: 0,1:16:05.28,1:16:06.32,Default,,0000,0000,0000,,is 45 degrees.
Dialogue: 0,1:16:06.32,1:16:07.49,Default,,0000,0000,0000,,PROFESSOR: How can you say?
Dialogue: 0,1:16:07.49,1:16:09.36,Default,,0000,0000,0000,,STUDENT: Because the\Nslope of that line is 1.
Dialogue: 0,1:16:09.36,1:16:12.61,Default,,0000,0000,0000,,PROFESSOR: Right, so say,\Nnow I'm going to observe z
Dialogue: 0,1:16:12.61,1:16:13.66,Default,,0000,0000,0000,,was a as well.
Dialogue: 0,1:16:13.66,1:16:18.58,Default,,0000,0000,0000,,
Dialogue: 0,1:16:18.58,1:16:23.06,Default,,0000,0000,0000,,So that means it's a\Nright isosceles triangle.
Dialogue: 0,1:16:23.06,1:16:25.10,Default,,0000,0000,0000,,If it's a right\Nisosceles triangle,
Dialogue: 0,1:16:25.10,1:16:27.70,Default,,0000,0000,0000,,this is 45 degree angle.
Dialogue: 0,1:16:27.70,1:16:37.29,Default,,0000,0000,0000,,So this is from d phi from\N0 to pi over 4, excellent.
Dialogue: 0,1:16:37.29,1:16:41.01,Default,,0000,0000,0000,,Finally, the only one that gives\Nus a little bit of a headache
Dialogue: 0,1:16:41.01,1:16:46.80,Default,,0000,0000,0000,,but not too much of a\Nheadache is the radius r.
Dialogue: 0,1:16:46.80,1:16:48.50,Default,,0000,0000,0000,,Should I change the color?
Dialogue: 0,1:16:48.50,1:16:51.48,Default,,0000,0000,0000,,No, I'll leave it\Nr dr. So we have
Dialogue: 0,1:16:51.48,1:17:01.78,Default,,0000,0000,0000,,to think a little bit of\Nthe meaning of our rays.
Dialogue: 0,1:17:01.78,1:17:04.28,Default,,0000,0000,0000,,Drawing vertical strips\Nor horizontal strips
Dialogue: 0,1:17:04.28,1:17:06.78,Default,,0000,0000,0000,,or whatever strips\Nis not a good idea.
Dialogue: 0,1:17:06.78,1:17:08.76,Default,,0000,0000,0000,,When we are in\Nspherical coordinates,
Dialogue: 0,1:17:08.76,1:17:11.82,Default,,0000,0000,0000,,what do we need to draw?
Dialogue: 0,1:17:11.82,1:17:15.62,Default,,0000,0000,0000,,Rays, like rays of sun\Ncoming from a source.
Dialogue: 0,1:17:15.62,1:17:20.60,Default,,0000,0000,0000,,The source is here at the\Norigin in spherical coordinates.
Dialogue: 0,1:17:20.60,1:17:25.24,Default,,0000,0000,0000,,These are like rays of\Nsun that are free to move.
Dialogue: 0,1:17:25.24,1:17:26.59,Default,,0000,0000,0000,,But they bump.
Dialogue: 0,1:17:26.59,1:17:31.94,Default,,0000,0000,0000,,They just bump against\Nthe plane, the flat roof.
Dialogue: 0,1:17:31.94,1:17:36.24,Default,,0000,0000,0000,,So they would reflect if\Nthis were a physical problem.
Dialogue: 0,1:17:36.24,1:17:38.92,Default,,0000,0000,0000,,
Dialogue: 0,1:17:38.92,1:17:42.97,Default,,0000,0000,0000,,So definitely all\Nyour rays start at 0.
Dialogue: 0,1:17:42.97,1:17:44.73,Default,,0000,0000,0000,,So you have to put 0 here.
Dialogue: 0,1:17:44.73,1:17:49.31,Default,,0000,0000,0000,,But this is a question mark.
Dialogue: 0,1:17:49.31,1:17:50.19,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:17:50.19,1:17:57.55,Default,,0000,0000,0000,,
Dialogue: 0,1:17:57.55,1:17:59.02,Default,,0000,0000,0000,,STUDENT: a square root 2.
Dialogue: 0,1:17:59.02,1:18:01.99,Default,,0000,0000,0000,,
Dialogue: 0,1:18:01.99,1:18:03.98,Default,,0000,0000,0000,,PROFESSOR: No, it's\Nnot a fixed answer.
Dialogue: 0,1:18:03.98,1:18:07.19,Default,,0000,0000,0000,,So you have z will be a fixed.
Dialogue: 0,1:18:07.19,1:18:09.77,Default,,0000,0000,0000,,But who was z in\Nspherical coordinates?
Dialogue: 0,1:18:09.77,1:18:12.56,Default,,0000,0000,0000,,That was the only\Nthing you can ask.
Dialogue: 0,1:18:12.56,1:18:16.44,Default,,0000,0000,0000,,So z equals a is your\Ntradition that is the roof.
Dialogue: 0,1:18:16.44,1:18:18.46,Default,,0000,0000,0000,,STUDENT: That would\Nbe r [INAUDIBLE].
Dialogue: 0,1:18:18.46,1:18:28.05,Default,,0000,0000,0000,,PROFESSOR: Very good,\Nr cosine of phi,
Dialogue: 0,1:18:28.05,1:18:32.95,Default,,0000,0000,0000,,of the latitude\Nfrom the North Pole.
Dialogue: 0,1:18:32.95,1:18:33.95,Default,,0000,0000,0000,,This is 45.
Dialogue: 0,1:18:33.95,1:18:38.62,Default,,0000,0000,0000,,But I mean for a point like\Nthis, phi will be this phi.
Dialogue: 0,1:18:38.62,1:18:40.21,Default,,0000,0000,0000,,Do you guys understand?
Dialogue: 0,1:18:40.21,1:18:44.05,Default,,0000,0000,0000,,Phi could be any point where the\Npoint inside [INAUDIBLE], phi
Dialogue: 0,1:18:44.05,1:18:46.43,Default,,0000,0000,0000,,will be the latitude\Nfrom the North Pole.
Dialogue: 0,1:18:46.43,1:18:51.83,Default,,0000,0000,0000,,OK, so the way you do it\Nis r is between 0 and z
Dialogue: 0,1:18:51.83,1:18:56.65,Default,,0000,0000,0000,,over cosine phi.
Dialogue: 0,1:18:56.65,1:18:58.21,Default,,0000,0000,0000,,And that's the hard thing.
Dialogue: 0,1:18:58.21,1:19:01.51,Default,,0000,0000,0000,,Since z at the\Nroof is a, you have
Dialogue: 0,1:19:01.51,1:19:06.55,Default,,0000,0000,0000,,to put here a over-- a is\Nfixed, that 43 of yours,
Dialogue: 0,1:19:06.55,1:19:08.82,Default,,0000,0000,0000,,whatever it was-- cosine phi.
Dialogue: 0,1:19:08.82,1:19:13.94,Default,,0000,0000,0000,,
Dialogue: 0,1:19:13.94,1:19:17.87,Default,,0000,0000,0000,,So when you guys integrate\Nwith respect to r,
Dialogue: 0,1:19:17.87,1:19:24.33,Default,,0000,0000,0000,,assume this F will be 1,\Njust like you asked me, Alex.
Dialogue: 0,1:19:24.33,1:19:27.47,Default,,0000,0000,0000,,That would make my life\Neasier and would be good.
Dialogue: 0,1:19:27.47,1:19:29.46,Default,,0000,0000,0000,,When I integrate\Nwith respect to r,
Dialogue: 0,1:19:29.46,1:19:32.03,Default,,0000,0000,0000,,would it be hard\Nto solve a problem?
Dialogue: 0,1:19:32.03,1:19:33.20,Default,,0000,0000,0000,,Oh, not so hard.
Dialogue: 0,1:19:33.20,1:19:34.88,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:19:34.88,1:19:37.65,Default,,0000,0000,0000,,OK, integrate this\Nwith respect to r.
Dialogue: 0,1:19:37.65,1:19:39.55,Default,,0000,0000,0000,,We have r cubed.
Dialogue: 0,1:19:39.55,1:19:41.59,Default,,0000,0000,0000,,Integrate r squared.
Dialogue: 0,1:19:41.59,1:19:43.46,Default,,0000,0000,0000,,We have r cubed over 3, right?
Dialogue: 0,1:19:43.46,1:19:49.42,Default,,0000,0000,0000,,Let's do this, solve the\Nsame problem when F is 1.
Dialogue: 0,1:19:49.42,1:19:54.90,Default,,0000,0000,0000,,
Dialogue: 0,1:19:54.90,1:20:02.56,Default,,0000,0000,0000,,Solve the same problem when\NF would be 1, for F equals 1.
Dialogue: 0,1:20:02.56,1:20:06.53,Default,,0000,0000,0000,,Then you get integral\Nfrom 0 to 2 pi,
Dialogue: 0,1:20:06.53,1:20:10.94,Default,,0000,0000,0000,,integral from 0 to pi\Nover 4, integral from 0
Dialogue: 0,1:20:10.94,1:20:21.54,Default,,0000,0000,0000,,to a over cosine phi, 1 r\Nsquared sine phi dr d phi d
Dialogue: 0,1:20:21.54,1:20:22.31,Default,,0000,0000,0000,,theta.
Dialogue: 0,1:20:22.31,1:20:25.84,Default,,0000,0000,0000,,The guy that sits on my\Nnerves is again theta.
Dialogue: 0,1:20:25.84,1:20:27.13,Default,,0000,0000,0000,,He's very nice.
Dialogue: 0,1:20:27.13,1:20:29.88,Default,,0000,0000,0000,,He can be eliminated\Nfrom the game.
Dialogue: 0,1:20:29.88,1:20:34.21,Default,,0000,0000,0000,,So 2 pi out, and I\Nwill focus my attention
Dialogue: 0,1:20:34.21,1:20:37.12,Default,,0000,0000,0000,,to the product of function.
Dialogue: 0,1:20:37.12,1:20:40.61,Default,,0000,0000,0000,,Well, OK, I have to\Nintegrate one at a time.
Dialogue: 0,1:20:40.61,1:20:43.21,Default,,0000,0000,0000,,So I integrate with\Nrespect to what?
Dialogue: 0,1:20:43.21,1:20:45.57,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:20:45.57,1:20:50.18,Default,,0000,0000,0000,,PROFESSOR: So I get r cubed\Nover 3, all right, the integral
Dialogue: 0,1:20:50.18,1:20:52.84,Default,,0000,0000,0000,,from 0 to pi over 4.
Dialogue: 0,1:20:52.84,1:20:54.77,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:20:54.77,1:20:59.34,Default,,0000,0000,0000,,PROFESSOR: r cubed over 3\Nbetween-- it's a little bit
Dialogue: 0,1:20:59.34,1:21:03.18,Default,,0000,0000,0000,,of a headache. r equals\Na over cosine phi.
Dialogue: 0,1:21:03.18,1:21:05.30,Default,,0000,0000,0000,,And I bet you my video\Ndoesn't see anything,
Dialogue: 0,1:21:05.30,1:21:07.84,Default,,0000,0000,0000,,so let me change the colors.
Dialogue: 0,1:21:07.84,1:21:10.76,Default,,0000,0000,0000,,r equals a over cosine phi.
Dialogue: 0,1:21:10.76,1:21:13.02,Default,,0000,0000,0000,,And r equals 0 down.
Dialogue: 0,1:21:13.02,1:21:14.78,Default,,0000,0000,0000,,That's the easy part.
Dialogue: 0,1:21:14.78,1:21:19.10,Default,,0000,0000,0000,,Inside I have r\Ncubed over 3, right?
Dialogue: 0,1:21:19.10,1:21:25.64,Default,,0000,0000,0000,,All right, and sine\Nphi, and all I'm
Dialogue: 0,1:21:25.64,1:21:31.76,Default,,0000,0000,0000,,left with is a phi\Nintegration, is an integration
Dialogue: 0,1:21:31.76,1:21:33.55,Default,,0000,0000,0000,,with respect to phi.
Dialogue: 0,1:21:33.55,1:21:35.100,Default,,0000,0000,0000,,Let's see-- yes, sir.
Dialogue: 0,1:21:35.100,1:21:36.98,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:21:36.98,1:21:40.56,Default,,0000,0000,0000,,PROFESSOR: Well, I should be\Nable to manage with this guy.
Dialogue: 0,1:21:40.56,1:21:46.58,Default,,0000,0000,0000,,
Dialogue: 0,1:21:46.58,1:21:47.45,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:21:47.45,1:22:05.18,Default,,0000,0000,0000,,
Dialogue: 0,1:22:05.18,1:22:07.29,Default,,0000,0000,0000,,PROFESSOR: I'm writing\Njust as you said, OK?
Dialogue: 0,1:22:07.29,1:22:13.64,Default,,0000,0000,0000,,
Dialogue: 0,1:22:13.64,1:22:16.15,Default,,0000,0000,0000,,Now, how much of a headache\Ndo you think this is?
Dialogue: 0,1:22:16.15,1:22:18.30,Default,,0000,0000,0000,,STUDENT: It's not much\Nof one, because it's
Dialogue: 0,1:22:18.30,1:22:20.62,Default,,0000,0000,0000,,the same as a tangent\Ntimes the secant squared
Dialogue: 0,1:22:20.62,1:22:22.86,Default,,0000,0000,0000,,with a constant pulled out.
Dialogue: 0,1:22:22.86,1:22:26.08,Default,,0000,0000,0000,,STUDENT: So psi and\Ncosine don't [INAUDIBLE].
Dialogue: 0,1:22:26.08,1:22:34.01,Default,,0000,0000,0000,,Tangents will give\Nyou 1 over cosine--
Dialogue: 0,1:22:34.01,1:22:36.51,Default,,0000,0000,0000,,PROFESSOR: What's the simplest\Nway to do it without thinking
Dialogue: 0,1:22:36.51,1:22:39.93,Default,,0000,0000,0000,,of tangent and cotangent, huh?
Dialogue: 0,1:22:39.93,1:22:41.25,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:22:41.25,1:22:43.52,Default,,0000,0000,0000,,PROFESSOR: Instead, a\Nu substitution there?
Dialogue: 0,1:22:43.52,1:22:45.25,Default,,0000,0000,0000,,What is the u substitution?
Dialogue: 0,1:22:45.25,1:22:47.63,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:22:47.63,1:22:49.53,Default,,0000,0000,0000,,PROFESSOR: Is this good?
Dialogue: 0,1:22:49.53,1:22:50.97,Default,,0000,0000,0000,,STUDENT: No.
Dialogue: 0,1:22:50.97,1:22:52.17,Default,,0000,0000,0000,,PROFESSOR: No?
Dialogue: 0,1:22:52.17,1:22:54.100,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:22:54.100,1:22:57.70,Default,,0000,0000,0000,,PROFESSOR: It's\Nu to the minus 3.
Dialogue: 0,1:22:57.70,1:22:59.88,Default,,0000,0000,0000,,And that's OK.
Dialogue: 0,1:22:59.88,1:23:05.74,Default,,0000,0000,0000,,So I have 2 pi a cubed\Nover 3 [INAUDIBLE]
Dialogue: 0,1:23:05.74,1:23:09.78,Default,,0000,0000,0000,,because they are in my way\Nthere making my life miserable,
Dialogue: 0,1:23:09.78,1:23:11.12,Default,,0000,0000,0000,,integral.
Dialogue: 0,1:23:11.12,1:23:18.10,Default,,0000,0000,0000,,And then I have u to the\Nminus 3 times-- for du
Dialogue: 0,1:23:18.10,1:23:22.32,Default,,0000,0000,0000,,I get a minus that\Nthat is sort of ugh.
Dialogue: 0,1:23:22.32,1:23:26.04,Default,,0000,0000,0000,,I have to invent the minus, and\NI have to invent the minus here
Dialogue: 0,1:23:26.04,1:23:27.99,Default,,0000,0000,0000,,in front as well.
Dialogue: 0,1:23:27.99,1:23:32.08,Default,,0000,0000,0000,,So they will compensate\Nfor one another.
Dialogue: 0,1:23:32.08,1:23:34.68,Default,,0000,0000,0000,,And I'll say du.
Dialogue: 0,1:23:34.68,1:23:39.71,Default,,0000,0000,0000,,But these limit points, of\Ncourse I can do them by myself.
Dialogue: 0,1:23:39.71,1:23:41.68,Default,,0000,0000,0000,,I don't need your help.
Dialogue: 0,1:23:41.68,1:23:45.83,Default,,0000,0000,0000,,But I pretend that\NI need your help.
Dialogue: 0,1:23:45.83,1:23:48.33,Default,,0000,0000,0000,,What will be u when phi is 0?
Dialogue: 0,1:23:48.33,1:23:49.15,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,1:23:49.15,1:23:51.84,Default,,0000,0000,0000,,PROFESSOR: 1.
Dialogue: 0,1:23:51.84,1:23:56.82,Default,,0000,0000,0000,,What will be u when\Nphi is pi over 4?
Dialogue: 0,1:23:56.82,1:23:57.70,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:23:57.70,1:24:02.09,Default,,0000,0000,0000,,
Dialogue: 0,1:24:02.09,1:24:04.46,Default,,0000,0000,0000,,PROFESSOR: And from now on\Nyou should be able to do this.
Dialogue: 0,1:24:04.46,1:24:14.67,Default,,0000,0000,0000,,So I have minus 2 pi a cubed\Nover 3 times-- I integrate.
Dialogue: 0,1:24:14.67,1:24:17.85,Default,,0000,0000,0000,,So I add the power, I add\Nthe 1, and I add the 1.
Dialogue: 0,1:24:17.85,1:24:20.45,Default,,0000,0000,0000,,So you have u to the\Nminus 2 over minus 2.
Dialogue: 0,1:24:20.45,1:24:27.26,Default,,0000,0000,0000,,Are you guys with\Nme-- between u equals
Dialogue: 0,1:24:27.26,1:24:30.63,Default,,0000,0000,0000,,1 and u equals root 2 over 2.
Dialogue: 0,1:24:30.63,1:24:33.01,Default,,0000,0000,0000,,I promise you if\Nyou have something
Dialogue: 0,1:24:33.01,1:24:35.06,Default,,0000,0000,0000,,like that in the final\Nand you stop here,
Dialogue: 0,1:24:35.06,1:24:38.03,Default,,0000,0000,0000,,I'm not going to be blaming you.
Dialogue: 0,1:24:38.03,1:24:41.41,Default,,0000,0000,0000,,I'll say, very good, leave\Nit there, I don't care.
Dialogue: 0,1:24:41.41,1:24:44.92,Default,,0000,0000,0000,,Because from this\Npoint on, what follows
Dialogue: 0,1:24:44.92,1:24:47.94,Default,,0000,0000,0000,,is just routine algebra.
Dialogue: 0,1:24:47.94,1:24:52.47,Default,,0000,0000,0000,,So we have-- I hate this.
Dialogue: 0,1:24:52.47,1:24:55.26,Default,,0000,0000,0000,,I'm not a calculator.
Dialogue: 0,1:24:55.26,1:24:59.88,Default,,0000,0000,0000,,But it's better for me to write\N1 over root 2, like you said.
Dialogue: 0,1:24:59.88,1:25:03.15,Default,,0000,0000,0000,,Because in that case, the\Nsquare will be 1 over 2.
Dialogue: 0,1:25:03.15,1:25:06.18,Default,,0000,0000,0000,,And when I invert 1\Nover 2, I get a 2.
Dialogue: 0,1:25:06.18,1:25:10.55,Default,,0000,0000,0000,,So I have 2 over minus 2.
Dialogue: 0,1:25:10.55,1:25:11.67,Default,,0000,0000,0000,,Are you guys with me again?
Dialogue: 0,1:25:11.67,1:25:14.22,Default,,0000,0000,0000,,So I'm thinking the\Nsame-- 1 over root 2.
Dialogue: 0,1:25:14.22,1:25:16.68,Default,,0000,0000,0000,,Square it, you have 1 over 2.
Dialogue: 0,1:25:16.68,1:25:20.57,Default,,0000,0000,0000,,Take it as a minus,\Nyou have exactly 2.
Dialogue: 0,1:25:20.57,1:25:24.13,Default,,0000,0000,0000,,And you have 2 over\Nminus-- is this a minus?
Dialogue: 0,1:25:24.13,1:25:26.86,Default,,0000,0000,0000,,I'm so silly, look\Nat me, minus 2.
Dialogue: 0,1:25:26.86,1:25:29.29,Default,,0000,0000,0000,,STUDENT: It's a cubed\Nover 3, not over 2.
Dialogue: 0,1:25:29.29,1:25:31.59,Default,,0000,0000,0000,,PROFESSOR: It's going to be--
Dialogue: 0,1:25:31.59,1:25:33.68,Default,,0000,0000,0000,,STUDENT: You've got an a\Ncubed over 2 right there.
Dialogue: 0,1:25:33.68,1:25:35.63,Default,,0000,0000,0000,,And it was--
Dialogue: 0,1:25:35.63,1:25:36.60,Default,,0000,0000,0000,,PROFESSOR: Huh?
Dialogue: 0,1:25:36.60,1:25:39.04,Default,,0000,0000,0000,,STUDENT: You just wrote\N2 pi a cubed over 2.
Dialogue: 0,1:25:39.04,1:25:41.00,Default,,0000,0000,0000,,It's a cubed over 3.
Dialogue: 0,1:25:41.00,1:25:43.37,Default,,0000,0000,0000,,PROFESSOR: Yes,\Nit's my silliness.
Dialogue: 0,1:25:43.37,1:25:46.42,Default,,0000,0000,0000,,I looked, and I say\Nthis instead of that.
Dialogue: 0,1:25:46.42,1:25:48.00,Default,,0000,0000,0000,,Thank you so much.
Dialogue: 0,1:25:48.00,1:25:49.79,Default,,0000,0000,0000,,What do I have here?
Dialogue: 0,1:25:49.79,1:25:51.62,Default,,0000,0000,0000,,1 over minus 2.
Dialogue: 0,1:25:51.62,1:25:54.84,Default,,0000,0000,0000,,In the end, what does this mean?
Dialogue: 0,1:25:54.84,1:25:56.24,Default,,0000,0000,0000,,Let's see, what does this mean?
Dialogue: 0,1:25:56.24,1:26:00.92,Default,,0000,0000,0000,,When I plug in, I subtract.
Dialogue: 0,1:26:00.92,1:26:02.24,Default,,0000,0000,0000,,This is what?
Dialogue: 0,1:26:02.24,1:26:07.88,Default,,0000,0000,0000,,This is minus 1 plus\N1/2 is minus 1/2.
Dialogue: 0,1:26:07.88,1:26:12.25,Default,,0000,0000,0000,,But that minus\Nshould not scare me.
Dialogue: 0,1:26:12.25,1:26:14.05,Default,,0000,0000,0000,,Because of course\Na minus in a volume
Dialogue: 0,1:26:14.05,1:26:15.86,Default,,0000,0000,0000,,would be completely wrong.
Dialogue: 0,1:26:15.86,1:26:18.23,Default,,0000,0000,0000,,But I have a minus from before.
Dialogue: 0,1:26:18.23,1:26:31.94,Default,,0000,0000,0000,,So it's plus 2 pi times a\Ncubed over 3, and times 1/2.
Dialogue: 0,1:26:31.94,1:26:35.27,Default,,0000,0000,0000,,So in the end, the\Nanswer, the total answer,
Dialogue: 0,1:26:35.27,1:26:37.32,Default,,0000,0000,0000,,would be answered what?
Dialogue: 0,1:26:37.32,1:26:39.79,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:26:39.79,1:26:45.10,Default,,0000,0000,0000,,PROFESSOR: Pi a cubed\Nover 3, pi a cubed over 3.
Dialogue: 0,1:26:45.10,1:26:46.56,Default,,0000,0000,0000,,It looks very-- huh?
Dialogue: 0,1:26:46.56,1:26:48.34,Default,,0000,0000,0000,,It looks pretty.
Dialogue: 0,1:26:48.34,1:26:53.48,Default,,0000,0000,0000,,Actually yes, it looks\Npretty because-- now,
Dialogue: 0,1:26:53.48,1:26:55.90,Default,,0000,0000,0000,,OK, I'm asking you a question.
Dialogue: 0,1:26:55.90,1:26:59.75,Default,,0000,0000,0000,,Would we have done\Nthat without calculus?
Dialogue: 0,1:26:59.75,1:27:06.20,Default,,0000,0000,0000,,If somebody told you [INAUDIBLE]\Nit has a volume of some cone,
Dialogue: 0,1:27:06.20,1:27:08.59,Default,,0000,0000,0000,,what's the volume of a cone?
Dialogue: 0,1:27:08.59,1:27:13.06,Default,,0000,0000,0000,,Area of the base times\Nthe height divided by 3.
Dialogue: 0,1:27:13.06,1:27:17.47,Default,,0000,0000,0000,,So you could have very nicely\Ncheated on me on the exam
Dialogue: 0,1:27:17.47,1:27:21.01,Default,,0000,0000,0000,,by saying, you\Nhave this cone that
Dialogue: 0,1:27:21.01,1:27:26.19,Default,,0000,0000,0000,,has pi is squared times a-- pi\Nis squared times a-- divided
Dialogue: 0,1:27:26.19,1:27:29.58,Default,,0000,0000,0000,,by 3 equals pi cubed over 3.
Dialogue: 0,1:27:29.58,1:27:32.88,Default,,0000,0000,0000,,When can you not\Ncheat on this problem?
Dialogue: 0,1:27:32.88,1:27:35.00,Default,,0000,0000,0000,,STUDENT: When you say,\Nyou've got to do it with a--
Dialogue: 0,1:27:35.00,1:27:38.31,Default,,0000,0000,0000,,PROFESSOR: Exactly, when I\Nsay, do it with a-- well,
Dialogue: 0,1:27:38.31,1:27:45.25,Default,,0000,0000,0000,,I can say, OK, if we say, set up\Nthe integral and write it down,
Dialogue: 0,1:27:45.25,1:27:47.10,Default,,0000,0000,0000,,you set up the integral\Nand write it down.
Dialogue: 0,1:27:47.10,1:27:50.38,Default,,0000,0000,0000,,If we say, set up the\Nintegral and compute it,
Dialogue: 0,1:27:50.38,1:27:53.91,Default,,0000,0000,0000,,you set up the integral,\Nyou fake the computation,
Dialogue: 0,1:27:53.91,1:27:55.26,Default,,0000,0000,0000,,and you come up with this.
Dialogue: 0,1:27:55.26,1:27:58.40,Default,,0000,0000,0000,,
Dialogue: 0,1:27:58.40,1:28:02.11,Default,,0000,0000,0000,,If we say, set up the integral\Nand show all your work,
Dialogue: 0,1:28:02.11,1:28:03.49,Default,,0000,0000,0000,,then you're in trouble.
Dialogue: 0,1:28:03.49,1:28:09.18,Default,,0000,0000,0000,,But I'm going to try to advocate\Nthat for a simple problem, that
Dialogue: 0,1:28:09.18,1:28:11.13,Default,,0000,0000,0000,,is actually elementary.
Dialogue: 0,1:28:11.13,1:28:15.72,Default,,0000,0000,0000,,One should not have\Nto show all the work.
Dialogue: 0,1:28:15.72,1:28:19.11,Default,,0000,0000,0000,,All right, but keep in mind\Nwhen you have 2 minuses
Dialogue: 0,1:28:19.11,1:28:22.58,Default,,0000,0000,0000,,like that-- that reminds me.
Dialogue: 0,1:28:22.58,1:28:28.79,Default,,0000,0000,0000,,So there was a professor whose\Nsink didn't work anymore.
Dialogue: 0,1:28:28.79,1:28:32.06,Default,,0000,0000,0000,,And he asked for a plumber\Nto come to his house.
Dialogue: 0,1:28:32.06,1:28:33.47,Default,,0000,0000,0000,,He was a math professor.
Dialogue: 0,1:28:33.47,1:28:36.53,Default,,0000,0000,0000,,So the plumber comes to his\Nhouse and fixes this, and says,
Dialogue: 0,1:28:36.53,1:28:37.58,Default,,0000,0000,0000,,what else is wrong?
Dialogue: 0,1:28:37.58,1:28:40.12,Default,,0000,0000,0000,,Fixes the toilet, fixes\Neverything in the house,
Dialogue: 0,1:28:40.12,1:28:44.21,Default,,0000,0000,0000,,and then he shows the\Nprofessor the bill.
Dialogue: 0,1:28:44.21,1:28:50.06,Default,,0000,0000,0000,,So the guy said, oh my god, this\Nis 1/3 of my monthly salary.
Dialogue: 0,1:28:50.06,1:28:53.58,Default,,0000,0000,0000,,So the plumber said,\Nyeah, I mean, really?
Dialogue: 0,1:28:53.58,1:28:54.62,Default,,0000,0000,0000,,You're a smart guy.
Dialogue: 0,1:28:54.62,1:28:55.41,Default,,0000,0000,0000,,You're a professor.
Dialogue: 0,1:28:55.41,1:28:57.26,Default,,0000,0000,0000,,You make that little money?
Dialogue: 0,1:28:57.26,1:28:58.85,Default,,0000,0000,0000,,Yeah, really.
Dialogue: 0,1:28:58.85,1:28:59.99,Default,,0000,0000,0000,,I'm so sorry for you.
Dialogue: 0,1:28:59.99,1:29:03.78,Default,,0000,0000,0000,,Why don't you apply\Nto our company
Dialogue: 0,1:29:03.78,1:29:06.85,Default,,0000,0000,0000,,and become a plumber if you're\Ninterested, if you crave money?
Dialogue: 0,1:29:06.85,1:29:09.00,Default,,0000,0000,0000,,No, of course, I need\Nmoney desperately.
Dialogue: 0,1:29:09.00,1:29:12.98,Default,,0000,0000,0000,,I have five children\Nand a wife [INAUDIBLE].
Dialogue: 0,1:29:12.98,1:29:15.46,Default,,0000,0000,0000,,OK, he applies.
Dialogue: 0,1:29:15.46,1:29:16.83,Default,,0000,0000,0000,,And he says, pay attention.
Dialogue: 0,1:29:16.83,1:29:21.24,Default,,0000,0000,0000,,Don't write that you are a\Nprofessor or you have a PhD.
Dialogue: 0,1:29:21.24,1:29:23.82,Default,,0000,0000,0000,,Just say you just finished\Nhigh school or say,
Dialogue: 0,1:29:23.82,1:29:25.39,Default,,0000,0000,0000,,I didn't finish high school.
Dialogue: 0,1:29:25.39,1:29:27.46,Default,,0000,0000,0000,,So he writes, I didn't\Nfinish high school.
Dialogue: 0,1:29:27.46,1:29:28.51,Default,,0000,0000,0000,,I went to 10th grade.
Dialogue: 0,1:29:28.51,1:29:29.78,Default,,0000,0000,0000,,They accept him.
Dialogue: 0,1:29:29.78,1:29:32.07,Default,,0000,0000,0000,,They give him a job.
Dialogue: 0,1:29:32.07,1:29:34.77,Default,,0000,0000,0000,,And they say, this\Nis your salary.
Dialogue: 0,1:29:34.77,1:29:36.37,Default,,0000,0000,0000,,But there is something new.
Dialogue: 0,1:29:36.37,1:29:40.15,Default,,0000,0000,0000,,Everybody has to\Nfinish high school.
Dialogue: 0,1:29:40.15,1:29:42.84,Default,,0000,0000,0000,,So they have to\Ntake AP Calculus.
Dialogue: 0,1:29:42.84,1:29:44.63,Default,,0000,0000,0000,,So he goes, oh my god.
Dialogue: 0,1:29:44.63,1:29:45.63,Default,,0000,0000,0000,,They all go.
Dialogue: 0,1:29:45.63,1:29:49.64,Default,,0000,0000,0000,,And there comes a TA from\Nthe community college.
Dialogue: 0,1:29:49.64,1:29:51.41,Default,,0000,0000,0000,,The class was full.
Dialogue: 0,1:29:51.41,1:29:56.26,Default,,0000,0000,0000,,He tries to solve a\Nproblem-- with calculus
Dialogue: 0,1:29:56.26,1:30:00.84,Default,,0000,0000,0000,,compute the area inside\Nthis disc of radius a.
Dialogue: 0,1:30:00.84,1:30:03.75,Default,,0000,0000,0000,,So the TA-- OK, I did this.
Dialogue: 0,1:30:03.75,1:30:07.39,Default,,0000,0000,0000,,I got minus pi a squared.
Dialogue: 0,1:30:07.39,1:30:11.67,Default,,0000,0000,0000,,And the professor says,\NOK, you cannot get that.
Dialogue: 0,1:30:11.67,1:30:13.85,Default,,0000,0000,0000,,Let me explain to you.
Dialogue: 0,1:30:13.85,1:30:15.77,Default,,0000,0000,0000,,He goes, I don't know\Nwhere he made a mistake.
Dialogue: 0,1:30:15.77,1:30:21.54,Default,,0000,0000,0000,,Because I still get-- where\Nis minus pi a squared?
Dialogue: 0,1:30:21.54,1:30:22.97,Default,,0000,0000,0000,,I don't see where\Nthe mistake is.
Dialogue: 0,1:30:22.97,1:30:26.63,Default,,0000,0000,0000,,And then the whole\Nclass, 12, 15-- reverse
Dialogue: 0,1:30:26.63,1:30:30.04,Default,,0000,0000,0000,,the integral limits.
Dialogue: 0,1:30:30.04,1:30:34.42,Default,,0000,0000,0000,,Change the integral limits\Nand you'll get it right.
Dialogue: 0,1:30:34.42,1:30:41.42,Default,,0000,0000,0000,,So we can all pretend that\Nwe want to do something else
Dialogue: 0,1:30:41.42,1:30:45.92,Default,,0000,0000,0000,,and we didn't finish high school\Nand we'll get a lot more money.
Dialogue: 0,1:30:45.92,1:30:47.93,Default,,0000,0000,0000,,The person who came to\Nfix my air conditioner
Dialogue: 0,1:30:47.93,1:30:55.40,Default,,0000,0000,0000,,said that he actually\Nmakes about $100 an hour.
Dialogue: 0,1:30:55.40,1:30:57.37,Default,,0000,0000,0000,,And I was thinking, wow.
Dialogue: 0,1:30:57.37,1:30:59.91,Default,,0000,0000,0000,,Wow, I'll never get there.
Dialogue: 0,1:30:59.91,1:31:01.75,Default,,0000,0000,0000,,But that's impressive.
Dialogue: 0,1:31:01.75,1:31:05.27,Default,,0000,0000,0000,,Just changing some\Nthings and fix,
Dialogue: 0,1:31:05.27,1:31:08.12,Default,,0000,0000,0000,,press the button, $100 an hour.
Dialogue: 0,1:31:08.12,1:31:11.44,Default,,0000,0000,0000,,STUDENT: But they don't\Nwork full time. [INAUDIBLE].
Dialogue: 0,1:31:11.44,1:31:15.47,Default,,0000,0000,0000,,PROFESSOR: Yeah, and I think\Nthey are paid by the job.
Dialogue: 0,1:31:15.47,1:31:19.00,Default,,0000,0000,0000,,But in any case, whether\Nit's a simple job
Dialogue: 0,1:31:19.00,1:31:21.36,Default,,0000,0000,0000,,and they just-- there\Nis a contact that's
Dialogue: 0,1:31:21.36,1:31:25.81,Default,,0000,0000,0000,,missing or something\Ntrivial, they still
Dialogue: 0,1:31:25.81,1:31:27.76,Default,,0000,0000,0000,,charge a lot of money.
Dialogue: 0,1:31:27.76,1:31:30.19,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:31:30.19,1:31:31.65,Default,,0000,0000,0000,,PROFESSOR: A professor?
Dialogue: 0,1:31:31.65,1:31:33.59,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:31:33.59,1:31:34.56,Default,,0000,0000,0000,,PROFESSOR: What?
Dialogue: 0,1:31:34.56,1:31:37.69,Default,,0000,0000,0000,,[LAUGHING] No.
Dialogue: 0,1:31:37.69,1:31:38.94,Default,,0000,0000,0000,,STUDENT: I know the professor.
Dialogue: 0,1:31:38.94,1:31:40.08,Default,,0000,0000,0000,,I won't tell you who.
Dialogue: 0,1:31:40.08,1:31:41.58,Default,,0000,0000,0000,,PROFESSOR: OK, I\Ndon't want to know.
Dialogue: 0,1:31:41.58,1:31:42.94,Default,,0000,0000,0000,,I don't want to know.
Dialogue: 0,1:31:42.94,1:31:44.76,Default,,0000,0000,0000,,But anyway, it's interesting.
Dialogue: 0,1:31:44.76,1:31:46.56,Default,,0000,0000,0000,,STUDENT: But he doesn't\Ndo in the college.
Dialogue: 0,1:31:46.56,1:31:50.96,Default,,0000,0000,0000,,He does outside the college\Nby just advising it.
Dialogue: 0,1:31:50.96,1:31:53.53,Default,,0000,0000,0000,,PROFESSOR: Oh, you mean\Nlike consulting or tutoring
Dialogue: 0,1:31:53.53,1:31:54.50,Default,,0000,0000,0000,,or stuff like that?
Dialogue: 0,1:31:54.50,1:31:56.93,Default,,0000,0000,0000,,STUDENT: Not tutoring,\Nconsulting for the--
Dialogue: 0,1:31:56.93,1:31:59.36,Default,,0000,0000,0000,,PROFESSOR: Consulting.
Dialogue: 0,1:31:59.36,1:32:02.76,Default,,0000,0000,0000,,Actually, I bet that\Nif we did tutoring,
Dialogue: 0,1:32:02.76,1:32:06.44,Default,,0000,0000,0000,,which we don't have time for,\Nwe would make a lot of money.
Dialogue: 0,1:32:06.44,1:32:08.76,Default,,0000,0000,0000,,But the nature of\Nmy job, for example,
Dialogue: 0,1:32:08.76,1:32:11.81,Default,,0000,0000,0000,,is that I work about\N60 hours a week, 65,
Dialogue: 0,1:32:11.81,1:32:14.79,Default,,0000,0000,0000,,and I will not have any time\Nleft to do other things,
Dialogue: 0,1:32:14.79,1:32:18.26,Default,,0000,0000,0000,,like consulting,\Ntutoring, and stuff.
Dialogue: 0,1:32:18.26,1:32:20.44,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:32:20.44,1:32:22.23,Default,,0000,0000,0000,,PROFESSOR: I don't need\Nnormally that much.
Dialogue: 0,1:32:22.23,1:32:24.71,Default,,0000,0000,0000,,I don't crave money that much.
Dialogue: 0,1:32:24.71,1:32:27.05,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:32:27.05,1:32:28.92,Default,,0000,0000,0000,,PROFESSOR: I have a\Nfriend who got a masters.
Dialogue: 0,1:32:28.92,1:32:31.16,Default,,0000,0000,0000,,She didn't get a PhD.
Dialogue: 0,1:32:31.16,1:32:34.63,Default,,0000,0000,0000,,She got an offer from\Nthis-- I told you about her.
Dialogue: 0,1:32:34.63,1:32:36.12,Default,,0000,0000,0000,,She moved to California.
Dialogue: 0,1:32:36.12,1:32:38.61,Default,,0000,0000,0000,,She was a single mom.
Dialogue: 0,1:32:38.61,1:32:42.14,Default,,0000,0000,0000,,She earns a lot of\Nmoney working for Pixar.
Dialogue: 0,1:32:42.14,1:32:44.79,Default,,0000,0000,0000,,And she helped with all\Nthe animation things.
Dialogue: 0,1:32:44.79,1:32:47.86,Default,,0000,0000,0000,,
Dialogue: 0,1:32:47.86,1:32:52.31,Default,,0000,0000,0000,,It was about 15 years\Nago that she started.
Dialogue: 0,1:32:52.31,1:32:54.19,Default,,0000,0000,0000,,And it was really hard.
Dialogue: 0,1:32:54.19,1:33:00.78,Default,,0000,0000,0000,,We were all on Toy Story\Nand that kind of-- what
Dialogue: 0,1:33:00.78,1:33:03.65,Default,,0000,0000,0000,,was that called?
Dialogue: 0,1:33:03.65,1:33:07.82,Default,,0000,0000,0000,,There were two\Nrendering algorithms,
Dialogue: 0,1:33:07.82,1:33:09.21,Default,,0000,0000,0000,,rendering algorithms.
Dialogue: 0,1:33:09.21,1:33:13.86,Default,,0000,0000,0000,,Two masters students\Nwere interested in that.
Dialogue: 0,1:33:13.86,1:33:16.24,Default,,0000,0000,0000,,They got in immediately.
Dialogue: 0,1:33:16.24,1:33:19.41,Default,,0000,0000,0000,,To be hired, I think\Na post-doc with a PhD
Dialogue: 0,1:33:19.41,1:33:23.15,Default,,0000,0000,0000,,was making about $40,000.
Dialogue: 0,1:33:23.15,1:33:24.80,Default,,0000,0000,0000,,That was my offer.
Dialogue: 0,1:33:24.80,1:33:31.12,Default,,0000,0000,0000,,My first offer was a post-doc\Nat Urbana-Champaign for $38,000
Dialogue: 0,1:33:31.12,1:33:34.11,Default,,0000,0000,0000,,while she was at the hundred\Nand something thousand
Dialogue: 0,1:33:34.11,1:33:36.98,Default,,0000,0000,0000,,dollars to start with\Nworking at Disney.
Dialogue: 0,1:33:36.98,1:33:41.26,Default,,0000,0000,0000,,Imagine-- with just a masters,\Nno aspiration for a PhD
Dialogue: 0,1:33:41.26,1:33:42.67,Default,,0000,0000,0000,,whatever.
Dialogue: 0,1:33:42.67,1:33:46.07,Default,,0000,0000,0000,,So in a way, if you're\Nthinking of doing this,
Dialogue: 0,1:33:46.07,1:33:50.88,Default,,0000,0000,0000,,a masters in mathematics\Nis probably paying off.
Dialogue: 0,1:33:50.88,1:33:54.34,Default,,0000,0000,0000,,Because it opens a\Nlot of doors for you.
Dialogue: 0,1:33:54.34,1:33:55.80,Default,,0000,0000,0000,,And that's just in general.
Dialogue: 0,1:33:55.80,1:33:58.87,Default,,0000,0000,0000,,I mean, masters in engineering\Nopens a lot of doors.
Dialogue: 0,1:33:58.87,1:34:02.22,Default,,0000,0000,0000,,But in a way, you\Npay a price after
Dialogue: 0,1:34:02.22,1:34:05.53,Default,,0000,0000,0000,,if you want to start\Neven further, get a PhD,
Dialogue: 0,1:34:05.53,1:34:06.57,Default,,0000,0000,0000,,stay in academia.
Dialogue: 0,1:34:06.57,1:34:08.47,Default,,0000,0000,0000,,Then you pay a price.
Dialogue: 0,1:34:08.47,1:34:10.54,Default,,0000,0000,0000,,And if you want to\Naugment your salary,
Dialogue: 0,1:34:10.54,1:34:16.36,Default,,0000,0000,0000,,you really have to be very\Ngood and accomplish some--
Dialogue: 0,1:34:16.36,1:34:19.16,Default,,0000,0000,0000,,get [INAUDIBLE]\Ntwo or three times
Dialogue: 0,1:34:19.16,1:34:22.65,Default,,0000,0000,0000,,and get higher up\Neach [INAUDIBLE].
Dialogue: 0,1:34:22.65,1:34:27.50,Default,,0000,0000,0000,,But we all struggle\Nwith these issues.
Dialogue: 0,1:34:27.50,1:34:31.42,Default,,0000,0000,0000,,It's a lot of work.
Dialogue: 0,1:34:31.42,1:34:34.36,Default,,0000,0000,0000,,But having a masters\Nin math is not so hard.
Dialogue: 0,1:34:34.36,1:34:39.06,Default,,0000,0000,0000,,If you like math,\Nit's easy to get it.
Dialogue: 0,1:34:39.06,1:34:39.83,Default,,0000,0000,0000,,It's a pleasure.
Dialogue: 0,1:34:39.83,1:34:41.54,Default,,0000,0000,0000,,It's not a lot of hours.
Dialogue: 0,1:34:41.54,1:34:45.98,Default,,0000,0000,0000,,I think in 36 hours in most\Nschools you can get a masters.
Dialogue: 0,1:34:45.98,1:34:49.60,Default,,0000,0000,0000,,And it's doable.
Dialogue: 0,1:34:49.60,1:35:02.49,Default,,0000,0000,0000,,All right, let's go back to\Nreview Chapter 11 briefly here.
Dialogue: 0,1:35:02.49,1:35:11.92,Default,,0000,0000,0000,,
Dialogue: 0,1:35:11.92,1:35:13.90,Default,,0000,0000,0000,,Is this on the midterm?
Dialogue: 0,1:35:13.90,1:35:16.38,Default,,0000,0000,0000,,No, but it's going\Nto be on the final.
Dialogue: 0,1:35:16.38,1:35:23.33,Default,,0000,0000,0000,,
Dialogue: 0,1:35:23.33,1:35:33.18,Default,,0000,0000,0000,,Assume you have a\Nx equals u plus v,
Dialogue: 0,1:35:33.18,1:35:47.39,Default,,0000,0000,0000,,y equals u minus v. Write\Nthe following derivative.
Dialogue: 0,1:35:47.39,1:35:57.05,Default,,0000,0000,0000,,
Dialogue: 0,1:35:57.05,1:36:14.01,Default,,0000,0000,0000,,dx/dv where u of t equals\Nt squared and v equals t.
Dialogue: 0,1:36:14.01,1:36:31.35,Default,,0000,0000,0000,,Do these both directly\Nand by writing a chain
Dialogue: 0,1:36:31.35,1:36:38.28,Default,,0000,0000,0000,,rule for the values you have.
Dialogue: 0,1:36:38.28,1:36:43.53,Default,,0000,0000,0000,,
Dialogue: 0,1:36:43.53,1:36:45.91,Default,,0000,0000,0000,,OK, how do we do this directly?
Dialogue: 0,1:36:45.91,1:36:47.91,Default,,0000,0000,0000,,It's probably the simplest way.
Dialogue: 0,1:36:47.91,1:36:56.91,Default,,0000,0000,0000,,
Dialogue: 0,1:36:56.91,1:37:02.17,Default,,0000,0000,0000,,Replace u by t squared, replace\Nv by t and see what you have.
Dialogue: 0,1:37:02.17,1:37:03.22,Default,,0000,0000,0000,,So 1, directly.
Dialogue: 0,1:37:03.22,1:37:07.81,Default,,0000,0000,0000,,
Dialogue: 0,1:37:07.81,1:37:12.27,Default,,0000,0000,0000,,X of t equals t squared plus t.
Dialogue: 0,1:37:12.27,1:37:16.41,Default,,0000,0000,0000,,y of t equals t squared minus t.
Dialogue: 0,1:37:16.41,1:37:23.66,Default,,0000,0000,0000,,
Dialogue: 0,1:37:23.66,1:37:24.93,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:37:24.93,1:37:32.86,Default,,0000,0000,0000,,So it's a piece of cake.\Ndx/dt equals 2t plus 1.
Dialogue: 0,1:37:32.86,1:37:37.94,Default,,0000,0000,0000,,Unfortunately, this is just\Nthe first part of the problem.
Dialogue: 0,1:37:37.94,1:37:41.41,Default,,0000,0000,0000,,And it's actually\N[INAUDIBLE] to show the chain
Dialogue: 0,1:37:41.41,1:37:43.52,Default,,0000,0000,0000,,rule for the mappings we have.
Dialogue: 0,1:37:43.52,1:37:45.29,Default,,0000,0000,0000,,And what mappings do we have?
Dialogue: 0,1:37:45.29,1:37:54.46,Default,,0000,0000,0000,,We have a map from t\Nto u of t and v of t.
Dialogue: 0,1:37:54.46,1:38:00.34,Default,,0000,0000,0000,,And then again, from u of t and\Nv of t to x of t and y of t.
Dialogue: 0,1:38:00.34,1:38:04.76,Default,,0000,0000,0000,,And the transformation is\Nwhat? x equals u plus v,
Dialogue: 0,1:38:04.76,1:38:10.07,Default,,0000,0000,0000,,y equals u minus v. And we have\Nanother transformation here.
Dialogue: 0,1:38:10.07,1:38:15.14,Default,,0000,0000,0000,,So how do you write dx/dt?
Dialogue: 0,1:38:15.14,1:38:20.55,Default,,0000,0000,0000,,x is a function\Nof u and v, right?
Dialogue: 0,1:38:20.55,1:38:25.33,Default,,0000,0000,0000,,So first you say\Nthat dx/dv round,
Dialogue: 0,1:38:25.33,1:38:28.18,Default,,0000,0000,0000,,which means we do it\Nwith the first variable.
Dialogue: 0,1:38:28.18,1:38:31.21,Default,,0000,0000,0000,,I'll write it for\Nyou to see better,
Dialogue: 0,1:38:31.21,1:38:39.12,Default,,0000,0000,0000,,that initially your x and y\Nwere functions of u and v.
Dialogue: 0,1:38:39.12,1:38:51.25,Default,,0000,0000,0000,,Times-- what is that?\Ndv/dt plus dx/du.
Dialogue: 0,1:38:51.25,1:38:52.83,Default,,0000,0000,0000,,You can change the order.
Dialogue: 0,1:38:52.83,1:38:55.56,Default,,0000,0000,0000,,If you didn't like\Nthat I started with v,
Dialogue: 0,1:38:55.56,1:39:00.10,Default,,0000,0000,0000,,I could have started with\Nthe u, and the u, and the v,
Dialogue: 0,1:39:00.10,1:39:00.72,Default,,0000,0000,0000,,and the v here.
Dialogue: 0,1:39:00.72,1:39:03.74,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,1:39:03.74,1:39:06.13,Default,,0000,0000,0000,,Guys, do you mind, really?
Dialogue: 0,1:39:06.13,1:39:07.64,Default,,0000,0000,0000,,v, v. u, u.
Dialogue: 0,1:39:07.64,1:39:08.70,Default,,0000,0000,0000,,Shooting cowboys?
Dialogue: 0,1:39:08.70,1:39:10.100,Default,,0000,0000,0000,,Doesn't matter, remember just\Nthat they're [INAUDIBLE].
Dialogue: 0,1:39:10.100,1:39:14.03,Default,,0000,0000,0000,,
Dialogue: 0,1:39:14.03,1:39:15.46,Default,,0000,0000,0000,,D sorry, d.
Dialogue: 0,1:39:15.46,1:39:18.16,Default,,0000,0000,0000,,Because where there\Nis no other variable,
Dialogue: 0,1:39:18.16,1:39:22.56,Default,,0000,0000,0000,,we would put v. So dx/dt?
Dialogue: 0,1:39:22.56,1:39:25.46,Default,,0000,0000,0000,,Lets see if we get\Nthe same answer.
Dialogue: 0,1:39:25.46,1:39:26.32,Default,,0000,0000,0000,,We should.
Dialogue: 0,1:39:26.32,1:39:27.81,Default,,0000,0000,0000,,What is dx/dt?
Dialogue: 0,1:39:27.81,1:39:30.91,Default,,0000,0000,0000,,1, from here.
Dialogue: 0,1:39:30.91,1:39:32.38,Default,,0000,0000,0000,,What is dv/dt?
Dialogue: 0,1:39:32.38,1:39:37.99,Default,,0000,0000,0000,,
Dialogue: 0,1:39:37.99,1:39:42.64,Default,,0000,0000,0000,,1 plus dx/du.
Dialogue: 0,1:39:42.64,1:39:46.14,Default,,0000,0000,0000,,1, du/dt.
Dialogue: 0,1:39:46.14,1:39:47.79,Default,,0000,0000,0000,,2t.
Dialogue: 0,1:39:47.79,1:39:50.08,Default,,0000,0000,0000,,If we were to do\Nthe same thing-- so
Dialogue: 0,1:39:50.08,1:39:52.10,Default,,0000,0000,0000,,we got the same answer.
Dialogue: 0,1:39:52.10,1:39:56.14,Default,,0000,0000,0000,,If you want to do\Nthe same thing,
Dialogue: 0,1:39:56.14,1:40:01.91,Default,,0000,0000,0000,,quickly with respect\Nto say dy/dt,
Dialogue: 0,1:40:01.91,1:40:05.79,Default,,0000,0000,0000,,suppose that most finals\Nask you to do both.
Dialogue: 0,1:40:05.79,1:40:08.14,Default,,0000,0000,0000,,I have students\Nwho didn't finish
Dialogue: 0,1:40:08.14,1:40:13.34,Default,,0000,0000,0000,,because they didn't\Nhave the time to finish,
Dialogue: 0,1:40:13.34,1:40:14.92,Default,,0000,0000,0000,,but that was just my policy.
Dialogue: 0,1:40:14.92,1:40:17.70,Default,,0000,0000,0000,,When I grade it,\NI gave them 100%,
Dialogue: 0,1:40:17.70,1:40:20.64,Default,,0000,0000,0000,,no matter if they\Nstopped here, because I
Dialogue: 0,1:40:20.64,1:40:22.89,Default,,0000,0000,0000,,said you prove to me that\Nyou know the chain rule.
Dialogue: 0,1:40:22.89,1:40:26.73,Default,,0000,0000,0000,,Why would I punish you further?
Dialogue: 0,1:40:26.73,1:40:27.96,Default,,0000,0000,0000,,So that's what I do.
Dialogue: 0,1:40:27.96,1:40:32.28,Default,,0000,0000,0000,,But I want you to do it\Nnow, without my help.
Dialogue: 0,1:40:32.28,1:40:34.67,Default,,0000,0000,0000,,Both ways, dy/dt.
Dialogue: 0,1:40:34.67,1:40:39.50,Default,,0000,0000,0000,,First you do it\Nwith the chain rule.
Dialogue: 0,1:40:39.50,1:40:42.39,Default,,0000,0000,0000,,First you write those\Nthree [INAUDIBLE].
Dialogue: 0,1:40:42.39,1:40:47.73,Default,,0000,0000,0000,,
Dialogue: 0,1:40:47.73,1:40:48.68,Default,,0000,0000,0000,,dy del y.
Dialogue: 0,1:40:48.68,1:40:51.58,Default,,0000,0000,0000,,
Dialogue: 0,1:40:51.58,1:40:56.42,Default,,0000,0000,0000,,del u, du/dt, plus del y.
Dialogue: 0,1:40:56.42,1:40:59.05,Default,,0000,0000,0000,,del v, dv/dt.
Dialogue: 0,1:40:59.05,1:41:03.77,Default,,0000,0000,0000,,I'm not going to write it\Ndown, you write it down.
Dialogue: 0,1:41:03.77,1:41:05.88,Default,,0000,0000,0000,,What I'm going to\Nwrite down is what
Dialogue: 0,1:41:05.88,1:41:07.88,Default,,0000,0000,0000,,you tell me the numbers are.
Dialogue: 0,1:41:07.88,1:41:13.18,Default,,0000,0000,0000,,
Dialogue: 0,1:41:13.18,1:41:13.81,Default,,0000,0000,0000,,For everything.
Dialogue: 0,1:41:13.81,1:41:19.72,Default,,0000,0000,0000,,
Dialogue: 0,1:41:19.72,1:41:24.68,Default,,0000,0000,0000,,STUDENT: Dy divided by d
Dialogue: 0,1:41:24.68,1:41:29.64,Default,,0000,0000,0000,,PROFESSOR: Or just give me\Nthe final answer in terms of
Dialogue: 0,1:41:29.64,1:41:30.14,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,1:41:30.14,1:41:42.09,Default,,0000,0000,0000,,
Dialogue: 0,1:41:42.09,1:41:47.70,Default,,0000,0000,0000,,What are the two [INAUDIBLE]?
Dialogue: 0,1:41:47.70,1:41:49.67,Default,,0000,0000,0000,,Tell me.
Dialogue: 0,1:41:49.67,1:41:56.58,Default,,0000,0000,0000,,Tell me, this times this,\Nplus this times that.
Dialogue: 0,1:41:56.58,1:41:58.17,Default,,0000,0000,0000,,What?
Dialogue: 0,1:41:58.17,1:41:59.37,Default,,0000,0000,0000,,So let's write down.
Dialogue: 0,1:41:59.37,1:42:00.89,Default,,0000,0000,0000,,Let's write it down together.
Dialogue: 0,1:42:00.89,1:42:16.26,Default,,0000,0000,0000,,dy/du, du/dt, plus dy/dv dv/dt.
Dialogue: 0,1:42:16.26,1:42:16.76,Default,,0000,0000,0000,,Alright.
Dialogue: 0,1:42:16.76,1:42:30.48,Default,,0000,0000,0000,,
Dialogue: 0,1:42:30.48,1:42:34.54,Default,,0000,0000,0000,,1 This is 1.
Dialogue: 0,1:42:34.54,1:42:35.77,Default,,0000,0000,0000,,How much is dy/dt?
Dialogue: 0,1:42:35.77,1:42:38.55,Default,,0000,0000,0000,,
Dialogue: 0,1:42:38.55,1:42:40.14,Default,,0000,0000,0000,,Or du/dt, I'm sorry.
Dialogue: 0,1:42:40.14,1:42:44.94,Default,,0000,0000,0000,,I said dy, it's du/dt.
Dialogue: 0,1:42:44.94,1:42:50.61,Default,,0000,0000,0000,,Plus minus 1, excellent.
Dialogue: 0,1:42:50.61,1:42:54.03,Default,,0000,0000,0000,,Times 1.
Dialogue: 0,1:42:54.03,1:42:56.07,Default,,0000,0000,0000,,Of course you would have\Ndone the same thing,
Dialogue: 0,1:42:56.07,1:42:59.64,Default,,0000,0000,0000,,by plugging in the\Nvariables and saying well,
Dialogue: 0,1:42:59.64,1:43:04.77,Default,,0000,0000,0000,,I have y, which is this\Nis t squared, this is t,
Dialogue: 0,1:43:04.77,1:43:09.27,Default,,0000,0000,0000,,and I have t squared minus\Nt prime is 2t minus 1.
Dialogue: 0,1:43:09.27,1:43:13.69,Default,,0000,0000,0000,,That's a simpler way\Nto verify [INAUDIBLE].
Dialogue: 0,1:43:13.69,1:43:14.47,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:43:14.47,1:43:20.07,Default,,0000,0000,0000,,So remember to do that,\Nhave this in mind,
Dialogue: 0,1:43:20.07,1:43:25.13,Default,,0000,0000,0000,,because on the final you may\Nhave something like that.
Dialogue: 0,1:43:25.13,1:43:28.62,Default,,0000,0000,0000,,As we keep going in\Nthe month of April,
Dialogue: 0,1:43:28.62,1:43:32.62,Default,,0000,0000,0000,,I'm going to do as much review\Nas possible for the final.
Dialogue: 0,1:43:32.62,1:43:37.47,Default,,0000,0000,0000,,Mark a star, or F, not the grade\NF, but F around for the final,
Dialogue: 0,1:43:37.47,1:43:45.15,Default,,0000,0000,0000,,put F and circle there to say\Nreview this for the final.
Dialogue: 0,1:43:45.15,1:43:48.95,Default,,0000,0000,0000,,And since we are still\Nin chapter 11 review,
Dialogue: 0,1:43:48.95,1:43:56.87,Default,,0000,0000,0000,,we'll do another problem\Nof F, final review
Dialogue: 0,1:43:56.87,1:44:02.40,Default,,0000,0000,0000,,that I didn't put on the midterm\Nbut it may be on the final.
Dialogue: 0,1:44:02.40,1:44:06.96,Default,,0000,0000,0000,,Let's say given the\Nconstraint x squared
Dialogue: 0,1:44:06.96,1:44:13.42,Default,,0000,0000,0000,,plus y squared plus z squared\Nequals 5, compute z sub x
Dialogue: 0,1:44:13.42,1:44:16.66,Default,,0000,0000,0000,,and z sub y.
Dialogue: 0,1:44:16.66,1:44:18.08,Default,,0000,0000,0000,,How do you do that?
Dialogue: 0,1:44:18.08,1:44:20.55,Default,,0000,0000,0000,,What is this called,\Nactually, and why is it
Dialogue: 0,1:44:20.55,1:44:22.72,Default,,0000,0000,0000,,so important for the final?
Dialogue: 0,1:44:22.72,1:44:24.22,Default,,0000,0000,0000,,It's called implicit\Ndifferentiation
Dialogue: 0,1:44:24.22,1:44:28.98,Default,,0000,0000,0000,,and it appears on almost every\Nfinal, at least once a year,
Dialogue: 0,1:44:28.98,1:44:31.77,Default,,0000,0000,0000,,so there is always\Na big possibility
Dialogue: 0,1:44:31.77,1:44:35.88,Default,,0000,0000,0000,,that you are going to\Nsee something like that.
Dialogue: 0,1:44:35.88,1:44:41.81,Default,,0000,0000,0000,,I taught you how to think in\Nterms of implicit functions.
Dialogue: 0,1:44:41.81,1:44:49.73,Default,,0000,0000,0000,,If you think of z as\Na function of x and y.
Dialogue: 0,1:44:49.73,1:44:52.34,Default,,0000,0000,0000,,That's a way of changing\Nyour perspective.
Dialogue: 0,1:44:52.34,1:44:56.91,Default,,0000,0000,0000,,So you say, OK, I\Nunderstand that z
Dialogue: 0,1:44:56.91,1:45:01.25,Default,,0000,0000,0000,,has to be viewed as a\Nfunction of x and y.
Dialogue: 0,1:45:01.25,1:45:05.18,Default,,0000,0000,0000,,I'm just changing\Nmy perspective.
Dialogue: 0,1:45:05.18,1:45:10.09,Default,,0000,0000,0000,,STUDENT: For that one,\Nwouldn't you just solve for z?
Dialogue: 0,1:45:10.09,1:45:11.23,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,1:45:11.23,1:45:15.88,Default,,0000,0000,0000,,Solving for z would make\Nyour life a lot harder.
Dialogue: 0,1:45:15.88,1:45:17.94,Default,,0000,0000,0000,,The point of\Nimplicit functions is
Dialogue: 0,1:45:17.94,1:45:20.47,Default,,0000,0000,0000,,that you don't separate them.
Dialogue: 0,1:45:20.47,1:45:22.57,Default,,0000,0000,0000,,If you're going\Nto separate them,
Dialogue: 0,1:45:22.57,1:45:26.04,Default,,0000,0000,0000,,you have to separately\Nintegrate these.
Dialogue: 0,1:45:26.04,1:45:27.92,Default,,0000,0000,0000,,And it's a headache.
Dialogue: 0,1:45:27.92,1:45:30.85,Default,,0000,0000,0000,,It's easier-- actually\Nit's a good question.
Dialogue: 0,1:45:30.85,1:45:36.44,Default,,0000,0000,0000,,It's easier to do z sub x,\Nz sub y without splitting it
Dialogue: 0,1:45:36.44,1:45:37.38,Default,,0000,0000,0000,,into two cases.
Dialogue: 0,1:45:37.38,1:45:39.96,Default,,0000,0000,0000,,
Dialogue: 0,1:45:39.96,1:45:42.98,Default,,0000,0000,0000,,Step two.
Dialogue: 0,1:45:42.98,1:45:44.56,Default,,0000,0000,0000,,Differentiate this\Nwith respect to x.
Dialogue: 0,1:45:44.56,1:45:47.52,Default,,0000,0000,0000,,What do we have?
Dialogue: 0,1:45:47.52,1:45:56.14,Default,,0000,0000,0000,,2x plus 0 plus the chain rule--\Ndon't write the chain rule.
Dialogue: 0,1:45:56.14,1:46:00.42,Default,,0000,0000,0000,,2 jumping down, it jumped down.
Dialogue: 0,1:46:00.42,1:46:06.34,Default,,0000,0000,0000,,2z times-- cover the\N2 with your hand.
Dialogue: 0,1:46:06.34,1:46:07.80,Default,,0000,0000,0000,,z sub x, very good.
Dialogue: 0,1:46:07.80,1:46:11.73,Default,,0000,0000,0000,,z prime with respect\Nto x equals zero.
Dialogue: 0,1:46:11.73,1:46:13.79,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:46:13.79,1:46:18.26,Default,,0000,0000,0000,,So z sub x, step three.
Dialogue: 0,1:46:18.26,1:46:19.40,Default,,0000,0000,0000,,And the last step.
Dialogue: 0,1:46:19.40,1:46:22.79,Default,,0000,0000,0000,,See sub x will be what?
Dialogue: 0,1:46:22.79,1:46:23.62,Default,,0000,0000,0000,,Pull it out.
Dialogue: 0,1:46:23.62,1:46:26.26,Default,,0000,0000,0000,,Pull this 2 out.
Dialogue: 0,1:46:26.26,1:46:31.60,Default,,0000,0000,0000,,Minus x over z.
Dialogue: 0,1:46:31.60,1:46:36.25,Default,,0000,0000,0000,,
Dialogue: 0,1:46:36.25,1:46:40.06,Default,,0000,0000,0000,,The other one is symmetric.
Dialogue: 0,1:46:40.06,1:46:43.51,Default,,0000,0000,0000,,Alex said let's be smart\Nand not do the whole thing
Dialogue: 0,1:46:43.51,1:46:44.39,Default,,0000,0000,0000,,all over again.
Dialogue: 0,1:46:44.39,1:46:46.85,Default,,0000,0000,0000,,Look at beautiful\Nsymmetric polynomial.
Dialogue: 0,1:46:46.85,1:46:49.99,Default,,0000,0000,0000,,You would have to be a\Nlittle bit careful with when
Dialogue: 0,1:46:49.99,1:46:53.85,Default,,0000,0000,0000,,you have a 1 here and\Ny would have a 2 here.
Dialogue: 0,1:46:53.85,1:46:56.91,Default,,0000,0000,0000,,It wouldn't be\Nsymmetric in x and y.
Dialogue: 0,1:46:56.91,1:47:00.18,Default,,0000,0000,0000,,But here, if you reverse\Nthe roles of x and y,
Dialogue: 0,1:47:00.18,1:47:02.02,Default,,0000,0000,0000,,it's not a big deal.
Dialogue: 0,1:47:02.02,1:47:02.89,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:47:02.89,1:47:07.26,Default,,0000,0000,0000,,Here we are. z sub y\Nequals minus y over z.
Dialogue: 0,1:47:07.26,1:47:10.25,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,1:47:10.25,1:47:14.09,Default,,0000,0000,0000,,Keep this in mind\Nfor-- I also saw,
Dialogue: 0,1:47:14.09,1:47:18.61,Default,,0000,0000,0000,,when I was looking at\Nthe [INAUDIBLE] library
Dialogue: 0,1:47:18.61,1:47:21.12,Default,,0000,0000,0000,,files, [INAUDIBLE].
Dialogue: 0,1:47:21.12,1:47:25.36,Default,,0000,0000,0000,,I also saw exams, and I was\Nlooking at your reviews there.
Dialogue: 0,1:47:25.36,1:47:27.75,Default,,0000,0000,0000,,I was looking at [INAUDIBLE].
Dialogue: 0,1:47:27.75,1:47:32.25,Default,,0000,0000,0000,,The University of Houston has\Na very beautiful online, free
Dialogue: 0,1:47:32.25,1:47:36.74,Default,,0000,0000,0000,,library of calculus 1\Nand calculus 2 exams
Dialogue: 0,1:47:36.74,1:47:39.68,Default,,0000,0000,0000,,that I found very useful.
Dialogue: 0,1:47:39.68,1:47:44.10,Default,,0000,0000,0000,,Now, one of them--\Nlisten to me so you
Dialogue: 0,1:47:44.10,1:47:47.33,Default,,0000,0000,0000,,don't fall through this crack.
Dialogue: 0,1:47:47.33,1:47:54.30,Default,,0000,0000,0000,,On the Cal 2 exam, they\Nwrote something like that.
Dialogue: 0,1:47:54.30,1:48:00.21,Default,,0000,0000,0000,,
Dialogue: 0,1:48:00.21,1:48:05.85,Default,,0000,0000,0000,,You don't have to write 1 over\Nx squared, and then compute.
Dialogue: 0,1:48:05.85,1:48:09.12,Default,,0000,0000,0000,,You just say, OK, if the natural\Npart of the of the argument
Dialogue: 0,1:48:09.12,1:48:13.43,Default,,0000,0000,0000,,is 5, then the\Nargument is a constant.
Dialogue: 0,1:48:13.43,1:48:15.40,Default,,0000,0000,0000,,And I don't care\Nwhat constant it
Dialogue: 0,1:48:15.40,1:48:18.30,Default,,0000,0000,0000,,is, it it's something\Nthat prime will give me 0,
Dialogue: 0,1:48:18.30,1:48:20.09,Default,,0000,0000,0000,,it's the same problem.
Dialogue: 0,1:48:20.09,1:48:21.39,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:48:21.39,1:48:26.04,Default,,0000,0000,0000,,So in that case, I'm going\Nhave just what kind of change?
Dialogue: 0,1:48:26.04,1:48:28.50,Default,,0000,0000,0000,,This will be to the 5.
Dialogue: 0,1:48:28.50,1:48:30.74,Default,,0000,0000,0000,,And I still have 0.
Dialogue: 0,1:48:30.74,1:48:32.85,Default,,0000,0000,0000,,It's the same answer.
Dialogue: 0,1:48:32.85,1:48:37.22,Default,,0000,0000,0000,,They just wanted to play\Ngames, and you can play games.
Dialogue: 0,1:48:37.22,1:48:40.76,Default,,0000,0000,0000,,For example, you can make this.
Dialogue: 0,1:48:40.76,1:48:43.45,Default,,0000,0000,0000,,If you really have\Na working mind,
Dialogue: 0,1:48:43.45,1:48:50.51,Default,,0000,0000,0000,,and most mathematicians do,\Ngive this to your students.
Dialogue: 0,1:48:50.51,1:48:54.30,Default,,0000,0000,0000,,I mean, most people\Nfreak out so bad
Dialogue: 0,1:48:54.30,1:48:58.69,Default,,0000,0000,0000,,when they see that, the\Nwon't even touch it.
Dialogue: 0,1:48:58.69,1:49:02.22,Default,,0000,0000,0000,,It's just all in the head.
Dialogue: 0,1:49:02.22,1:49:12.62,Default,,0000,0000,0000,,Remember that log in base\N17 of a would be what?
Dialogue: 0,1:49:12.62,1:49:16.26,Default,,0000,0000,0000,,STUDENT: If it's a\Nconstant, it's to the 17th.
Dialogue: 0,1:49:16.26,1:49:17.26,Default,,0000,0000,0000,,PROFESSOR: Who knows?
Dialogue: 0,1:49:17.26,1:49:19.26,Default,,0000,0000,0000,,STUDENT: What do you\Nmean, you don't do that?
Dialogue: 0,1:49:19.26,1:49:23.44,Default,,0000,0000,0000,,PROFESSOR: No, no, expressed\Nin terms of natural logs.
Dialogue: 0,1:49:23.44,1:49:24.40,Default,,0000,0000,0000,,STUDENT: Natural log?
Dialogue: 0,1:49:24.40,1:49:26.88,Default,,0000,0000,0000,,The natural log of a\Nover natural log of 17.
Dialogue: 0,1:49:26.88,1:49:27.76,Default,,0000,0000,0000,,PROFESSOR: Very good.
Dialogue: 0,1:49:27.76,1:49:29.68,Default,,0000,0000,0000,,So what does this matter?
Dialogue: 0,1:49:29.68,1:49:33.38,Default,,0000,0000,0000,,In the end, you multiply 2, you\Ndo the derivative, you still
Dialogue: 0,1:49:33.38,1:49:34.95,Default,,0000,0000,0000,,get the same answer.
Dialogue: 0,1:49:34.95,1:49:38.14,Default,,0000,0000,0000,,Some people are trying\Nto make things scarier
Dialogue: 0,1:49:38.14,1:49:40.61,Default,,0000,0000,0000,,than they are, just to impress.
Dialogue: 0,1:49:40.61,1:49:44.62,Default,,0000,0000,0000,,When you think of the\Nproblem, it's a piece of cake.
Dialogue: 0,1:49:44.62,1:49:47.59,Default,,0000,0000,0000,,So don't be afraid of it.
Dialogue: 0,1:49:47.59,1:50:05.49,Default,,0000,0000,0000,,
Dialogue: 0,1:50:05.49,1:50:16.57,Default,,0000,0000,0000,,Oh, by the way, the final\Nexam-- so the midterm would
Dialogue: 0,1:50:16.57,1:50:24.11,Default,,0000,0000,0000,,be 10 problems pus 1 extra one.
Dialogue: 0,1:50:24.11,1:50:27.09,Default,,0000,0000,0000,,
Dialogue: 0,1:50:27.09,1:50:29.21,Default,,0000,0000,0000,,And did I tell\Nyou how much time?
Dialogue: 0,1:50:29.21,1:50:34.75,Default,,0000,0000,0000,,It's going to be approximately--\NI say, in actual time.
Dialogue: 0,1:50:34.75,1:50:35.38,Default,,0000,0000,0000,,Needed time.
Dialogue: 0,1:50:35.38,1:50:39.52,Default,,0000,0000,0000,,
Dialogue: 0,1:50:39.52,1:50:43.63,Default,,0000,0000,0000,,For average student,\Nit'll be about 40 minutes.
Dialogue: 0,1:50:43.63,1:50:53.24,Default,,0000,0000,0000,,Allowed time one\Nhour and 40 minutes.
Dialogue: 0,1:50:53.24,1:50:57.60,Default,,0000,0000,0000,,So you have from 12:10 to 1:50.
Dialogue: 0,1:50:57.60,1:51:01.00,Default,,0000,0000,0000,,
Dialogue: 0,1:51:01.00,1:51:06.10,Default,,0000,0000,0000,,On the final, just a guess,\Nabout 15-16 problems.
Dialogue: 0,1:51:06.10,1:51:09.80,Default,,0000,0000,0000,,
Dialogue: 0,1:51:09.80,1:51:11.82,Default,,0000,0000,0000,,Two hours and a half.
Dialogue: 0,1:51:11.82,1:51:15.98,Default,,0000,0000,0000,,
Dialogue: 0,1:51:15.98,1:51:18.38,Default,,0000,0000,0000,,STUDENT: Is that allowed time?
Dialogue: 0,1:51:18.38,1:51:21.45,Default,,0000,0000,0000,,PROFESSOR: Not allowed time.
Dialogue: 0,1:51:21.45,1:51:24.65,Default,,0000,0000,0000,,If I manage to review very\Nwell with you on these concepts
Dialogue: 0,1:51:24.65,1:51:29.71,Default,,0000,0000,0000,,guys, I guarantee you're not\Ngoing to need more than 1.5.
Dialogue: 0,1:51:29.71,1:51:31.45,Default,,0000,0000,0000,,This is the allowed time.
Dialogue: 0,1:51:31.45,1:51:34.40,Default,,0000,0000,0000,,The allowed time for somebody\Nwho hasn't practiced enough.
Dialogue: 0,1:51:34.40,1:51:38.71,Default,,0000,0000,0000,,
Dialogue: 0,1:51:38.71,1:51:41.00,Default,,0000,0000,0000,,Let me ask you what you\Nthink would be good.
Dialogue: 0,1:51:41.00,1:51:44.22,Default,,0000,0000,0000,,
Dialogue: 0,1:51:44.22,1:51:45.89,Default,,0000,0000,0000,,I have a bunch of finals.
Dialogue: 0,1:51:45.89,1:51:50.41,Default,,0000,0000,0000,,All the finals for Cal 3\Nlook very similar in nature.
Dialogue: 0,1:51:50.41,1:51:54.25,Default,,0000,0000,0000,,The same kind of topics\Nas the ones I review.
Dialogue: 0,1:51:54.25,1:51:57.36,Default,,0000,0000,0000,,I would like to know\Nwhat you would prefer.
Dialogue: 0,1:51:57.36,1:52:02.99,Default,,0000,0000,0000,,I would have two or\Nthree finals to give you.
Dialogue: 0,1:52:02.99,1:52:05.84,Default,,0000,0000,0000,,Would you prefer that you\Ntry them yourselves first,
Dialogue: 0,1:52:05.84,1:52:08.05,Default,,0000,0000,0000,,and then I give\Nyou the solutions?
Dialogue: 0,1:52:08.05,1:52:08.84,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,1:52:08.84,1:52:11.26,Default,,0000,0000,0000,,PROFESSOR: Or I give you the\Nsolutions from the beginning?
Dialogue: 0,1:52:11.26,1:52:17.40,Default,,0000,0000,0000,,I'll give you the\Nsolutions anyway, but--
Dialogue: 0,1:52:17.40,1:52:19.23,Default,,0000,0000,0000,,STUDENT: Can it just\Nbe on a separate sheet,
Dialogue: 0,1:52:19.23,1:52:20.96,Default,,0000,0000,0000,,where we could go through--
Dialogue: 0,1:52:20.96,1:52:23.77,Default,,0000,0000,0000,,PROFESSOR: No, no, they are\Nalready on a separate sheet.
Dialogue: 0,1:52:23.77,1:52:28.44,Default,,0000,0000,0000,,For example, I have Fall\N2013, or Spring 2012.
Dialogue: 0,1:52:28.44,1:52:30.20,Default,,0000,0000,0000,,They are from\Ndifferent semesters.
Dialogue: 0,1:52:30.20,1:52:31.48,Default,,0000,0000,0000,,They are all very similar.
Dialogue: 0,1:52:31.48,1:52:35.29,Default,,0000,0000,0000,,So I'll give you-- I have\Ntwo files on this blog.
Dialogue: 0,1:52:35.29,1:52:37.46,Default,,0000,0000,0000,,The exam itself\Nand the solutions.
Dialogue: 0,1:52:37.46,1:52:43.04,Default,,0000,0000,0000,,I'll give you the exam, I'll\Nlet you work if for two weeks,
Dialogue: 0,1:52:43.04,1:52:44.58,Default,,0000,0000,0000,,and then I'll give\Nyou the solutions.
Dialogue: 0,1:52:44.58,1:52:45.60,Default,,0000,0000,0000,,How about that?
Dialogue: 0,1:52:45.60,1:52:48.73,Default,,0000,0000,0000,,Put you'll work on it, you\Ndon't cheat on me and any way.
Dialogue: 0,1:52:48.73,1:52:52.19,Default,,0000,0000,0000,,Because working things\Nyourself, you're learning.
Dialogue: 0,1:52:52.19,1:52:55.65,Default,,0000,0000,0000,,If you expect other people\Nto feed you the solutions,
Dialogue: 0,1:52:55.65,1:52:57.63,Default,,0000,0000,0000,,you're not learning as much.
Dialogue: 0,1:52:57.63,1:53:01.58,Default,,0000,0000,0000,,You are learning some, but\Nyou're not learning as much.
Dialogue: 0,1:53:01.58,1:53:03.55,Default,,0000,0000,0000,,OK, it's getting ready.
Dialogue: 0,1:53:03.55,1:53:05.53,Default,,0000,0000,0000,,I have a few more\Nthings to tell you.
Dialogue: 0,1:53:05.53,1:53:19.86,Default,,0000,0000,0000,,
Dialogue: 0,1:53:19.86,1:53:22.19,Default,,0000,0000,0000,,Chapter 13, necessary reminders.
Dialogue: 0,1:53:22.19,1:53:30.88,Default,,0000,0000,0000,,
Dialogue: 0,1:53:30.88,1:53:34.76,Default,,0000,0000,0000,,The gradient is very important.
Dialogue: 0,1:53:34.76,1:53:45.79,Default,,0000,0000,0000,,Gradient of a function\Nf from r 2 to r.
Dialogue: 0,1:53:45.79,1:53:51.78,Default,,0000,0000,0000,,We write that as z equals\Nf of x and y, usually.
Dialogue: 0,1:53:51.78,1:53:53.13,Default,,0000,0000,0000,,And what was the gradient?
Dialogue: 0,1:53:53.13,1:53:55.06,Default,,0000,0000,0000,,This is good review\Nfor the midterm,
Dialogue: 0,1:53:55.06,1:53:58.95,Default,,0000,0000,0000,,but that's the beginning\Nof section 13.1.
Dialogue: 0,1:53:58.95,1:54:01.39,Default,,0000,0000,0000,,So I'm actually\Ndoing two things,
Dialogue: 0,1:54:01.39,1:54:03.86,Default,,0000,0000,0000,,I'm giving you the\Nbeginning of section 13.1,
Dialogue: 0,1:54:03.86,1:54:06.52,Default,,0000,0000,0000,,while doing review\Nfor the final.
Dialogue: 0,1:54:06.52,1:54:12.02,Default,,0000,0000,0000,,
Dialogue: 0,1:54:12.02,1:54:18.16,Default,,0000,0000,0000,,You have gradient of f of x,\Ny-- some people are ask me,
Dialogue: 0,1:54:18.16,1:54:22.44,Default,,0000,0000,0000,,do you prefer that I\Nwrite on the exams,
Dialogue: 0,1:54:22.44,1:54:27.51,Default,,0000,0000,0000,,on the midterm, on the\Nfinal a granular bracket?
Dialogue: 0,1:54:27.51,1:54:34.57,Default,,0000,0000,0000,,Or do you prefer I write this in\Nthis form in the standard base
Dialogue: 0,1:54:34.57,1:54:35.23,Default,,0000,0000,0000,,i, j.
Dialogue: 0,1:54:35.23,1:54:36.76,Default,,0000,0000,0000,,Standard [INAUDIBLE].
Dialogue: 0,1:54:36.76,1:54:38.53,Default,,0000,0000,0000,,It doesn't make a difference.
Dialogue: 0,1:54:38.53,1:54:40.75,Default,,0000,0000,0000,,In linear algebra,\Nyou would have
Dialogue: 0,1:54:40.75,1:54:43.37,Default,,0000,0000,0000,,to say what bases you are using.
Dialogue: 0,1:54:43.37,1:54:45.99,Default,,0000,0000,0000,,But in calculus, we\Nassume that you are using
Dialogue: 0,1:54:45.99,1:54:48.46,Default,,0000,0000,0000,,the bases which is 1, 0, 0, 1.
Dialogue: 0,1:54:48.46,1:54:51.11,Default,,0000,0000,0000,,
Dialogue: 0,1:54:51.11,1:54:55.92,Default,,0000,0000,0000,,So you have space in a plane.
Dialogue: 0,1:54:55.92,1:54:58.87,Default,,0000,0000,0000,,I'm indifferent.
Dialogue: 0,1:54:58.87,1:55:01.73,Default,,0000,0000,0000,,This is OK, you can\Nuse whatever you like.
Dialogue: 0,1:55:01.73,1:55:05.55,Default,,0000,0000,0000,,If you have a function of\Nthree variables, of course
Dialogue: 0,1:55:05.55,1:55:06.71,Default,,0000,0000,0000,,you have a gradient.
Dialogue: 0,1:55:06.71,1:55:09.76,Default,,0000,0000,0000,,
Dialogue: 0,1:55:09.76,1:55:16.41,Default,,0000,0000,0000,,But I prefer to write f sub x,\Ni plus f sub y, j plus f sub z,
Dialogue: 0,1:55:16.41,1:55:19.16,Default,,0000,0000,0000,,the beginning of some ck.
Dialogue: 0,1:55:19.16,1:55:31.45,Default,,0000,0000,0000,,
Dialogue: 0,1:55:31.45,1:55:33.90,Default,,0000,0000,0000,,Has anybody heard of\Ndivergence before?
Dialogue: 0,1:55:33.90,1:55:35.86,Default,,0000,0000,0000,,What is divergence?
Dialogue: 0,1:55:35.86,1:55:37.83,Default,,0000,0000,0000,,Gradient is something\Nyou've heard before.
Dialogue: 0,1:55:37.83,1:55:40.78,Default,,0000,0000,0000,,But divergence, have you\Never heard of divergence?
Dialogue: 0,1:55:40.78,1:55:47.01,Default,,0000,0000,0000,,
Dialogue: 0,1:55:47.01,1:55:58.95,Default,,0000,0000,0000,,Maybe in mechanical engineering,\Nhave you heard of it before?
Dialogue: 0,1:55:58.95,1:56:00.89,Default,,0000,0000,0000,,No?
Dialogue: 0,1:56:00.89,1:56:02.26,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:56:02.26,1:56:05.64,Default,,0000,0000,0000,,Suppose that you\Nhave a function,
Dialogue: 0,1:56:05.64,1:56:08.05,Default,,0000,0000,0000,,and that is a vector\Nvalue function.
Dialogue: 0,1:56:08.05,1:56:13.30,Default,,0000,0000,0000,,
Dialogue: 0,1:56:13.30,1:56:14.61,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,1:56:14.61,1:56:21.41,Default,,0000,0000,0000,,A vector in itself will\Nhave coordinates at x, y.
Dialogue: 0,1:56:21.41,1:56:29.00,Default,,0000,0000,0000,,And it's assumed that\Nwill be f1 of x, y-- no,
Dialogue: 0,1:56:29.00,1:56:31.48,Default,,0000,0000,0000,,this is not a vector,\Nthat's scalar.
Dialogue: 0,1:56:31.48,1:56:35.12,Default,,0000,0000,0000,,Times i, plus f2 x, y, j.
Dialogue: 0,1:56:35.12,1:56:38.77,Default,,0000,0000,0000,,And somebody, one of you\Nactually showed me-- of course
Dialogue: 0,1:56:38.77,1:56:44.86,Default,,0000,0000,0000,,in mechanics-- you were\Nusing divergence in that.
Dialogue: 0,1:56:44.86,1:56:49.36,Default,,0000,0000,0000,,And I feel bad that I was\Nnot the first maybe for some
Dialogue: 0,1:56:49.36,1:56:53.22,Default,,0000,0000,0000,,of you, I was not the first to\Ntell you what divergence means.
Dialogue: 0,1:56:53.22,1:57:00.30,Default,,0000,0000,0000,,Divergence f, assuming that f\Nwould be missing one function.
Dialogue: 0,1:57:00.30,1:57:01.53,Default,,0000,0000,0000,,What does this mean?
Dialogue: 0,1:57:01.53,1:57:06.13,Default,,0000,0000,0000,,It means that it's differential,\Nbut its derivatives
Dialogue: 0,1:57:06.13,1:57:07.00,Default,,0000,0000,0000,,are continuous.
Dialogue: 0,1:57:07.00,1:57:09.67,Default,,0000,0000,0000,,
Dialogue: 0,1:57:09.67,1:57:13.50,Default,,0000,0000,0000,,We did note that this\Nis the diff of f.
Dialogue: 0,1:57:13.50,1:57:18.47,Default,,0000,0000,0000,,But in engineering, they denoted\Nmost of the time like that.
Dialogue: 0,1:57:18.47,1:57:22.38,Default,,0000,0000,0000,,There's not a lot of symbols,\Nbut you saw the gradient
Dialogue: 0,1:57:22.38,1:57:24.37,Default,,0000,0000,0000,,with a little dot after that.
Dialogue: 0,1:57:24.37,1:57:29.85,Default,,0000,0000,0000,,
Dialogue: 0,1:57:29.85,1:57:32.48,Default,,0000,0000,0000,,If you don't put the dot,\Nit doesn't make sense
Dialogue: 0,1:57:32.48,1:57:34.24,Default,,0000,0000,0000,,with what I'm saying.
Dialogue: 0,1:57:34.24,1:57:37.48,Default,,0000,0000,0000,,So pay attention to the dot.
Dialogue: 0,1:57:37.48,1:57:38.14,Default,,0000,0000,0000,,Alright.
Dialogue: 0,1:57:38.14,1:57:39.03,Default,,0000,0000,0000,,What does this mean?
Dialogue: 0,1:57:39.03,1:57:44.24,Default,,0000,0000,0000,,It means that you\Nhave the derivative
Dialogue: 0,1:57:44.24,1:57:47.56,Default,,0000,0000,0000,,of the first component\Nwith respect to x.
Dialogue: 0,1:57:47.56,1:57:50.84,Default,,0000,0000,0000,,
Dialogue: 0,1:57:50.84,1:57:55.10,Default,,0000,0000,0000,,Plus it's going to\Nbe a value function,
Dialogue: 0,1:57:55.10,1:57:58.07,Default,,0000,0000,0000,,the derivative of the second\Ncomponent with respect to y.
Dialogue: 0,1:57:58.07,1:58:03.87,Default,,0000,0000,0000,,
Dialogue: 0,1:58:03.87,1:58:08.44,Default,,0000,0000,0000,,How do you generalize\Nfor higher powers?
Dialogue: 0,1:58:08.44,1:58:18.07,Default,,0000,0000,0000,,What if you have a function--\Nassume you have a function
Dialogue: 0,1:58:18.07,1:58:20.03,Default,,0000,0000,0000,,f that looks like that.
Dialogue: 0,1:58:20.03,1:58:30.63,Default,,0000,0000,0000,,If x1, x2, x n variables,\Ni plus the last one will be
Dialogue: 0,1:58:30.63,1:58:35.23,Default,,0000,0000,0000,,a [INAUDIBLE] of x1, x2\Nx n variables times--
Dialogue: 0,1:58:35.23,1:58:40.37,Default,,0000,0000,0000,,
Dialogue: 0,1:58:40.37,1:58:42.32,Default,,0000,0000,0000,,eij doesn't make any sense.
Dialogue: 0,1:58:42.32,1:58:46.70,Default,,0000,0000,0000,,So e1, e2, e n would\Nbe the standard bases.
Dialogue: 0,1:58:46.70,1:58:54.51,Default,,0000,0000,0000,,[INAUDIBLE] doesn't\Nmake the [INAUDIBLE]
Dialogue: 0,1:58:54.51,1:59:02.94,Default,,0000,0000,0000,,for a computer scientist,\Nan ordered set of components
Dialogue: 0,1:59:02.94,1:59:04.81,Default,,0000,0000,0000,,and values.
Dialogue: 0,1:59:04.81,1:59:08.30,Default,,0000,0000,0000,,And would be 7, 17, 29.
Dialogue: 0,1:59:08.30,1:59:10.10,Default,,0000,0000,0000,,Some natural numbers.
Dialogue: 0,1:59:10.10,1:59:18.85,Default,,0000,0000,0000,,So all these values are taken\Nin r, with every r x is in on.
Dialogue: 0,1:59:18.85,1:59:21.83,Default,,0000,0000,0000,,What do you think that\Nthe divergence of u
Dialogue: 0,1:59:21.83,1:59:24.32,Default,,0000,0000,0000,,would be in that case?
Dialogue: 0,1:59:24.32,1:59:27.79,Default,,0000,0000,0000,,If you were to generalize y.
Dialogue: 0,1:59:27.79,1:59:32.76,Default,,0000,0000,0000,,First component, prime with\Nrespect to the first variable.
Dialogue: 0,1:59:32.76,1:59:35.75,Default,,0000,0000,0000,,Alright.
Dialogue: 0,1:59:35.75,1:59:41.74,Default,,0000,0000,0000,,Only plus second component with\Nrespect to the second variable,
Dialogue: 0,1:59:41.74,1:59:43.35,Default,,0000,0000,0000,,and so on.
Dialogue: 0,1:59:43.35,1:59:46.01,Default,,0000,0000,0000,,Last component with respect\Nto the last variable.
Dialogue: 0,1:59:46.01,1:59:49.45,Default,,0000,0000,0000,,So that would be the\Ngeneral definition.
Dialogue: 0,1:59:49.45,1:59:58.48,Default,,0000,0000,0000,,And now I'm asking you, assume\Nx that somebody gives you
Dialogue: 0,1:59:58.48,2:00:03.21,Default,,0000,0000,0000,,a function, f of x, y.
Dialogue: 0,2:00:03.21,2:00:04.80,Default,,0000,0000,0000,,And with domain in the plane.
Dialogue: 0,2:00:04.80,2:00:10.13,Default,,0000,0000,0000,,
Dialogue: 0,2:00:10.13,2:00:14.18,Default,,0000,0000,0000,,And f is c1.
Dialogue: 0,2:00:14.18,2:00:18.20,Default,,0000,0000,0000,,[INAUDIBLE] with\Ncontinuous radius.
Dialogue: 0,2:00:18.20,2:00:20.18,Default,,0000,0000,0000,,Actually no, I want more.
Dialogue: 0,2:00:20.18,2:00:23.16,Default,,0000,0000,0000,,I want c2.
Dialogue: 0,2:00:23.16,2:00:34.13,Default,,0000,0000,0000,,So twice differential bond,\Nwith continuous variables.
Dialogue: 0,2:00:34.13,2:00:39.11,Default,,0000,0000,0000,,
Dialogue: 0,2:00:39.11,2:00:45.23,Default,,0000,0000,0000,,Compute a.
Dialogue: 0,2:00:45.23,2:00:46.04,Default,,0000,0000,0000,,Gradient double f.
Dialogue: 0,2:00:46.04,2:00:48.75,Default,,0000,0000,0000,,
Dialogue: 0,2:00:48.75,2:00:53.54,Default,,0000,0000,0000,,b, divergence of gradient\Nof f, which you can also
Dialogue: 0,2:00:53.54,2:01:00.28,Default,,0000,0000,0000,,write divergence like engineers\Ndo, or gradient to our left.
Dialogue: 0,2:01:00.28,2:01:07.66,Default,,0000,0000,0000,,Do you know what name that's\Nthe last thing we need today,
Dialogue: 0,2:01:07.66,2:01:14.80,Default,,0000,0000,0000,,the name for this operator.
Dialogue: 0,2:01:14.80,2:01:15.78,Default,,0000,0000,0000,,Underlined here.
Dialogue: 0,2:01:15.78,2:01:21.69,Default,,0000,0000,0000,,
Dialogue: 0,2:01:21.69,2:01:25.10,Default,,0000,0000,0000,,So what would be a good\Nname for this kind?
Dialogue: 0,2:01:25.10,2:01:29.16,Default,,0000,0000,0000,,I'm curious if any of you\Nknow if from engineering.
Dialogue: 0,2:01:29.16,2:01:30.62,Default,,0000,0000,0000,,But we will see.
Dialogue: 0,2:01:30.62,2:01:39.84,Default,,0000,0000,0000,,
Dialogue: 0,2:01:39.84,2:01:44.33,Default,,0000,0000,0000,,So we are in 13.
Dialogue: 0,2:01:44.33,2:01:50.69,Default,,0000,0000,0000,,a will be the gradient of\Nf, that's a piece of cake.
Dialogue: 0,2:01:50.69,2:01:52.10,Default,,0000,0000,0000,,She only wants the\Ndefinition, let
Dialogue: 0,2:01:52.10,2:01:57.52,Default,,0000,0000,0000,,me give her the definition\Nof f sub xi, plus f sub y j.
Dialogue: 0,2:01:57.52,2:01:59.02,Default,,0000,0000,0000,,And if we don't\Nknow what those are,
Dialogue: 0,2:01:59.02,2:02:02.40,Default,,0000,0000,0000,,this is the variable\Nwith respect to x.
Dialogue: 0,2:02:02.40,2:02:07.72,Default,,0000,0000,0000,,And then for dy, df/dy j.
Dialogue: 0,2:02:07.72,2:02:08.22,Default,,0000,0000,0000,,Good.
Dialogue: 0,2:02:08.22,2:02:11.58,Default,,0000,0000,0000,,So we know what a gradient is.
Dialogue: 0,2:02:11.58,2:02:15.47,Default,,0000,0000,0000,,What will this divergence\Nwith the gradient be?
Dialogue: 0,2:02:15.47,2:02:17.41,Default,,0000,0000,0000,,That sounds really weird.
Dialogue: 0,2:02:17.41,2:02:21.79,Default,,0000,0000,0000,,
Dialogue: 0,2:02:21.79,2:02:25.32,Default,,0000,0000,0000,,According to this\Ndefinition, we have
Dialogue: 0,2:02:25.32,2:02:30.08,Default,,0000,0000,0000,,to see what big\NF1 and big F2 are.
Dialogue: 0,2:02:30.08,2:02:32.56,Default,,0000,0000,0000,,Or, big F1 and big F2.
Dialogue: 0,2:02:32.56,2:02:35.55,Default,,0000,0000,0000,,I'm going to take\Nthem in breaths.
Dialogue: 0,2:02:35.55,2:02:38.53,Default,,0000,0000,0000,,Big F1, and big F2.
Dialogue: 0,2:02:38.53,2:02:42.01,Default,,0000,0000,0000,,
Dialogue: 0,2:02:42.01,2:02:48.07,Default,,0000,0000,0000,,The components of the vector,\Nyou apply divergence to it.
Dialogue: 0,2:02:48.07,2:02:50.07,Default,,0000,0000,0000,,So now that I'm finishing,\Nwhat do I have to do?
Dialogue: 0,2:02:50.07,2:02:53.27,Default,,0000,0000,0000,,Somebody tell me.
Dialogue: 0,2:02:53.27,2:02:57.57,Default,,0000,0000,0000,,So yeah, I can write it\Nf sub x plus f sub y,
Dialogue: 0,2:02:57.57,2:03:03.47,Default,,0000,0000,0000,,and that shows that you are\Nfast, and very [INAUDIBLE].
Dialogue: 0,2:03:03.47,2:03:05.60,Default,,0000,0000,0000,,I can also write\Nit like this, which
Dialogue: 0,2:03:05.60,2:03:11.38,Default,,0000,0000,0000,,is what I meant-- this is what\Nthe book shows first course.
Dialogue: 0,2:03:11.38,2:03:12.83,Default,,0000,0000,0000,,This is the same thing.
Dialogue: 0,2:03:12.83,2:03:15.38,Default,,0000,0000,0000,,
Dialogue: 0,2:03:15.38,2:03:21.05,Default,,0000,0000,0000,,Now I really doubt that\Nsomebody knows that,
Dialogue: 0,2:03:21.05,2:03:25.33,Default,,0000,0000,0000,,but I want to give a\Ndollar to the person who
Dialogue: 0,2:03:25.33,2:03:27.06,Default,,0000,0000,0000,,would know the name of this.
Dialogue: 0,2:03:27.06,2:03:30.04,Default,,0000,0000,0000,,
Dialogue: 0,2:03:30.04,2:03:33.00,Default,,0000,0000,0000,,Let me see if I have a dollar.
Dialogue: 0,2:03:33.00,2:03:37.96,Default,,0000,0000,0000,,
Dialogue: 0,2:03:37.96,2:03:40.88,Default,,0000,0000,0000,,Maybe I have $0.35 and a candy.
Dialogue: 0,2:03:40.88,2:03:44.33,Default,,0000,0000,0000,,Does anybody know\Nthe name of this?
Dialogue: 0,2:03:44.33,2:03:55.14,Default,,0000,0000,0000,,Maybe I can help you a little\Nbit. $0.25 $0.85, $0.95.
Dialogue: 0,2:03:55.14,2:03:57.09,Default,,0000,0000,0000,,Do you know what this is?
Dialogue: 0,2:03:57.09,2:04:01.46,Default,,0000,0000,0000,,I'll give you a hint, because I\Nknow in mechanical engineering,
Dialogue: 0,2:04:01.46,2:04:03.03,Default,,0000,0000,0000,,I already introduced this.
Dialogue: 0,2:04:03.03,2:04:08.96,Default,,0000,0000,0000,,And some physics classes and\Nwe would try angle in front,
Dialogue: 0,2:04:08.96,2:04:15.02,Default,,0000,0000,0000,,and we did all of this\Ntriangle operators in the way.
Dialogue: 0,2:04:15.02,2:04:18.97,Default,,0000,0000,0000,,And we can play a game.
Dialogue: 0,2:04:18.97,2:04:26.89,Default,,0000,0000,0000,,It's a letter that\Nstarts with L.
Dialogue: 0,2:04:26.89,2:04:30.46,Default,,0000,0000,0000,,But $0.95 we have\Ntwo more minutes.
Dialogue: 0,2:04:30.46,2:04:32.70,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]?
Dialogue: 0,2:04:32.70,2:04:33.28,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,2:04:33.28,2:04:38.26,Default,,0000,0000,0000,,You are getting close though,\Nbecause-- [INTERPOSING VOICES]
Dialogue: 0,2:04:38.26,2:04:41.23,Default,,0000,0000,0000,,What kind of operator is this?
Dialogue: 0,2:04:41.23,2:04:43.98,Default,,0000,0000,0000,,You're getting close, $0.95.
Dialogue: 0,2:04:43.98,2:04:46.44,Default,,0000,0000,0000,,Tomorrow, I don't need this.
Dialogue: 0,2:04:46.44,2:04:47.86,Default,,0000,0000,0000,,When I go to the\Nairports, I don't
Dialogue: 0,2:04:47.86,2:04:49.47,Default,,0000,0000,0000,,like to have coins with me.
Dialogue: 0,2:04:49.47,2:04:56.99,Default,,0000,0000,0000,,
Dialogue: 0,2:04:56.99,2:04:58.87,Default,,0000,0000,0000,,STUDENT: Laplace?
Dialogue: 0,2:04:58.87,2:04:59.57,Default,,0000,0000,0000,,PROFESSOR: $0.95!
Dialogue: 0,2:04:59.57,2:05:02.05,Default,,0000,0000,0000,,I wish I had a dollar.
Dialogue: 0,2:05:02.05,2:05:04.53,Default,,0000,0000,0000,,Yes, this is the famous\NLaplace operator.
Dialogue: 0,2:05:04.53,2:05:06.02,Default,,0000,0000,0000,,Laplace was a mathematician.
Dialogue: 0,2:05:06.02,2:05:11.50,Default,,0000,0000,0000,,
Dialogue: 0,2:05:11.50,2:05:13.57,Default,,0000,0000,0000,,And remember it.
Dialogue: 0,2:05:13.57,2:05:16.61,Default,,0000,0000,0000,,If you take-- how\Nmany of you-- you all
Dialogue: 0,2:05:16.61,2:05:19.79,Default,,0000,0000,0000,,have to take differential\Nequations, right?
Dialogue: 0,2:05:19.79,2:05:21.51,Default,,0000,0000,0000,,They will kill you with that.
Dialogue: 0,2:05:21.51,2:05:23.94,Default,,0000,0000,0000,,You're going to see\Nthis all the time.
Dialogue: 0,2:05:23.94,2:05:25.91,Default,,0000,0000,0000,,This Laplace operator\Nis really famous.
Dialogue: 0,2:05:25.91,2:05:28.65,Default,,0000,0000,0000,,
Dialogue: 0,2:05:28.65,2:05:31.34,Default,,0000,0000,0000,,I will tell you more\Nwhen I come back.
Dialogue: 0,2:05:31.34,2:05:34.53,Default,,0000,0000,0000,,I'm going to see you on Tuesday.
Dialogue: 0,2:05:34.53,2:05:36.86,Default,,0000,0000,0000,,We'll knock out the midterm.
Dialogue: 0,2:05:36.86,2:05:40.60,Default,,0000,0000,0000,,For you, the people who feel\Noverly prepared for midterm
Dialogue: 0,2:05:40.60,2:05:45.54,Default,,0000,0000,0000,,can go ahead and\Nread section 13.1
Dialogue: 0,2:05:45.54,2:05:48.84,Default,,0000,0000,0000,,and see a little bit\Nabout Laplace's operator.
Dialogue: 0,2:05:48.84,2:05:51.29,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,2:05:51.29,2:06:45.86,Default,,0000,0000,0000,,