[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:02.02,Default,,0000,0000,0000,,PROFESSOR: I'll\Ngo over the exam.
Dialogue: 0,0:00:02.02,0:00:04.17,Default,,0000,0000,0000,,It's good review for\Nthe final, and it's
Dialogue: 0,0:00:04.17,0:00:09.53,Default,,0000,0000,0000,,a good feedback for you in\Ncase you have questions.
Dialogue: 0,0:00:09.53,0:00:15.07,Default,,0000,0000,0000,,I do not change grades,\NI do not curve your exam.
Dialogue: 0,0:00:15.07,0:00:19.49,Default,,0000,0000,0000,,I do not make adjustments\Nafter I give you the grade.
Dialogue: 0,0:00:19.49,0:00:22.19,Default,,0000,0000,0000,,Therefore, it's very\Nimportant for me
Dialogue: 0,0:00:22.19,0:00:25.38,Default,,0000,0000,0000,,to explain why you\Ngot what you got.
Dialogue: 0,0:00:25.38,0:00:30.30,Default,,0000,0000,0000,,Not everybody did\Nwell on this exam.
Dialogue: 0,0:00:30.30,0:00:35.06,Default,,0000,0000,0000,,Most people did\Npretty good, and I'm
Dialogue: 0,0:00:35.06,0:00:41.80,Default,,0000,0000,0000,,quite happy with what I see\Nas a average for the class.
Dialogue: 0,0:00:41.80,0:00:44.12,Default,,0000,0000,0000,,However, there are\Nmany open questions
Dialogue: 0,0:00:44.12,0:00:47.92,Default,,0000,0000,0000,,from many people, things\Nthey didn't quite understand,
Dialogue: 0,0:00:47.92,0:00:51.18,Default,,0000,0000,0000,,and I would like\Nto discuss those.
Dialogue: 0,0:00:51.18,0:00:54.42,Default,,0000,0000,0000,,
Dialogue: 0,0:00:54.42,0:01:09.16,Default,,0000,0000,0000,,First of all, the midterm\Nexam was 11 questions.
Dialogue: 0,0:01:09.16,0:01:16.48,Default,,0000,0000,0000,,10 were mandatory, so\Nthe maximum possible
Dialogue: 0,0:01:16.48,0:01:20.80,Default,,0000,0000,0000,,percentage-wise was 110%.
Dialogue: 0,0:01:20.80,0:01:25.40,Default,,0000,0000,0000,,So for somebody who\Ndid perfectly fine,
Dialogue: 0,0:01:25.40,0:01:26.43,Default,,0000,0000,0000,,they would have 110%.
Dialogue: 0,0:01:26.43,0:01:29.13,Default,,0000,0000,0000,,
Dialogue: 0,0:01:29.13,0:01:33.81,Default,,0000,0000,0000,,There is one person\Nonly who got the high.
Dialogue: 0,0:01:33.81,0:01:36.75,Default,,0000,0000,0000,,I didn't disclose\Nhis name, but I would
Dialogue: 0,0:01:36.75,0:01:39.58,Default,,0000,0000,0000,,like to say congratulations.
Dialogue: 0,0:01:39.58,0:01:46.83,Default,,0000,0000,0000,,And I'm going to go ahead and\Nsolve each problem with you,
Dialogue: 0,0:01:46.83,0:01:48.28,Default,,0000,0000,0000,,for you.
Dialogue: 0,0:01:48.28,0:01:53.60,Default,,0000,0000,0000,,So you have the\Nfunction f of x, y,
Dialogue: 0,0:01:53.60,0:01:55.95,Default,,0000,0000,0000,,to be x squared minus y squared.
Dialogue: 0,0:01:55.95,0:02:01.80,Default,,0000,0000,0000,,And the differential was f\Nsub x dx plus f sub y dy,
Dialogue: 0,0:02:01.80,0:02:06.24,Default,,0000,0000,0000,,whihc is 2x dx, minus 2y dy.
Dialogue: 0,0:02:06.24,0:02:09.07,Default,,0000,0000,0000,,
Dialogue: 0,0:02:09.07,0:02:11.19,Default,,0000,0000,0000,,That was something very easy.
Dialogue: 0,0:02:11.19,0:02:18.32,Default,,0000,0000,0000,,It was not supposed to give you\Nany headache, and most of you
Dialogue: 0,0:02:18.32,0:02:21.73,Default,,0000,0000,0000,,did a fine job on this one.
Dialogue: 0,0:02:21.73,0:02:24.44,Default,,0000,0000,0000,,What created some\Nproblems to most students
Dialogue: 0,0:02:24.44,0:02:26.65,Default,,0000,0000,0000,,was the second problem, though.
Dialogue: 0,0:02:26.65,0:02:29.99,Default,,0000,0000,0000,,And I sorry to hear\Nthat, sorry to see that.
Dialogue: 0,0:02:29.99,0:02:33.46,Default,,0000,0000,0000,,Find the directional\Nderivative of a function,
Dialogue: 0,0:02:33.46,0:02:35.93,Default,,0000,0000,0000,,the same function as before.
Dialogue: 0,0:02:35.93,0:02:39.89,Default,,0000,0000,0000,,
Dialogue: 0,0:02:39.89,0:02:45.52,Default,,0000,0000,0000,,So I have taken advantage\Nof the previous problem,
Dialogue: 0,0:02:45.52,0:02:50.49,Default,,0000,0000,0000,,in order to make\Nyour do time shorter.
Dialogue: 0,0:02:50.49,0:02:56.44,Default,,0000,0000,0000,,At the point p of coordinates\Nx equals 0, y equals 1,
Dialogue: 0,0:02:56.44,0:02:59.42,Default,,0000,0000,0000,,you will have a direction\Ngiven by the vector.
Dialogue: 0,0:02:59.42,0:03:07.09,Default,,0000,0000,0000,,What does it mean\Ndirection given?
Dialogue: 0,0:03:07.09,0:03:14.70,Default,,0000,0000,0000,,Analyze that direction given\Nby the vector means what?
Dialogue: 0,0:03:14.70,0:03:20.84,Default,,0000,0000,0000,,Not the vector i minus j is y,\Nbecause it's not a unit vector.
Dialogue: 0,0:03:20.84,0:03:24.69,Default,,0000,0000,0000,,What is the corresponding\Ndirection given by it?
Dialogue: 0,0:03:24.69,0:03:27.28,Default,,0000,0000,0000,,A corresponding\Ndirection given by it
Dialogue: 0,0:03:27.28,0:03:32.60,Default,,0000,0000,0000,,is 1 over square root of 2i\Nminus 1 over square root of 2j.
Dialogue: 0,0:03:32.60,0:03:36.90,Default,,0000,0000,0000,,So it's a collinear vector--\Nthat is, unit varies.
Dialogue: 0,0:03:36.90,0:03:38.40,Default,,0000,0000,0000,,Say it again, [INAUDIBLE]?
Dialogue: 0,0:03:38.40,0:03:43.80,Default,,0000,0000,0000,,The direction u represents\Na collinear vector.
Dialogue: 0,0:03:43.80,0:03:47.21,Default,,0000,0000,0000,,So pointing in the\Nsame direction as v,
Dialogue: 0,0:03:47.21,0:03:49.14,Default,,0000,0000,0000,,but it has to be unitary.
Dialogue: 0,0:03:49.14,0:03:49.64,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:03:49.64,0:03:54.51,Default,,0000,0000,0000,,Because the definition of\Nthe directional derivative
Dialogue: 0,0:03:54.51,0:03:59.85,Default,,0000,0000,0000,,is a function along the\Ndirection u at the point p,
Dialogue: 0,0:03:59.85,0:04:06.26,Default,,0000,0000,0000,,was given by the formula\Npartial derivative at p
Dialogue: 0,0:04:06.26,0:04:10.35,Default,,0000,0000,0000,,and at 1, plus partial\Nderivative at p times 2.
Dialogue: 0,0:04:10.35,0:04:12.42,Default,,0000,0000,0000,,Did I expect to\Nwrite all this down?
Dialogue: 0,0:04:12.42,0:04:15.35,Default,,0000,0000,0000,,Yes, I did, as I\Nshowed you last time.
Dialogue: 0,0:04:15.35,0:04:19.47,Default,,0000,0000,0000,,So you have 2x evaluated\Nat-- what is x?
Dialogue: 0,0:04:19.47,0:04:20.24,Default,,0000,0000,0000,,0.
Dialogue: 0,0:04:20.24,0:04:24.49,Default,,0000,0000,0000,,1 times u1.
Dialogue: 0,0:04:24.49,0:04:31.44,Default,,0000,0000,0000,,This is u1 minus 2y,\Nevaluated at 0, 1 times
Dialogue: 0,0:04:31.44,0:04:36.23,Default,,0000,0000,0000,,minus 1 over root\N2, which is u2.
Dialogue: 0,0:04:36.23,0:04:45.54,Default,,0000,0000,0000,,
Dialogue: 0,0:04:45.54,0:04:48.12,Default,,0000,0000,0000,,Well that means the\Nfirst term goes away,
Dialogue: 0,0:04:48.12,0:04:50.42,Default,,0000,0000,0000,,because this is going to be 0.
Dialogue: 0,0:04:50.42,0:04:54.25,Default,,0000,0000,0000,,And after the second\Nterm, you have a plus.
Dialogue: 0,0:04:54.25,0:04:57.57,Default,,0000,0000,0000,,y is 1, thank god, that's easy.
Dialogue: 0,0:04:57.57,0:05:01.67,Default,,0000,0000,0000,,2 over square root of\N2, the answer is root 2.
Dialogue: 0,0:05:01.67,0:05:05.05,Default,,0000,0000,0000,,So any other answer would\Nnormally receiving a 0.
Dialogue: 0,0:05:05.05,0:05:10.61,Default,,0000,0000,0000,,The answer was b, square root 2.
Dialogue: 0,0:05:10.61,0:05:17.79,Default,,0000,0000,0000,,Now, on number three, the\Nfunction given is different.
Dialogue: 0,0:05:17.79,0:05:24.88,Default,,0000,0000,0000,,f of x, y equals e to the xy.
Dialogue: 0,0:05:24.88,0:05:29.33,Default,,0000,0000,0000,,
Dialogue: 0,0:05:29.33,0:05:34.60,Default,,0000,0000,0000,,And they say, the\Ngradient of this function
Dialogue: 0,0:05:34.60,0:05:37.01,Default,,0000,0000,0000,,is at an arbitrary point.
Dialogue: 0,0:05:37.01,0:05:38.46,Default,,0000,0000,0000,,Say is again?
Dialogue: 0,0:05:38.46,0:05:42.15,Default,,0000,0000,0000,,The gradient of this function\Nis at an arbitrary point.
Dialogue: 0,0:05:42.15,0:05:44.62,Default,,0000,0000,0000,,That was only part\Nof the problem.
Dialogue: 0,0:05:44.62,0:05:48.58,Default,,0000,0000,0000,,A little bit of credit for\Njust writing the gradient.
Dialogue: 0,0:05:48.58,0:05:52.05,Default,,0000,0000,0000,,This is actually\Neasy, a piece of cake.
Dialogue: 0,0:05:52.05,0:05:53.04,Default,,0000,0000,0000,,y is that.
Dialogue: 0,0:05:53.04,0:05:57.50,Default,,0000,0000,0000,,You have f sub x\Ni plus f sub y j.
Dialogue: 0,0:05:57.50,0:06:11.13,Default,,0000,0000,0000,,It equals y to the xy\Ni plus x to the xyj.
Dialogue: 0,0:06:11.13,0:06:12.55,Default,,0000,0000,0000,,That's very good.
Dialogue: 0,0:06:12.55,0:06:18.69,Default,,0000,0000,0000,,
Dialogue: 0,0:06:18.69,0:06:20.67,Default,,0000,0000,0000,,Alright, OK?
Dialogue: 0,0:06:20.67,0:06:39.66,Default,,0000,0000,0000,,Then, which direction-- it's\Njust the gradient right?
Dialogue: 0,0:06:39.66,0:06:41.41,Default,,0000,0000,0000,,The direction corresponds\Nto the gradient.
Dialogue: 0,0:06:41.41,0:06:43.71,Default,,0000,0000,0000,,They don't ask you for the u.
Dialogue: 0,0:06:43.71,0:06:45.58,Default,,0000,0000,0000,,Actually, you don't need the u.
Dialogue: 0,0:06:45.58,0:06:49.04,Default,,0000,0000,0000,,You just need the tangent\Nplane in this case.
Dialogue: 0,0:06:49.04,0:06:53.26,Default,,0000,0000,0000,,And if you know the equation\Nof the tangent plane,
Dialogue: 0,0:06:53.26,0:06:56.98,Default,,0000,0000,0000,,as I told you to remember that,\Nthat would be very helpful.
Dialogue: 0,0:06:56.98,0:06:58.96,Default,,0000,0000,0000,,Write your answer in\Nthe space provided.
Dialogue: 0,0:06:58.96,0:07:01.33,Default,,0000,0000,0000,,So what did I expect you to do?
Dialogue: 0,0:07:01.33,0:07:09.22,Default,,0000,0000,0000,,First this, and then write\Nthe equation z minus z0
Dialogue: 0,0:07:09.22,0:07:17.11,Default,,0000,0000,0000,,equals f sub x times x minus\Nx0, plus f sub y, y minus y0.
Dialogue: 0,0:07:17.11,0:07:22.27,Default,,0000,0000,0000,,Here at the point p,\Nevaluate it at the point p.
Dialogue: 0,0:07:22.27,0:07:24.47,Default,,0000,0000,0000,,But attention, what\Nis the point p?
Dialogue: 0,0:07:24.47,0:07:29.21,Default,,0000,0000,0000,,Well, p is the origin,\Nbecause we say at the origin.
Dialogue: 0,0:07:29.21,0:07:32.25,Default,,0000,0000,0000,,Oh, so that makes things easier.
Dialogue: 0,0:07:32.25,0:07:33.43,Default,,0000,0000,0000,,I'm not done.
Dialogue: 0,0:07:33.43,0:07:37.10,Default,,0000,0000,0000,,Half of the problem\Nis still coming.
Dialogue: 0,0:07:37.10,0:07:40.83,Default,,0000,0000,0000,,If you did until this point, I\Ncan only give you 5 out of 10
Dialogue: 0,0:07:40.83,0:07:41.82,Default,,0000,0000,0000,,or something like that.
Dialogue: 0,0:07:41.82,0:07:44.54,Default,,0000,0000,0000,,
Dialogue: 0,0:07:44.54,0:07:48.62,Default,,0000,0000,0000,,Many people made\Na mistake at z0.
Dialogue: 0,0:07:48.62,0:07:53.06,Default,,0000,0000,0000,,Attention guys, you plus\Nin that 0, you don't get 0.
Dialogue: 0,0:07:53.06,0:07:55.81,Default,,0000,0000,0000,,For god's sake, it's 1, right?
Dialogue: 0,0:07:55.81,0:07:58.72,Default,,0000,0000,0000,,So z0 is 1.
Dialogue: 0,0:07:58.72,0:08:01.89,Default,,0000,0000,0000,,Now you're getting\Nthe sense that you
Dialogue: 0,0:08:01.89,0:08:07.40,Default,,0000,0000,0000,,have z minus 1 equals\Nf sub x, computed as 0,
Dialogue: 0,0:08:07.40,0:08:10.65,Default,,0000,0000,0000,,0 will be 0, lucky you.
Dialogue: 0,0:08:10.65,0:08:18.38,Default,,0000,0000,0000,,f sub y computed at 0, 0, you\Nwere expected to say that.
Dialogue: 0,0:08:18.38,0:08:23.54,Default,,0000,0000,0000,,So the answer for this\Nproblem was z equals 1.
Dialogue: 0,0:08:23.54,0:08:28.45,Default,,0000,0000,0000,,Still, if you messed up, I\Ngave some partial credit,
Dialogue: 0,0:08:28.45,0:08:32.54,Default,,0000,0000,0000,,because I didn't want to punish\Nyou too much, too harshly.
Dialogue: 0,0:08:32.54,0:08:36.38,Default,,0000,0000,0000,,
Dialogue: 0,0:08:36.38,0:08:42.87,Default,,0000,0000,0000,,On number four, find\Nthe direction u-- now
Dialogue: 0,0:08:42.87,0:08:45.87,Default,,0000,0000,0000,,you're using number\Nthree, so I should not
Dialogue: 0,0:08:45.87,0:08:50.05,Default,,0000,0000,0000,,erase number three completely.
Dialogue: 0,0:08:50.05,0:08:52.26,Default,,0000,0000,0000,,On number four, you\Nuse number three, so
Dialogue: 0,0:08:52.26,0:08:54.63,Default,,0000,0000,0000,,the same type of function.
Dialogue: 0,0:08:54.63,0:08:58.30,Default,,0000,0000,0000,,But it says find the\Ndirection in which
Dialogue: 0,0:08:58.30,0:09:05.82,Default,,0000,0000,0000,,this function increases most\Nrapidly, at the point 1, 1.
Dialogue: 0,0:09:05.82,0:09:06.74,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:09:06.74,0:09:16.32,Default,,0000,0000,0000,,So what do you do? you compute\Nthe gradient at the point 1,1,
Dialogue: 0,0:09:16.32,0:09:20.01,Default,,0000,0000,0000,,and you say, this\Nis a piece of cake.
Dialogue: 0,0:09:20.01,0:09:23.41,Default,,0000,0000,0000,,It's going to be ei plus ej, ee.
Dialogue: 0,0:09:23.41,0:09:33.13,Default,,0000,0000,0000,,
Dialogue: 0,0:09:33.13,0:09:35.68,Default,,0000,0000,0000,,Wonderful.
Dialogue: 0,0:09:35.68,0:09:40.16,Default,,0000,0000,0000,,So what you do is a u is the\Ngradient f over the length
Dialogue: 0,0:09:40.16,0:09:43.34,Default,,0000,0000,0000,,of the gradient of f at p.
Dialogue: 0,0:09:43.34,0:09:48.73,Default,,0000,0000,0000,,Which is ee divided\Nby the length of it.
Dialogue: 0,0:09:48.73,0:09:52.12,Default,,0000,0000,0000,,But you say, I don't have\Nto compute the length of it.
Dialogue: 0,0:09:52.12,0:09:56.67,Default,,0000,0000,0000,,I know what is pulling\Nyour two e's is what?
Dialogue: 0,0:09:56.67,0:10:00.60,Default,,0000,0000,0000,,And no matter what you have here\Nyou get the same unique result.
Dialogue: 0,0:10:00.60,0:10:02.84,Default,,0000,0000,0000,,Remember we talked\Nabout that uniqueness?
Dialogue: 0,0:10:02.84,0:10:04.78,Default,,0000,0000,0000,,This is what I\Ntried to emphasize,
Dialogue: 0,0:10:04.78,0:10:10.07,Default,,0000,0000,0000,,that you can have 77,\Nee, 99, 55, 100 and 100.
Dialogue: 0,0:10:10.07,0:10:13.77,Default,,0000,0000,0000,,If you divide by the norm,\Nyou still get the same answer.
Dialogue: 0,0:10:13.77,0:10:17.62,Default,,0000,0000,0000,,Not 11, but 1 over\Nroot 2, 1 over root 2.
Dialogue: 0,0:10:17.62,0:10:19.49,Default,,0000,0000,0000,,So no matter what\Nyou had there--
Dialogue: 0,0:10:19.49,0:10:22.47,Default,,0000,0000,0000,,it could have had a million,\Nor something instead of e,
Dialogue: 0,0:10:22.47,0:10:24.42,Default,,0000,0000,0000,,you still have the same u.
Dialogue: 0,0:10:24.42,0:10:29.58,Default,,0000,0000,0000,,
Dialogue: 0,0:10:29.58,0:10:31.30,Default,,0000,0000,0000,,Yes, put it back.
Dialogue: 0,0:10:31.30,0:10:32.94,Default,,0000,0000,0000,,Give yourself\Npoints, modify that.
Dialogue: 0,0:10:32.94,0:10:35.84,Default,,0000,0000,0000,,
Dialogue: 0,0:10:35.84,0:10:37.63,Default,,0000,0000,0000,,OK, so let me tell you.
Dialogue: 0,0:10:37.63,0:10:40.80,Default,,0000,0000,0000,,Normally I should penalize,\Nbecause I say write the answer
Dialogue: 0,0:10:40.80,0:10:41.95,Default,,0000,0000,0000,,in the space provided.
Dialogue: 0,0:10:41.95,0:10:45.16,Default,,0000,0000,0000,,And thank god you had\Nenough space, right?
Dialogue: 0,0:10:45.16,0:10:48.33,Default,,0000,0000,0000,,Look, this person wrote--\NI shouldn't show you who
Dialogue: 0,0:10:48.33,0:10:50.16,Default,,0000,0000,0000,,he is, he's not in here anyway.
Dialogue: 0,0:10:50.16,0:10:52.66,Default,,0000,0000,0000,,He has space and he\Nprovided last year
Dialogue: 0,0:10:52.66,0:10:54.71,Default,,0000,0000,0000,,with no square root\Nof 2, because only two
Dialogue: 0,0:10:54.71,0:10:57.22,Default,,0000,0000,0000,,rows are enough to write that.
Dialogue: 0,0:10:57.22,0:10:59.71,Default,,0000,0000,0000,,It's OK, I understand\Nyou forgot to copy.
Dialogue: 0,0:10:59.71,0:11:02.04,Default,,0000,0000,0000,,My son did the same thing.
Dialogue: 0,0:11:02.04,0:11:03.99,Default,,0000,0000,0000,,He got a scantron at the UIL.
Dialogue: 0,0:11:03.99,0:11:07.52,Default,,0000,0000,0000,,Come to visit my son, I wanted\Nto kill him, but it's OK.
Dialogue: 0,0:11:07.52,0:11:10.32,Default,,0000,0000,0000,,He got all the answers\Nright, and then
Dialogue: 0,0:11:10.32,0:11:14.53,Default,,0000,0000,0000,,the teacher-- that reminds\Nme of a movie with Mr. Bean.
Dialogue: 0,0:11:14.53,0:11:17.26,Default,,0000,0000,0000,,So the teacher comes\Nto him and says,
Dialogue: 0,0:11:17.26,0:11:19.81,Default,,0000,0000,0000,,wow your scantron is blank.
Dialogue: 0,0:11:19.81,0:11:21.75,Default,,0000,0000,0000,,So what was I supposed to do?
Dialogue: 0,0:11:21.75,0:11:24.49,Default,,0000,0000,0000,,Adjust for all the answers\Nyou got in the box,
Dialogue: 0,0:11:24.49,0:11:25.80,Default,,0000,0000,0000,,put them in the scantron.
Dialogue: 0,0:11:25.80,0:11:26.30,Default,,0000,0000,0000,,Oh, really?
Dialogue: 0,0:11:26.30,0:11:27.56,Default,,0000,0000,0000,,So he goes quickly.
Dialogue: 0,0:11:27.56,0:11:30.74,Default,,0000,0000,0000,,And then he got only\N75% of them transferred.
Dialogue: 0,0:11:30.74,0:11:32.95,Default,,0000,0000,0000,,The rest of them\Nwere not transferred.
Dialogue: 0,0:11:32.95,0:11:34.30,Default,,0000,0000,0000,,I don't know what they did.
Dialogue: 0,0:11:34.30,0:11:36.69,Default,,0000,0000,0000,,I have no idea.
Dialogue: 0,0:11:36.69,0:11:40.14,Default,,0000,0000,0000,,But the professor would\Nhave given full credit,
Dialogue: 0,0:11:40.14,0:11:43.48,Default,,0000,0000,0000,,even for the answers\Nthat he had in the box.
Dialogue: 0,0:11:43.48,0:11:45.86,Default,,0000,0000,0000,,From what I understood,\Nthe rule for scantrons,
Dialogue: 0,0:11:45.86,0:11:49.11,Default,,0000,0000,0000,,exams like you I only say,\Nif you don't have them
Dialogue: 0,0:11:49.11,0:11:52.43,Default,,0000,0000,0000,,on the scantron,\Nthey don't count.
Dialogue: 0,0:11:52.43,0:11:56.18,Default,,0000,0000,0000,,This is very harsh,\Nbecause we don't do that.
Dialogue: 0,0:11:56.18,0:11:58.71,Default,,0000,0000,0000,,For example, the\Nfinal-- if you--
Dialogue: 0,0:11:58.71,0:12:02.94,Default,,0000,0000,0000,,that's why I'm trying\Nto read everything.
Dialogue: 0,0:12:02.94,0:12:07.98,Default,,0000,0000,0000,,Suppose you box you answer and\Nit's 1 over square root of 2.
Dialogue: 0,0:12:07.98,0:12:09.45,Default,,0000,0000,0000,,If that's the right answer.
Dialogue: 0,0:12:09.45,0:12:10.93,Default,,0000,0000,0000,,Then if have the\Nmultiple choice,
Dialogue: 0,0:12:10.93,0:12:14.38,Default,,0000,0000,0000,,and they forgot to circle\N1 over square root of 2.
Dialogue: 0,0:12:14.38,0:12:17.34,Default,,0000,0000,0000,,I still give you\N100% on that problem.
Dialogue: 0,0:12:17.34,0:12:20.30,Default,,0000,0000,0000,,Some professor do not.
Dialogue: 0,0:12:20.30,0:12:24.24,Default,,0000,0000,0000,,So this is at the latitude\Nat whoever makes the rules,
Dialogue: 0,0:12:24.24,0:12:29.17,Default,,0000,0000,0000,,or whoever writes the exam.
Dialogue: 0,0:12:29.17,0:12:29.67,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:12:29.67,0:12:32.18,Default,,0000,0000,0000,,
Dialogue: 0,0:12:32.18,0:12:39.40,Default,,0000,0000,0000,,So again for the final, even for\Nthe multiple choice problems,
Dialogue: 0,0:12:39.40,0:12:41.58,Default,,0000,0000,0000,,I still need the solutions.
Dialogue: 0,0:12:41.58,0:12:46.00,Default,,0000,0000,0000,,I'm going to ask you\Nto use a bluebook.
Dialogue: 0,0:12:46.00,0:12:50.88,Default,,0000,0000,0000,,Some professors do not\Nask you to use a bluebook.
Dialogue: 0,0:12:50.88,0:12:54.77,Default,,0000,0000,0000,,They say, as long as you can\Nwrite on the sheet, circle
Dialogue: 0,0:12:54.77,0:12:56.69,Default,,0000,0000,0000,,the answer, I'm fine.
Dialogue: 0,0:12:56.69,0:12:57.44,Default,,0000,0000,0000,,I'm not fine.
Dialogue: 0,0:12:57.44,0:13:00.01,Default,,0000,0000,0000,,I want to keep what's\Nin the bluebook.
Dialogue: 0,0:13:00.01,0:13:01.85,Default,,0000,0000,0000,,So buy-- how much is it?
Dialogue: 0,0:13:01.85,0:13:03.32,Default,,0000,0000,0000,,Like a dollar?
Dialogue: 0,0:13:03.32,0:13:06.26,Default,,0000,0000,0000,,Buy the books ahead of time,\Nmake sure you have them.
Dialogue: 0,0:13:06.26,0:13:13.12,Default,,0000,0000,0000,,Now, number five was a piece of\Ncake once you did number four.
Dialogue: 0,0:13:13.12,0:13:15.57,Default,,0000,0000,0000,,You have a question?
Dialogue: 0,0:13:15.57,0:13:18.33,Default,,0000,0000,0000,,STUDENT: What size bluebook\Ndo you need for the final?
Dialogue: 0,0:13:18.33,0:13:20.02,Default,,0000,0000,0000,,PROFESSOR: The big one.
Dialogue: 0,0:13:20.02,0:13:22.96,Default,,0000,0000,0000,,Bigger than that, right?
Dialogue: 0,0:13:22.96,0:13:27.91,Default,,0000,0000,0000,,The direction u for five.
Dialogue: 0,0:13:27.91,0:13:33.31,Default,,0000,0000,0000,,With the problem four was i\Nplus j over root 2, right?
Dialogue: 0,0:13:33.31,0:13:36.75,Default,,0000,0000,0000,,This is what you remember\Nthat you did in problem four.
Dialogue: 0,0:13:36.75,0:13:40.69,Default,,0000,0000,0000,,If you didn't do problem four,\Nyou cannot do problem five.
Dialogue: 0,0:13:40.69,0:13:44.27,Default,,0000,0000,0000,,Problem five says, this\Nis parallel to one line.
Dialogue: 0,0:13:44.27,0:13:49.71,Default,,0000,0000,0000,,This is parallel to--\Nwhat is i plus j?
Dialogue: 0,0:13:49.71,0:13:51.37,Default,,0000,0000,0000,,Of course, you don't\Nhave to draw that.
Dialogue: 0,0:13:51.37,0:13:52.83,Default,,0000,0000,0000,,I'm not expecting\Nyou to draw that.
Dialogue: 0,0:13:52.83,0:13:54.67,Default,,0000,0000,0000,,y equals x is the\Nfirst bisection.
Dialogue: 0,0:13:54.67,0:13:59.46,Default,,0000,0000,0000,,
Dialogue: 0,0:13:59.46,0:14:01.83,Default,,0000,0000,0000,,All you had to do was\Ncircle C, and that
Dialogue: 0,0:14:01.83,0:14:04.74,Default,,0000,0000,0000,,was-- once you circled\NC, you get full credit.
Dialogue: 0,0:14:04.74,0:14:09.22,Default,,0000,0000,0000,,If you don't do that, you\Ndon't get credit for anything.
Dialogue: 0,0:14:09.22,0:14:11.69,Default,,0000,0000,0000,,Now six.
Dialogue: 0,0:14:11.69,0:14:16.04,Default,,0000,0000,0000,,What is the maximum rate\Nof increase of the function
Dialogue: 0,0:14:16.04,0:14:22.03,Default,,0000,0000,0000,,z the same of your friend,\Nyour fellow z equals
Dialogue: 0,0:14:22.03,0:14:27.52,Default,,0000,0000,0000,,e to the xy at p0,\Ncoordinates 1, 1?
Dialogue: 0,0:14:27.52,0:14:33.52,Default,,0000,0000,0000,,Then the value of the\Nmaximum rate of change is?
Dialogue: 0,0:14:33.52,0:14:34.06,Default,,0000,0000,0000,,A noun.
Dialogue: 0,0:14:34.06,0:14:36.86,Default,,0000,0000,0000,,
Dialogue: 0,0:14:36.86,0:14:38.60,Default,,0000,0000,0000,,What's the simplest\Nway to do it?
Dialogue: 0,0:14:38.60,0:14:40.60,Default,,0000,0000,0000,,There are two ways to do it.
Dialogue: 0,0:14:40.60,0:14:42.96,Default,,0000,0000,0000,,One is the long way,\None is the short way.
Dialogue: 0,0:14:42.96,0:14:45.80,Default,,0000,0000,0000,,What's the short way, guys?
Dialogue: 0,0:14:45.80,0:14:51.07,Default,,0000,0000,0000,,Just compute the\Nlength of the gradient.
Dialogue: 0,0:14:51.07,0:14:54.62,Default,,0000,0000,0000,,The length of the\Ngradient at the point P.
Dialogue: 0,0:14:54.62,0:15:03.21,Default,,0000,0000,0000,,So you have whatever\Nthat was, ee in length.
Dialogue: 0,0:15:03.21,0:15:06.38,Default,,0000,0000,0000,,So the answer was e root 2.
Dialogue: 0,0:15:06.38,0:15:07.90,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,0:15:07.90,0:15:09.29,Default,,0000,0000,0000,,What was the long way?
Dialogue: 0,0:15:09.29,0:15:11.18,Default,,0000,0000,0000,,I saw somebody do it.
Dialogue: 0,0:15:11.18,0:15:14.11,Default,,0000,0000,0000,,This is a lot more\Nwork, but of course,
Dialogue: 0,0:15:14.11,0:15:18.11,Default,,0000,0000,0000,,would be to compute the\Ndirectional derivative
Dialogue: 0,0:15:18.11,0:15:20.51,Default,,0000,0000,0000,,at the point p\Nfor this function.
Dialogue: 0,0:15:20.51,0:15:24.03,Default,,0000,0000,0000,,In the direction of u,\Nwhere u is the gradient
Dialogue: 0,0:15:24.03,0:15:27.98,Default,,0000,0000,0000,,divided by the length.
Dialogue: 0,0:15:27.98,0:15:28.78,Default,,0000,0000,0000,,at the point p.
Dialogue: 0,0:15:28.78,0:15:30.74,Default,,0000,0000,0000,,And you get, of course,\Nthe same answer.
Dialogue: 0,0:15:30.74,0:15:31.24,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:15:31.24,0:15:35.70,Default,,0000,0000,0000,,Because we proved that actually\Nthe maximum rate of change
Dialogue: 0,0:15:35.70,0:15:38.67,Default,,0000,0000,0000,,represented directional\Nderivative exactly
Dialogue: 0,0:15:38.67,0:15:41.64,Default,,0000,0000,0000,,in the direction\Ngiven by the gradient.
Dialogue: 0,0:15:41.64,0:15:43.62,Default,,0000,0000,0000,,This is something we proved.
Dialogue: 0,0:15:43.62,0:15:48.08,Default,,0000,0000,0000,,One of the few things\Nwe proved in this class.
Dialogue: 0,0:15:48.08,0:15:50.06,Default,,0000,0000,0000,,Alright.
Dialogue: 0,0:15:50.06,0:15:54.02,Default,,0000,0000,0000,,So the answer was e root 2.
Dialogue: 0,0:15:54.02,0:15:55.50,Default,,0000,0000,0000,,Let's move on to number seven.
Dialogue: 0,0:15:55.50,0:16:01.94,Default,,0000,0000,0000,,Number seven-- and remind\Nme of your five points.
Dialogue: 0,0:16:01.94,0:16:07.38,Default,,0000,0000,0000,,Can you email me, so\NI have an Excel sheet,
Dialogue: 0,0:16:07.38,0:16:09.22,Default,,0000,0000,0000,,and I'll put it in.
Dialogue: 0,0:16:09.22,0:16:15.51,Default,,0000,0000,0000,,Consider the function f of x,\Ny e to the negative x squared,
Dialogue: 0,0:16:15.51,0:16:17.75,Default,,0000,0000,0000,,y squared.
Dialogue: 0,0:16:17.75,0:16:20.99,Default,,0000,0000,0000,,What can you tell me about\Nthis type of function?
Dialogue: 0,0:16:20.99,0:16:22.74,Default,,0000,0000,0000,,It's the headache function.
Dialogue: 0,0:16:22.74,0:16:25.62,Default,,0000,0000,0000,,If I would ask you to do\Nan anti-derivative of each
Dialogue: 0,0:16:25.62,0:16:28.62,Default,,0000,0000,0000,,of the negative squares,\Nyou would say Magdalene,
Dialogue: 0,0:16:28.62,0:16:31.58,Default,,0000,0000,0000,,didn't you say that\Nthis is impossible?
Dialogue: 0,0:16:31.58,0:16:38.98,Default,,0000,0000,0000,,While the anti-derivative\Nexists, it cannot be expressed.
Dialogue: 0,0:16:38.98,0:16:42.20,Default,,0000,0000,0000,,It cannot be expressed as\Nan elementary function.
Dialogue: 0,0:16:42.20,0:16:44.14,Default,,0000,0000,0000,,And that's a big headache.
Dialogue: 0,0:16:44.14,0:16:47.06,Default,,0000,0000,0000,,This problem is beautiful,\Nwhy is it beautiful?
Dialogue: 0,0:16:47.06,0:16:50.94,Default,,0000,0000,0000,,Because in the end,\Nit becomes magic.
Dialogue: 0,0:16:50.94,0:16:53.84,Default,,0000,0000,0000,,So it's a positive function.
Dialogue: 0,0:16:53.84,0:16:56.99,Default,,0000,0000,0000,,It's like a bell on top\Nof the church something.
Dialogue: 0,0:16:56.99,0:17:00.93,Default,,0000,0000,0000,,And then, you have to\Ncompute double integral
Dialogue: 0,0:17:00.93,0:17:06.37,Default,,0000,0000,0000,,over the unit disk of\Ncenters of 0 and radius 1.
Dialogue: 0,0:17:06.37,0:17:09.77,Default,,0000,0000,0000,,Of e to the negative x\Nsquared minus y squared dx/dy.
Dialogue: 0,0:17:09.77,0:17:12.28,Default,,0000,0000,0000,,
Dialogue: 0,0:17:12.28,0:17:15.62,Default,,0000,0000,0000,,Well then you say, well\NI've done this kind of thing
Dialogue: 0,0:17:15.62,0:17:19.83,Default,,0000,0000,0000,,before, but not with\NCartesian coordinates.
Dialogue: 0,0:17:19.83,0:17:24.88,Default,,0000,0000,0000,,We did it with the Jacobian\Nr, that changes everything
Dialogue: 0,0:17:24.88,0:17:28.79,Default,,0000,0000,0000,,into polar coordinates.
Dialogue: 0,0:17:28.79,0:17:34.13,Default,,0000,0000,0000,,So this guy becomes e\Nto the minus r squared.
Dialogue: 0,0:17:34.13,0:17:39.14,Default,,0000,0000,0000,,Each of the numbers are\Nsquared dr, d theta.
Dialogue: 0,0:17:39.14,0:17:42.00,Default,,0000,0000,0000,,D on the unit\N[INAUDIBLE] disk means
Dialogue: 0,0:17:42.00,0:17:44.73,Default,,0000,0000,0000,,the radius goes from 0 to 1.
Dialogue: 0,0:17:44.73,0:17:48.54,Default,,0000,0000,0000,,This is a blessing for us,\Nbecause it's easy data.
Dialogue: 0,0:17:48.54,0:17:51.03,Default,,0000,0000,0000,,Then we have 0 to 2 pi.
Dialogue: 0,0:17:51.03,0:17:53.79,Default,,0000,0000,0000,,You could have put\Nit in any order.
Dialogue: 0,0:17:53.79,0:17:58.87,Default,,0000,0000,0000,,For u, it's easier to close your\Neyes when it comes to theta.
Dialogue: 0,0:17:58.87,0:18:01.53,Default,,0000,0000,0000,,Say, theta is independent.
Dialogue: 0,0:18:01.53,0:18:05.35,Default,,0000,0000,0000,,He is like a partition\Nthat has to do nothing
Dialogue: 0,0:18:05.35,0:18:07.44,Default,,0000,0000,0000,,with what's inside here.
Dialogue: 0,0:18:07.44,0:18:10.71,Default,,0000,0000,0000,,So let's pull him\Nout of this picture.
Dialogue: 0,0:18:10.71,0:18:14.13,Default,,0000,0000,0000,,And he wants to live by himself.
Dialogue: 0,0:18:14.13,0:18:18.61,Default,,0000,0000,0000,,An integral from 0 to 2 pi of\Nd theta was of course 2 pi.
Dialogue: 0,0:18:18.61,0:18:22.10,Default,,0000,0000,0000,,He's happy to go\Nout, having fun.
Dialogue: 0,0:18:22.10,0:18:26.59,Default,,0000,0000,0000,,This guy inside has to\Nbe thoroughly computed.
Dialogue: 0,0:18:26.59,0:18:30.10,Default,,0000,0000,0000,,In the sense that you\Nperform the substitution.
Dialogue: 0,0:18:30.10,0:18:39.59,Default,,0000,0000,0000,,I was actually amused that half\Nof you did u equals r squared,
Dialogue: 0,0:18:39.59,0:18:42.84,Default,,0000,0000,0000,,and half of you did u\Nequals minus r squared.
Dialogue: 0,0:18:42.84,0:18:44.60,Default,,0000,0000,0000,,It really doesn't\Nmatter which one.
Dialogue: 0,0:18:44.60,0:18:46.94,Default,,0000,0000,0000,,But the problem is\Nthat some of you
Dialogue: 0,0:18:46.94,0:18:51.67,Default,,0000,0000,0000,,made a mess when you put the\Nlimit points back in place,
Dialogue: 0,0:18:51.67,0:18:53.64,Default,,0000,0000,0000,,and you made mistakes.
Dialogue: 0,0:18:53.64,0:18:56.49,Default,,0000,0000,0000,,Somebody even got\Nnegative answers,
Dialogue: 0,0:18:56.49,0:18:59.86,Default,,0000,0000,0000,,I was about to\Nfall off the chair.
Dialogue: 0,0:18:59.86,0:19:04.13,Default,,0000,0000,0000,,Of course, I was in a good\Nmood because it was a holiday,
Dialogue: 0,0:19:04.13,0:19:05.27,Default,,0000,0000,0000,,I graded them.
Dialogue: 0,0:19:05.27,0:19:09.04,Default,,0000,0000,0000,,Fortunately, I graded\Nthem over the break.
Dialogue: 0,0:19:09.04,0:19:12.33,Default,,0000,0000,0000,,So after I came\Nback from Georgia.
Dialogue: 0,0:19:12.33,0:19:18.68,Default,,0000,0000,0000,,I have minus r dr. rdr\Nwas minus a half du.
Dialogue: 0,0:19:18.68,0:19:21.36,Default,,0000,0000,0000,,
Dialogue: 0,0:19:21.36,0:19:24.66,Default,,0000,0000,0000,,This fellow is just\Ninto the u, and he's
Dialogue: 0,0:19:24.66,0:19:31.10,Default,,0000,0000,0000,,a blessing because\Nthe [INAUDIBLE]
Dialogue: 0,0:19:31.10,0:19:36.56,Default,,0000,0000,0000,,So into the u, however,\Ntake it between 1 and what?
Dialogue: 0,0:19:36.56,0:19:38.05,Default,,0000,0000,0000,,Not 0 and 1.
Dialogue: 0,0:19:38.05,0:19:41.52,Default,,0000,0000,0000,,But when you have 0\Nhere, you have 0 here.
Dialogue: 0,0:19:41.52,0:19:44.50,Default,,0000,0000,0000,,When you have 1,\Nyou have minus 1.
Dialogue: 0,0:19:44.50,0:19:48.47,Default,,0000,0000,0000,,So pay attention to\Nthat, otherwise, you
Dialogue: 0,0:19:48.47,0:19:52.44,Default,,0000,0000,0000,,get something that\Nmakes no sense.
Dialogue: 0,0:19:52.44,0:19:55.01,Default,,0000,0000,0000,,Times minus a half.
Dialogue: 0,0:19:55.01,0:19:59.48,Default,,0000,0000,0000,,That, you will have\Nto be careful about.
Dialogue: 0,0:19:59.48,0:19:59.98,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:19:59.98,0:20:04.10,Default,,0000,0000,0000,,Because there will be a minus\Nfrom here and here, in the end,
Dialogue: 0,0:20:04.10,0:20:05.31,Default,,0000,0000,0000,,the answer will be positive.
Dialogue: 0,0:20:05.31,0:20:08.27,Default,,0000,0000,0000,,And that's reminding me\Nof that city plumber joke
Dialogue: 0,0:20:08.27,0:20:11.93,Default,,0000,0000,0000,,when he doesn't pay attention\Nto the limits of integration.
Dialogue: 0,0:20:11.93,0:20:16.91,Default,,0000,0000,0000,,And you can get a minus\Nvolume, or a minus area.
Dialogue: 0,0:20:16.91,0:20:24.01,Default,,0000,0000,0000,,So e to the minus 1 minus 1.
Dialogue: 0,0:20:24.01,0:20:25.80,Default,,0000,0000,0000,,But that leaves a\Nnegative number,
Dialogue: 0,0:20:25.80,0:20:34.27,Default,,0000,0000,0000,,but when you multiply it by a\Nminus, you have 1 minus 1/e.
Dialogue: 0,0:20:34.27,0:20:35.06,Default,,0000,0000,0000,,1 minus 1/e.
Dialogue: 0,0:20:35.06,0:20:37.85,Default,,0000,0000,0000,,
Dialogue: 0,0:20:37.85,0:20:38.54,Default,,0000,0000,0000,,Good, thank god.
Dialogue: 0,0:20:38.54,0:20:41.13,Default,,0000,0000,0000,,This is a nice guy, less than 1.
Dialogue: 0,0:20:41.13,0:20:45.08,Default,,0000,0000,0000,,And this is key to your\Nanswer, because 2 goes away
Dialogue: 0,0:20:45.08,0:20:48.37,Default,,0000,0000,0000,,and pi stays in place\Nand this is less than pi.
Dialogue: 0,0:20:48.37,0:20:52.48,Default,,0000,0000,0000,,So the answer to this question\Nwas an answer less than pi.
Dialogue: 0,0:20:52.48,0:20:55.07,Default,,0000,0000,0000,,And if you didn't get\Nit, I'm very sorry,
Dialogue: 0,0:20:55.07,0:21:01.13,Default,,0000,0000,0000,,if you didn't get less than\Npi, you didn't get any points.
Dialogue: 0,0:21:01.13,0:21:04.52,Default,,0000,0000,0000,,But, there are enough chances\Nfor you to get another point.
Dialogue: 0,0:21:04.52,0:21:14.48,Default,,0000,0000,0000,,I was brokenhearted for 10\Npeople or more out of 25
Dialogue: 0,0:21:14.48,0:21:24.09,Default,,0000,0000,0000,,did not remember what I\Ntaught in class about the area
Dialogue: 0,0:21:24.09,0:21:28.49,Default,,0000,0000,0000,,of a collateral triangle.
Dialogue: 0,0:21:28.49,0:21:31.90,Default,,0000,0000,0000,,And it broke my heart,\Nand I was about to cry,
Dialogue: 0,0:21:31.90,0:21:35.32,Default,,0000,0000,0000,,but I said, c'mon, they'll\Ndo it better in the final.
Dialogue: 0,0:21:35.32,0:21:39.22,Default,,0000,0000,0000,,Honestly, I was\Nso brokenhearted.
Dialogue: 0,0:21:39.22,0:21:41.70,Default,,0000,0000,0000,,So this is 1, 0, 0.
Dialogue: 0,0:21:41.70,0:21:42.62,Default,,0000,0000,0000,,This was 0, 1, 0.
Dialogue: 0,0:21:42.62,0:21:44.94,Default,,0000,0000,0000,,This was 0, 0, 1.
Dialogue: 0,0:21:44.94,0:21:49.33,Default,,0000,0000,0000,,On number eight.
Dialogue: 0,0:21:49.33,0:21:50.32,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:21:50.32,0:22:03.11,Default,,0000,0000,0000,,
Dialogue: 0,0:22:03.11,0:22:04.09,Default,,0000,0000,0000,,Beautiful.
Dialogue: 0,0:22:04.09,0:22:10.38,Default,,0000,0000,0000,,It's an equilateral\Ntriangle, and the l side
Dialogue: 0,0:22:10.38,0:22:14.59,Default,,0000,0000,0000,,of that equilateral triangle\Nis the square root of 2.
Dialogue: 0,0:22:14.59,0:22:17.11,Default,,0000,0000,0000,,I even taught you how to cheat.
Dialogue: 0,0:22:17.11,0:22:17.98,Default,,0000,0000,0000,,That's why I was mad.
Dialogue: 0,0:22:17.98,0:22:22.12,Default,,0000,0000,0000,,I taught you how to cheat,\Nand you didn't take advantage.
Dialogue: 0,0:22:22.12,0:22:29.55,Default,,0000,0000,0000,,So the area was l squared,\Nsquare root of 2, 4.
Dialogue: 0,0:22:29.55,0:22:33.84,Default,,0000,0000,0000,,Which we did this together\Nin fifth or sixth grade
Dialogue: 0,0:22:33.84,0:22:40.62,Default,,0000,0000,0000,,by multiplying that height and\Nthe width and divided by 2.
Dialogue: 0,0:22:40.62,0:22:43.52,Default,,0000,0000,0000,,And then we came up with this\Nformula with the Pythagorean
Dialogue: 0,0:22:43.52,0:22:45.46,Default,,0000,0000,0000,,theorem in the classroom.
Dialogue: 0,0:22:45.46,0:22:49.13,Default,,0000,0000,0000,,If eligible to, you can\Nvery quickly get an answer.
Dialogue: 0,0:22:49.13,0:22:54.38,Default,,0000,0000,0000,,So that's going to be 2 root\N2 over 4, just root 3 over 2.
Dialogue: 0,0:22:54.38,0:22:56.84,Default,,0000,0000,0000,,And when I saw that people got\Nsomething else except root 3
Dialogue: 0,0:22:56.84,0:22:59.40,Default,,0000,0000,0000,,over 2, that broke my heart.
Dialogue: 0,0:22:59.40,0:23:01.24,Default,,0000,0000,0000,,Really.
Dialogue: 0,0:23:01.24,0:23:06.20,Default,,0000,0000,0000,,You have plenty of\Ntime to catch up
Dialogue: 0,0:23:06.20,0:23:07.66,Default,,0000,0000,0000,,with that being on your final.
Dialogue: 0,0:23:07.66,0:23:10.32,Default,,0000,0000,0000,,
Dialogue: 0,0:23:10.32,0:23:14.23,Default,,0000,0000,0000,,Did I expect you to really\Ndo the surface integral?
Dialogue: 0,0:23:14.23,0:23:20.22,Default,,0000,0000,0000,,Some people again, need to write\Nintegral over the shaded domain
Dialogue: 0,0:23:20.22,0:23:24.21,Default,,0000,0000,0000,,d a square root\Nof f sub x squared
Dialogue: 0,0:23:24.21,0:23:28.44,Default,,0000,0000,0000,,plus f sub y squared plus 1.
Dialogue: 0,0:23:28.44,0:23:31.10,Default,,0000,0000,0000,,That was the right track,\Nbecause this is root 3.
Dialogue: 0,0:23:31.10,0:23:37.61,Default,,0000,0000,0000,,And then the area, you get the\Narea of the 1 times 1 over 2,
Dialogue: 0,0:23:37.61,0:23:38.11,Default,,0000,0000,0000,,right?
Dialogue: 0,0:23:38.11,0:23:40.44,Default,,0000,0000,0000,,1/3 is the area.
Dialogue: 0,0:23:40.44,0:23:45.40,Default,,0000,0000,0000,,Root 3 gets out of this, so\Nyou have-- when you integrate,
Dialogue: 0,0:23:45.40,0:23:49.37,Default,,0000,0000,0000,,you have the area of the\Nshaded base that I have.
Dialogue: 0,0:23:49.37,0:23:51.35,Default,,0000,0000,0000,,And you get the same answer.
Dialogue: 0,0:23:51.35,0:23:54.33,Default,,0000,0000,0000,,No matter how you do\Nit, with calculators
Dialogue: 0,0:23:54.33,0:23:58.79,Default,,0000,0000,0000,,or without calculators, you\Nstill could have passed.
Dialogue: 0,0:23:58.79,0:24:02.54,Default,,0000,0000,0000,,Am I if you didn't\Nget the answer?
Dialogue: 0,0:24:02.54,0:24:03.08,Default,,0000,0000,0000,,No.
Dialogue: 0,0:24:03.08,0:24:04.84,Default,,0000,0000,0000,,Absolutely.
Dialogue: 0,0:24:04.84,0:24:08.99,Default,,0000,0000,0000,,But it hurts me as if, I\Ndon't know, a relative of mine
Dialogue: 0,0:24:08.99,0:24:13.73,Default,,0000,0000,0000,,messed up some task.
Dialogue: 0,0:24:13.73,0:24:16.93,Default,,0000,0000,0000,,That's why it's better that\Nyou don't know your students,
Dialogue: 0,0:24:16.93,0:24:21.01,Default,,0000,0000,0000,,because when you\Nknow your students,
Dialogue: 0,0:24:21.01,0:24:23.00,Default,,0000,0000,0000,,you know that they\Ncould have done better,
Dialogue: 0,0:24:23.00,0:24:24.13,Default,,0000,0000,0000,,because you know them.
Dialogue: 0,0:24:24.13,0:24:27.05,Default,,0000,0000,0000,,So we can say, OK,\Nit really hurts
Dialogue: 0,0:24:27.05,0:24:30.01,Default,,0000,0000,0000,,when you know that they messed\Nup, not because they are not
Dialogue: 0,0:24:30.01,0:24:34.58,Default,,0000,0000,0000,,smart or educated, but because\Nthey just either didn't
Dialogue: 0,0:24:34.58,0:24:37.62,Default,,0000,0000,0000,,pay attention or they\Nwere stressed out.
Dialogue: 0,0:24:37.62,0:24:41.84,Default,,0000,0000,0000,,However, my substitute,\Nthe guy who came here,
Dialogue: 0,0:24:41.84,0:24:43.22,Default,,0000,0000,0000,,was my Ph.D. student.
Dialogue: 0,0:24:43.22,0:24:47.56,Default,,0000,0000,0000,,He got a doctoral degree\Nmathematics with me last year.
Dialogue: 0,0:24:47.56,0:24:50.12,Default,,0000,0000,0000,,And he told me you were\Nnot stressed out at all.
Dialogue: 0,0:24:50.12,0:24:51.62,Default,,0000,0000,0000,,And I said, thank god.
Dialogue: 0,0:24:51.62,0:24:55.10,Default,,0000,0000,0000,,I'm glad that they were calm.
Dialogue: 0,0:24:55.10,0:24:58.24,Default,,0000,0000,0000,,And he said, I didn't\Nlook at the exam,
Dialogue: 0,0:24:58.24,0:25:01.20,Default,,0000,0000,0000,,but it seemed like they did very\Nwell and they were comfortable.
Dialogue: 0,0:25:01.20,0:25:02.69,Default,,0000,0000,0000,,And I was so happy.
Dialogue: 0,0:25:02.69,0:25:04.67,Default,,0000,0000,0000,,I was in Athens, Georgia.
Dialogue: 0,0:25:04.67,0:25:06.16,Default,,0000,0000,0000,,And reading this\Nemail I said, yay!
Dialogue: 0,0:25:06.16,0:25:07.64,Default,,0000,0000,0000,,Everybody's going to get an A!
Dialogue: 0,0:25:07.64,0:25:10.61,Default,,0000,0000,0000,,So I come home and\NI start grading it.
Dialogue: 0,0:25:10.61,0:25:16.06,Default,,0000,0000,0000,,I was sad to see that my\Nprediction was not correct.
Dialogue: 0,0:25:16.06,0:25:20.03,Default,,0000,0000,0000,,But anyway, [INAUDIBLE]\Nwith an average of B.
Dialogue: 0,0:25:20.03,0:25:21.93,Default,,0000,0000,0000,,For an honors class, it's OK.
Dialogue: 0,0:25:21.93,0:25:24.79,Default,,0000,0000,0000,,I just expected a lot better.
Dialogue: 0,0:25:24.79,0:25:29.56,Default,,0000,0000,0000,,And I know it's going to be\Na lot better in the final.
Dialogue: 0,0:25:29.56,0:25:32.43,Default,,0000,0000,0000,,Number nine.
Dialogue: 0,0:25:32.43,0:25:34.74,Default,,0000,0000,0000,,This was done by\Nalmost everybody,
Dialogue: 0,0:25:34.74,0:25:36.91,Default,,0000,0000,0000,,except for a few people who\Nmessed up on the limits.
Dialogue: 0,0:25:36.91,0:25:38.36,Default,,0000,0000,0000,,I don't know why.
Dialogue: 0,0:25:38.36,0:25:44.18,Default,,0000,0000,0000,,When they compute-- when they\Ndrew, they drew x squared,
Dialogue: 0,0:25:44.18,0:25:46.12,Default,,0000,0000,0000,,and they drew square root of xn.
Dialogue: 0,0:25:46.12,0:25:49.52,Default,,0000,0000,0000,,Of course, you were supposed--\Nthe answer was 0 to 1,
Dialogue: 0,0:25:49.52,0:25:50.98,Default,,0000,0000,0000,,integral of.
Dialogue: 0,0:25:50.98,0:25:56.14,Default,,0000,0000,0000,,Now, if you do first\Nx, you have x from y
Dialogue: 0,0:25:56.14,0:25:57.56,Default,,0000,0000,0000,,squared to square root of y.
Dialogue: 0,0:25:57.56,0:25:58.46,Default,,0000,0000,0000,,You guys with me?
Dialogue: 0,0:25:58.46,0:26:02.17,Default,,0000,0000,0000,,Because this is\Nsmaller than that.
Dialogue: 0,0:26:02.17,0:26:02.93,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:26:02.93,0:26:08.19,Default,,0000,0000,0000,,So you have 1 and dx dy equals\Nto integral from 0 to 1,
Dialogue: 0,0:26:08.19,0:26:11.69,Default,,0000,0000,0000,,integral x squared to\Nsquare root of x1 dy dx.
Dialogue: 0,0:26:11.69,0:26:17.19,Default,,0000,0000,0000,,
Dialogue: 0,0:26:17.19,0:26:24.27,Default,,0000,0000,0000,,Now, what a few people did--\Nand I just forgave them.
Dialogue: 0,0:26:24.27,0:26:29.22,Default,,0000,0000,0000,,They just-- one\Nput this like that.
Dialogue: 0,0:26:29.22,0:26:31.79,Default,,0000,0000,0000,,And here, he put root 2.
Dialogue: 0,0:26:31.79,0:26:33.38,Default,,0000,0000,0000,,Root y and y squared.
Dialogue: 0,0:26:33.38,0:26:34.60,Default,,0000,0000,0000,,Don't do that.
Dialogue: 0,0:26:34.60,0:26:37.50,Default,,0000,0000,0000,,It's like chasing that\Na positive number equals
Dialogue: 0,0:26:37.50,0:26:41.88,Default,,0000,0000,0000,,a negative number, which\Nis all complete nonsense.
Dialogue: 0,0:26:41.88,0:26:45.77,Default,,0000,0000,0000,,So the correct answer was\Nwe put y squared down,
Dialogue: 0,0:26:45.77,0:26:48.80,Default,,0000,0000,0000,,and square root of y\Nbecause this guy is
Dialogue: 0,0:26:48.80,0:26:54.00,Default,,0000,0000,0000,,bigger than this guy for\Nsomething between 0 and 1.
Dialogue: 0,0:26:54.00,0:26:54.96,Default,,0000,0000,0000,,Because I told you.
Dialogue: 0,0:26:54.96,0:27:04.13,Default,,0000,0000,0000,,Square root of 0.04 is bigger\Nthan the square of that.
Dialogue: 0,0:27:04.13,0:27:06.04,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:27:06.04,0:27:15.17,Default,,0000,0000,0000,,Now am I happy with that?
Dialogue: 0,0:27:15.17,0:27:16.45,Default,,0000,0000,0000,,I'm quite happy.
Dialogue: 0,0:27:16.45,0:27:21.58,Default,,0000,0000,0000,,In general, people understood\Nthe vertical strip method
Dialogue: 0,0:27:21.58,0:27:23.94,Default,,0000,0000,0000,,compared to the\Nhorizontal strip method.
Dialogue: 0,0:27:23.94,0:27:25.35,Default,,0000,0000,0000,,And why am I happy?
Dialogue: 0,0:27:25.35,0:27:30.11,Default,,0000,0000,0000,,Because I was asked by three\Npeople from other classes
Dialogue: 0,0:27:30.11,0:27:33.61,Default,,0000,0000,0000,,to help them, over\Nthere, on the corridor.
Dialogue: 0,0:27:33.61,0:27:35.22,Default,,0000,0000,0000,,And I asked them,\Nwho is your teacher?
Dialogue: 0,0:27:35.22,0:27:36.19,Default,,0000,0000,0000,,This and that.
Dialogue: 0,0:27:36.19,0:27:40.82,Default,,0000,0000,0000,,But we did not understand\Nreversing the order
Dialogue: 0,0:27:40.82,0:27:42.16,Default,,0000,0000,0000,,of integration in class.
Dialogue: 0,0:27:42.16,0:27:43.65,Default,,0000,0000,0000,,And I said, how come?
Dialogue: 0,0:27:43.65,0:27:45.64,Default,,0000,0000,0000,,Well, they didn't\Nexplain it very well.
Dialogue: 0,0:27:45.64,0:27:47.64,Default,,0000,0000,0000,,So I started\Nexplaining it to them.
Dialogue: 0,0:27:47.64,0:27:50.13,Default,,0000,0000,0000,,And then I realized that\Nit's a conflict of interest.
Dialogue: 0,0:27:50.13,0:27:52.62,Default,,0000,0000,0000,,I'm not allowed to do that.
Dialogue: 0,0:27:52.62,0:27:55.60,Default,,0000,0000,0000,,And then I go, oh my god, I\Ncannot do the homework for you.
Dialogue: 0,0:27:55.60,0:27:56.60,Default,,0000,0000,0000,,I'm not allowed.
Dialogue: 0,0:27:56.60,0:27:59.61,Default,,0000,0000,0000,,But I was already talking.
Dialogue: 0,0:27:59.61,0:28:04.35,Default,,0000,0000,0000,,So I said, guys, can you do it?
Dialogue: 0,0:28:04.35,0:28:05.12,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:28:05.12,0:28:07.06,Default,,0000,0000,0000,,I said, do you draw?
Dialogue: 0,0:28:07.06,0:28:07.95,Default,,0000,0000,0000,,Why would we draw?
Dialogue: 0,0:28:07.95,0:28:10.05,Default,,0000,0000,0000,,They didn't teach\Nus how to draw.
Dialogue: 0,0:28:10.05,0:28:13.24,Default,,0000,0000,0000,,I said, but how do you\Nknow about vertical strips
Dialogue: 0,0:28:13.24,0:28:14.46,Default,,0000,0000,0000,,and horizontal strips?
Dialogue: 0,0:28:14.46,0:28:15.44,Default,,0000,0000,0000,,No.
Dialogue: 0,0:28:15.44,0:28:17.89,Default,,0000,0000,0000,,And how do you do this?
Dialogue: 0,0:28:17.89,0:28:18.87,Default,,0000,0000,0000,,We don't know.
Dialogue: 0,0:28:18.87,0:28:21.83,Default,,0000,0000,0000,,We felt like we have\Nto figure it out.
Dialogue: 0,0:28:21.83,0:28:25.45,Default,,0000,0000,0000,,Without drawing, without\Nunderstanding how the vertical
Dialogue: 0,0:28:25.45,0:28:27.82,Default,,0000,0000,0000,,strips are drawn\Nbetween two functions,
Dialogue: 0,0:28:27.82,0:28:31.07,Default,,0000,0000,0000,,and how you switch\Nthe horizontal strips,
Dialogue: 0,0:28:31.07,0:28:33.34,Default,,0000,0000,0000,,you cannot do this\Nproblem, period.
Dialogue: 0,0:28:33.34,0:28:36.32,Default,,0000,0000,0000,,So if you don't have--\Nmaybe some people have
Dialogue: 0,0:28:36.32,0:28:39.18,Default,,0000,0000,0000,,enough imagination--\Nbut that's very rare--
Dialogue: 0,0:28:39.18,0:28:40.94,Default,,0000,0000,0000,,That they can close\Ntheir eyes and they
Dialogue: 0,0:28:40.94,0:28:44.96,Default,,0000,0000,0000,,can see a picture\Nwith their eyes closed
Dialogue: 0,0:28:44.96,0:28:46.05,Default,,0000,0000,0000,,and they can solve that.
Dialogue: 0,0:28:46.05,0:28:48.18,Default,,0000,0000,0000,,But that's not the way to learn.
Dialogue: 0,0:28:48.18,0:28:51.79,Default,,0000,0000,0000,,The way to learn is a very\Nvisual learning thing.
Dialogue: 0,0:28:51.79,0:28:54.46,Default,,0000,0000,0000,,So that's why we\Ndraw all the time.
Dialogue: 0,0:28:54.46,0:28:57.32,Default,,0000,0000,0000,,
Dialogue: 0,0:28:57.32,0:28:59.45,Default,,0000,0000,0000,,STUDENT: Professor, you\Ncan cheat these with Cal 2.
Dialogue: 0,0:28:59.45,0:29:00.45,Default,,0000,0000,0000,,PROFESSOR: Yes.
Dialogue: 0,0:29:00.45,0:29:02.44,Default,,0000,0000,0000,,You can do that with Cal 2.
Dialogue: 0,0:29:02.44,0:29:03.44,Default,,0000,0000,0000,,What's the problem?
Dialogue: 0,0:29:03.44,0:29:05.94,Default,,0000,0000,0000,,You have integral from 0 to 1.
Dialogue: 0,0:29:05.94,0:29:09.38,Default,,0000,0000,0000,,Square root of y\Nminus y squared.
Dialogue: 0,0:29:09.38,0:29:17.44,Default,,0000,0000,0000,,Well, they learn to\Ndo the other one.
Dialogue: 0,0:29:17.44,0:29:21.75,Default,,0000,0000,0000,,The one with square root x\Nminus x squared, 0,1 and so on.
Dialogue: 0,0:29:21.75,0:29:26.25,Default,,0000,0000,0000,,But they were told explicitly\Nto write-- the professor even
Dialogue: 0,0:29:26.25,0:29:29.77,Default,,0000,0000,0000,,left these empty and put\Nspaces, fill in the spaces.
Dialogue: 0,0:29:29.77,0:29:32.02,Default,,0000,0000,0000,,And they say, how the heck\Ndo we fill in those spaces?
Dialogue: 0,0:29:32.02,0:29:35.00,Default,,0000,0000,0000,,Plus the whiteboard problems\Nhave the empty spaces.
Dialogue: 0,0:29:35.00,0:29:36.99,Default,,0000,0000,0000,,And they couldn't\Nbelieve that at all.
Dialogue: 0,0:29:36.99,0:29:40.97,Default,,0000,0000,0000,,And one of them went to the\Ntutoring center and was lucky.
Dialogue: 0,0:29:40.97,0:29:43.13,Default,,0000,0000,0000,,Because he got--\Nthis is like when
Dialogue: 0,0:29:43.13,0:29:45.83,Default,,0000,0000,0000,,you go to a medical\Ndoctor, sometimes you
Dialogue: 0,0:29:45.83,0:29:48.61,Default,,0000,0000,0000,,are lucky and get a good\Ndoctor who takes care of you,
Dialogue: 0,0:29:48.61,0:29:49.98,Default,,0000,0000,0000,,figures out what\Nyour problem is.
Dialogue: 0,0:29:49.98,0:29:54.14,Default,,0000,0000,0000,,And sometimes, they give\Nyou the wrong medicine.
Dialogue: 0,0:29:54.14,0:29:58.01,Default,,0000,0000,0000,,So one of them got the right\Ntutor who knew how to explain
Dialogue: 0,0:29:58.01,0:29:59.86,Default,,0000,0000,0000,,and sort of knew something.
Dialogue: 0,0:29:59.86,0:30:04.18,Default,,0000,0000,0000,,But the other one got a tutor\Nwho never took Calculus 3
Dialogue: 0,0:30:04.18,0:30:09.04,Default,,0000,0000,0000,,and said, I don't know what the\Nheck these multiple snakes are.
Dialogue: 0,0:30:09.04,0:30:11.65,Default,,0000,0000,0000,,So I'm not going to\Nbe able to help you.
Dialogue: 0,0:30:11.65,0:30:14.99,Default,,0000,0000,0000,,So he was very disappointed.
Dialogue: 0,0:30:14.99,0:30:15.49,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:30:15.49,0:30:20.11,Default,,0000,0000,0000,,Compute the area of the domain\ND from the previous problem.
Dialogue: 0,0:30:20.11,0:30:23.16,Default,,0000,0000,0000,,This was something that\Nnobody's telling you,
Dialogue: 0,0:30:23.16,0:30:26.27,Default,,0000,0000,0000,,hey, you have to do it\Nwith the double snakes.
Dialogue: 0,0:30:26.27,0:30:28.72,Default,,0000,0000,0000,,You can do it with just\Nwith a simple snake
Dialogue: 0,0:30:28.72,0:30:31.11,Default,,0000,0000,0000,,and you're still fine.
Dialogue: 0,0:30:31.11,0:30:37.02,Default,,0000,0000,0000,,So in Calc 1-- this\NCalc 1, whatever it is.
Dialogue: 0,0:30:37.02,0:30:41.84,Default,,0000,0000,0000,,In Calc 1, you learn that\Nyou have to integrate this
Dialogue: 0,0:30:41.84,0:30:49.00,Default,,0000,0000,0000,,and you'll get 2/3 x\Nto the 3/2 minus 1/3 x
Dialogue: 0,0:30:49.00,0:30:56.01,Default,,0000,0000,0000,,cubed at x equals 1 minus\Nwhatever you have with 0.
Dialogue: 0,0:30:56.01,0:30:59.43,Default,,0000,0000,0000,,But at 0, you have 0, so\Nyou say, forget about it.
Dialogue: 0,0:30:59.43,0:31:05.29,Default,,0000,0000,0000,,And you have 2/3 minus 1/3\Nequals 1/3, then you're done.
Dialogue: 0,0:31:05.29,0:31:05.79,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:31:05.79,0:31:08.13,Default,,0000,0000,0000,,Did I expect you\Nto show me work?
Dialogue: 0,0:31:08.13,0:31:09.38,Default,,0000,0000,0000,,No.
Dialogue: 0,0:31:09.38,0:31:11.89,Default,,0000,0000,0000,,For everybody who\Nwrote 1.3-- and there
Dialogue: 0,0:31:11.89,0:31:13.99,Default,,0000,0000,0000,,were many people who\Ndid this mentally,
Dialogue: 0,0:31:13.99,0:31:17.80,Default,,0000,0000,0000,,and they came up with 1/3.
Dialogue: 0,0:31:17.80,0:31:20.90,Default,,0000,0000,0000,,They got 10 pionts\Non the problem.
Dialogue: 0,0:31:20.90,0:31:25.99,Default,,0000,0000,0000,,Finally, number 11.
Dialogue: 0,0:31:25.99,0:31:29.13,Default,,0000,0000,0000,,Without computing the\Nvolume inside the sphere,
Dialogue: 0,0:31:29.13,0:31:35.09,Default,,0000,0000,0000,,x squared plus y squared\Nplus z squared equals 2.
Dialogue: 0,0:31:35.09,0:31:37.85,Default,,0000,0000,0000,,
Dialogue: 0,0:31:37.85,0:31:42.86,Default,,0000,0000,0000,,Set up a triple integral\Ncorresponding to it
Dialogue: 0,0:31:42.86,0:31:44.75,Default,,0000,0000,0000,,in the space provided below.
Dialogue: 0,0:31:44.75,0:31:48.24,Default,,0000,0000,0000,,
Dialogue: 0,0:31:48.24,0:31:52.69,Default,,0000,0000,0000,,Some people, a few\Npeople, messed up.
Dialogue: 0,0:31:52.69,0:31:53.73,Default,,0000,0000,0000,,They forgot the Jacobian.
Dialogue: 0,0:31:53.73,0:31:57.22,Default,,0000,0000,0000,,So they put the 1 instead of\Nr squared [? side-side. ?]
Dialogue: 0,0:31:57.22,0:32:01.34,Default,,0000,0000,0000,,When you work in\Nthree components,
Dialogue: 0,0:32:01.34,0:32:04.29,Default,,0000,0000,0000,,they do fine setting\Nup the limits.
Dialogue: 0,0:32:04.29,0:32:05.99,Default,,0000,0000,0000,,[INAUDIBLE] 1 here.
Dialogue: 0,0:32:05.99,0:32:07.38,Default,,0000,0000,0000,,Don't look at it in the final.
Dialogue: 0,0:32:07.38,0:32:09.37,Default,,0000,0000,0000,,You can ruin your life this way.
Dialogue: 0,0:32:09.37,0:32:11.95,Default,,0000,0000,0000,,So we have r squared sine phi.
Dialogue: 0,0:32:11.95,0:32:16.15,Default,,0000,0000,0000,,Phi was the latitude\Nfrom the North Pole.
Dialogue: 0,0:32:16.15,0:32:18.51,Default,,0000,0000,0000,,it doesn't matter in\Nwhich order you do it.
Dialogue: 0,0:32:18.51,0:32:21.92,Default,,0000,0000,0000,,But I would do to\Ner b phi b theta.
Dialogue: 0,0:32:21.92,0:32:25.38,Default,,0000,0000,0000,,You tell me what the end\Npoints are, and we are done.
Dialogue: 0,0:32:25.38,0:32:26.37,Default,,0000,0000,0000,,STUDENT: From 0 to 5.
Dialogue: 0,0:32:26.37,0:32:27.36,Default,,0000,0000,0000,,PROFESSOR: 0 to--
Dialogue: 0,0:32:27.36,0:32:29.34,Default,,0000,0000,0000,,STUDENT: No, on the first one.
Dialogue: 0,0:32:29.34,0:32:31.32,Default,,0000,0000,0000,,PROFESSOR: 0 to--
Dialogue: 0,0:32:31.32,0:32:31.82,Default,,0000,0000,0000,,STUDENT: Dr?
Dialogue: 0,0:32:31.82,0:32:33.30,Default,,0000,0000,0000,,It's the square root of 2.
Dialogue: 0,0:32:33.30,0:32:34.29,Default,,0000,0000,0000,,PROFESSOR: Mm-hmm.
Dialogue: 0,0:32:34.29,0:32:35.78,Default,,0000,0000,0000,,STUDENT: And b theta--
Dialogue: 0,0:32:35.78,0:32:37.26,Default,,0000,0000,0000,,PROFESSOR: 0.
Dialogue: 0,0:32:37.26,0:32:38.74,Default,,0000,0000,0000,,2pi.
Dialogue: 0,0:32:38.74,0:32:41.54,Default,,0000,0000,0000,,And theta, all around.
Dialogue: 0,0:32:41.54,0:32:42.51,Default,,0000,0000,0000,,STUDENT: 2pi.
Dialogue: 0,0:32:42.51,0:32:45.88,Default,,0000,0000,0000,,PROFESSOR: Longitude\N360 meridian degrees.
Dialogue: 0,0:32:45.88,0:32:46.38,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:32:46.38,0:32:47.83,Default,,0000,0000,0000,,0 to 2pi.
Dialogue: 0,0:32:47.83,0:32:49.28,Default,,0000,0000,0000,,So good.
Dialogue: 0,0:32:49.28,0:32:50.33,Default,,0000,0000,0000,,So we are done.
Dialogue: 0,0:32:50.33,0:32:52.00,Default,,0000,0000,0000,,Did I expect you\Nto write it down?
Dialogue: 0,0:32:52.00,0:32:52.56,Default,,0000,0000,0000,,No.
Dialogue: 0,0:32:52.56,0:32:57.13,Default,,0000,0000,0000,,I had three people who\Nwere nice and wrote down 4.
Dialogue: 0,0:32:57.13,0:32:58.85,Default,,0000,0000,0000,,I mean, they actually\Ndid the work.
Dialogue: 0,0:32:58.85,0:33:00.81,Default,,0000,0000,0000,,Maybe they had\Nnothing better to do.
Dialogue: 0,0:33:00.81,0:33:02.40,Default,,0000,0000,0000,,I have no idea why.
Dialogue: 0,0:33:02.40,0:33:04.82,Default,,0000,0000,0000,,4pi i cubed over 3, right?
Dialogue: 0,0:33:04.82,0:33:09.81,Default,,0000,0000,0000,,And then they proved the formula\Nin general using the Jacobian.
Dialogue: 0,0:33:09.81,0:33:14.64,Default,,0000,0000,0000,,Using the formula, they got\Nthe correct formula for r
Dialogue: 0,0:33:14.64,0:33:16.09,Default,,0000,0000,0000,,equals square root of 2.
Dialogue: 0,0:33:16.09,0:33:18.03,Default,,0000,0000,0000,,And I was very happy.
Dialogue: 0,0:33:18.03,0:33:19.73,Default,,0000,0000,0000,,But did I ask you to do that?
Dialogue: 0,0:33:19.73,0:33:20.23,Default,,0000,0000,0000,,No.
Dialogue: 0,0:33:20.23,0:33:22.07,Default,,0000,0000,0000,,Did I give you extra credit.
Dialogue: 0,0:33:22.07,0:33:22.99,Default,,0000,0000,0000,,No.
Dialogue: 0,0:33:22.99,0:33:26.81,Default,,0000,0000,0000,,So all the extra credit\Nwas just one problem to
Dialogue: 0,0:33:26.81,0:33:30.12,Default,,0000,0000,0000,,asked to do exactly what\Nyou were told to do.
Dialogue: 0,0:33:30.12,0:33:32.96,Default,,0000,0000,0000,,
Dialogue: 0,0:33:32.96,0:33:36.49,Default,,0000,0000,0000,,I don't know about how\Nyou feel about this exam,
Dialogue: 0,0:33:36.49,0:33:39.15,Default,,0000,0000,0000,,but it wasn't a hard exam.
Dialogue: 0,0:33:39.15,0:33:41.14,Default,,0000,0000,0000,,It was not an easy exam.
Dialogue: 0,0:33:41.14,0:33:45.11,Default,,0000,0000,0000,,It was an exam that\Nwas supposed to test
Dialogue: 0,0:33:45.11,0:33:49.57,Default,,0000,0000,0000,,what you learned until now\Nall through the course.
Dialogue: 0,0:33:49.57,0:33:53.54,Default,,0000,0000,0000,,And that was the whole idea.
Dialogue: 0,0:33:53.54,0:33:55.89,Default,,0000,0000,0000,,I think you've\Nlearned very much,
Dialogue: 0,0:33:55.89,0:33:58.92,Default,,0000,0000,0000,,and I think you did fine,\Nthe majority of you.
Dialogue: 0,0:33:58.92,0:34:02.92,Default,,0000,0000,0000,,And that should ease\Nthe pressure on you
Dialogue: 0,0:34:02.92,0:34:05.42,Default,,0000,0000,0000,,when it comes to\Npreparing for the final.
Dialogue: 0,0:34:05.42,0:34:10.08,Default,,0000,0000,0000,,I was thinking last night, I'm\Ngoing to send you, probably
Dialogue: 0,0:34:10.08,0:34:13.12,Default,,0000,0000,0000,,by email or in-person\Nin class, two
Dialogue: 0,0:34:13.12,0:34:17.05,Default,,0000,0000,0000,,or three samples of the\Nfinal from old finals
Dialogue: 0,0:34:17.05,0:34:22.45,Default,,0000,0000,0000,,that inspire us when\Nwe write the final.
Dialogue: 0,0:34:22.45,0:34:25.89,Default,,0000,0000,0000,,A few of us will provide\Nproblems and comments
Dialogue: 0,0:34:25.89,0:34:29.32,Default,,0000,0000,0000,,and suggestions when we write\Nout the departmental final.
Dialogue: 0,0:34:29.32,0:34:33.25,Default,,0000,0000,0000,,But the final will be\Ndepartmental for all sections.
Dialogue: 0,0:34:33.25,0:34:37.67,Default,,0000,0000,0000,,I don't expect more than\N15 problems on the final.
Dialogue: 0,0:34:37.67,0:34:44.26,Default,,0000,0000,0000,,I have yet to think and\Ndecide if I want to [? lift ?]
Dialogue: 0,0:34:44.26,0:34:46.04,Default,,0000,0000,0000,,probably the same policy.
Dialogue: 0,0:34:46.04,0:34:47.89,Default,,0000,0000,0000,,I mean, the final is\Nthe same for everybody.
Dialogue: 0,0:34:47.89,0:34:51.54,Default,,0000,0000,0000,,But the policy about how\Nto give partial credit
Dialogue: 0,0:34:51.54,0:34:54.09,Default,,0000,0000,0000,,or not give partial\Ncredit. [INAUDIBLE].
Dialogue: 0,0:34:54.09,0:34:57.61,Default,,0000,0000,0000,,And I already decided that\NI'm going to read everything,
Dialogue: 0,0:34:57.61,0:35:00.79,Default,,0000,0000,0000,,so in case that you mess up\Nat the end with your miracle
Dialogue: 0,0:35:00.79,0:35:02.51,Default,,0000,0000,0000,,answer, you still\Nget partial credit
Dialogue: 0,0:35:02.51,0:35:05.94,Default,,0000,0000,0000,,for your integrals\N[INAUDIBLE] shown.
Dialogue: 0,0:35:05.94,0:35:10.36,Default,,0000,0000,0000,,Also, one of those 15 problems.\Nmight be for extra credit.
Dialogue: 0,0:35:10.36,0:35:12.55,Default,,0000,0000,0000,,I have to think a\Nlittle bit better
Dialogue: 0,0:35:12.55,0:35:17.58,Default,,0000,0000,0000,,how-- what is the maximum\Nweight I want to put.
Dialogue: 0,0:35:17.58,0:35:22.60,Default,,0000,0000,0000,,What I would say, since I never\N[INAUDIBLE] open a homework,
Dialogue: 0,0:35:22.60,0:35:27.03,Default,,0000,0000,0000,,and I never curve\Nexams, I would think
Dialogue: 0,0:35:27.03,0:35:33.71,Default,,0000,0000,0000,,I could make 110% as\Nthe possible maximum.
Dialogue: 0,0:35:33.71,0:35:37.10,Default,,0000,0000,0000,,In this case, you\Nhave some cushion
Dialogue: 0,0:35:37.10,0:35:42.36,Default,,0000,0000,0000,,to make a mistake or two and\Nstill get a perfect score.
Dialogue: 0,0:35:42.36,0:35:43.67,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:35:43.67,0:35:46.84,Default,,0000,0000,0000,,I'm going to move\Non to a new chapter.
Dialogue: 0,0:35:46.84,0:35:49.30,Default,,0000,0000,0000,,I have actually\Nmoved on already,
Dialogue: 0,0:35:49.30,0:35:51.75,Default,,0000,0000,0000,,but nobody believed me.
Dialogue: 0,0:35:51.75,0:35:56.17,Default,,0000,0000,0000,,Last time, I started Chapter 13.
Dialogue: 0,0:35:56.17,0:36:04.20,Default,,0000,0000,0000,,Chapter 13 is a mixture of\Nmathematics and physics.
Dialogue: 0,0:36:04.20,0:36:07.08,Default,,0000,0000,0000,,You will be surprised\Nhow many things
Dialogue: 0,0:36:07.08,0:36:10.75,Default,,0000,0000,0000,,are coming from solid\Nmechanics, fluid mechanics.
Dialogue: 0,0:36:10.75,0:36:12.22,Default,,0000,0000,0000,,Yes, Regan.
Dialogue: 0,0:36:12.22,0:36:14.67,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:36:14.67,0:36:16.33,Default,,0000,0000,0000,,PROFESSOR: For a job?
Dialogue: 0,0:36:16.33,0:36:18.24,Default,,0000,0000,0000,,You want me to come with you?
Dialogue: 0,0:36:18.24,0:36:19.67,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,0:36:19.67,0:36:22.06,Default,,0000,0000,0000,,STUDENT: Because I tried\Nto talk to you [INAUDIBLE].
Dialogue: 0,0:36:22.06,0:36:24.93,Default,,0000,0000,0000,,PROFESSOR: Yes, yes.
Dialogue: 0,0:36:24.93,0:36:26.84,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:36:26.84,0:36:28.29,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:36:28.29,0:36:29.86,Default,,0000,0000,0000,,And you have to sign up.
Dialogue: 0,0:36:29.86,0:36:32.13,Default,,0000,0000,0000,,Start a [? sheet, ?] attend\N[? the sheet, ?] and sign
Dialogue: 0,0:36:32.13,0:36:34.78,Default,,0000,0000,0000,,your name and good luck\Nwith the interview.
Dialogue: 0,0:36:34.78,0:36:37.18,Default,,0000,0000,0000,,You should have told me before!
Dialogue: 0,0:36:37.18,0:36:39.11,Default,,0000,0000,0000,,I could have said\Na prayer for you.
Dialogue: 0,0:36:39.11,0:36:41.03,Default,,0000,0000,0000,,This things are very stressful!
Dialogue: 0,0:36:41.03,0:36:43.43,Default,,0000,0000,0000,,I remember my own interviews.
Dialogue: 0,0:36:43.43,0:36:44.22,Default,,0000,0000,0000,,There were several.
Dialogue: 0,0:36:44.22,0:36:47.67,Default,,0000,0000,0000,,I didn't know anything about it,\Nand my hands were all sweaty.
Dialogue: 0,0:36:47.67,0:36:49.96,Default,,0000,0000,0000,,And you know you should never\Nshake hands with somebody
Dialogue: 0,0:36:49.96,0:36:51.95,Default,,0000,0000,0000,,when your hands are sweaty.
Dialogue: 0,0:36:51.95,0:36:54.94,Default,,0000,0000,0000,,You have to do like this first.
Dialogue: 0,0:36:54.94,0:36:57.43,Default,,0000,0000,0000,,Be confident and\Ndon't be nervous.
Dialogue: 0,0:36:57.43,0:36:59.42,Default,,0000,0000,0000,,Don't sweat or anything.
Dialogue: 0,0:36:59.42,0:37:01.41,Default,,0000,0000,0000,,Because they can see that.
Dialogue: 0,0:37:01.41,0:37:01.91,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:37:01.91,0:37:05.61,Default,,0000,0000,0000,,You just be yourself.
Dialogue: 0,0:37:05.61,0:37:06.83,Default,,0000,0000,0000,,Do you have earrings?
Dialogue: 0,0:37:06.83,0:37:10.07,Default,,0000,0000,0000,,Because after my\Nseveral job interviews--
Dialogue: 0,0:37:10.07,0:37:13.50,Default,,0000,0000,0000,,those are good earrings-- I\Nwas told that I should never
Dialogue: 0,0:37:13.50,0:37:17.63,Default,,0000,0000,0000,,wear dangling earrings at the\Ninterviews, which I did not,
Dialogue: 0,0:37:17.63,0:37:18.93,Default,,0000,0000,0000,,because I didn't have any.
Dialogue: 0,0:37:18.93,0:37:20.99,Default,,0000,0000,0000,,But I love dangling earrings.
Dialogue: 0,0:37:20.99,0:37:25.97,Default,,0000,0000,0000,,And I was asking some academics\Nwhy that was [? our ?] problem.
Dialogue: 0,0:37:25.97,0:37:27.78,Default,,0000,0000,0000,,And they say they\Nare distracting.
Dialogue: 0,0:37:27.78,0:37:30.67,Default,,0000,0000,0000,,Because mathematicians\Nare like cats.
Dialogue: 0,0:37:30.67,0:37:32.29,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,0:37:32.29,0:37:33.79,Default,,0000,0000,0000,,PROFESSOR: --pendulum,\Nand then they
Dialogue: 0,0:37:33.79,0:37:36.44,Default,,0000,0000,0000,,get hypnotized by the dangling.
Dialogue: 0,0:37:36.44,0:37:37.32,Default,,0000,0000,0000,,So I don't know.
Dialogue: 0,0:37:37.32,0:37:40.91,Default,,0000,0000,0000,,I think most of the\Ninterviewers have some problems
Dialogue: 0,0:37:40.91,0:37:45.31,Default,,0000,0000,0000,,and they find some things\Ndistracting or annoying.
Dialogue: 0,0:37:45.31,0:37:46.77,Default,,0000,0000,0000,,Otherwise, I think you are fine.
Dialogue: 0,0:37:46.77,0:37:49.69,Default,,0000,0000,0000,,You're dressed fine\Nfor an interview.
Dialogue: 0,0:37:49.69,0:37:50.19,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:37:50.19,0:37:52.99,Default,,0000,0000,0000,,So now serious job.
Dialogue: 0,0:37:52.99,0:37:57.89,Default,,0000,0000,0000,,We have to remember some of\Nthe things we don't remember.
Dialogue: 0,0:37:57.89,0:38:03.74,Default,,0000,0000,0000,,Which are the gradient for\Na function of let's say
Dialogue: 0,0:38:03.74,0:38:05.21,Default,,0000,0000,0000,,three variables.
Dialogue: 0,0:38:05.21,0:38:08.13,Default,,0000,0000,0000,,Let's grow up a little bit.
Dialogue: 0,0:38:08.13,0:38:13.29,Default,,0000,0000,0000,,And that was what\Nthe vector field
Dialogue: 0,0:38:13.29,0:38:19.63,Default,,0000,0000,0000,,F sub xi plus F sub\N[? I j ?] plus F sub z k.
Dialogue: 0,0:38:19.63,0:38:20.17,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:38:20.17,0:38:24.35,Default,,0000,0000,0000,,At an arbitrary point\Nxyz in your domain.
Dialogue: 0,0:38:24.35,0:38:29.88,Default,,0000,0000,0000,,So where xyz is in some\Ndomain, you are in a potato.
Dialogue: 0,0:38:29.88,0:38:34.94,Default,,0000,0000,0000,,And the meaning of the gradient,\Nthe geometric meaning of this,
Dialogue: 0,0:38:34.94,0:38:36.91,Default,,0000,0000,0000,,doesn't look like a\Ntheta [INAUDIBLE].
Dialogue: 0,0:38:36.91,0:38:41.17,Default,,0000,0000,0000,,It's some sort of solid\Nthat it corresponds
Dialogue: 0,0:38:41.17,0:38:42.67,Default,,0000,0000,0000,,to a closed surface.
Dialogue: 0,0:38:42.67,0:38:46.53,Default,,0000,0000,0000,,And this closed surface\Nthat closes up on its own
Dialogue: 0,0:38:46.53,0:38:49.66,Default,,0000,0000,0000,,is having a hard\Ntime [INAUDIBLE].
Dialogue: 0,0:38:49.66,0:38:51.90,Default,,0000,0000,0000,,It has a normal.
Dialogue: 0,0:38:51.90,0:38:56.65,Default,,0000,0000,0000,,And this normal is given by\Nthe gradient of this function,
Dialogue: 0,0:38:56.65,0:38:59.53,Default,,0000,0000,0000,,we can increase\N[? it ?] like that.
Dialogue: 0,0:38:59.53,0:39:00.97,Default,,0000,0000,0000,,You remember that.
Dialogue: 0,0:39:00.97,0:39:02.97,Default,,0000,0000,0000,,And that was a long time ago.
Dialogue: 0,0:39:02.97,0:39:06.81,Default,,0000,0000,0000,,But you should\Nstill master that.
Dialogue: 0,0:39:06.81,0:39:11.98,Default,,0000,0000,0000,,Last time, I gave you\Nthe z equals f of xy,
Dialogue: 0,0:39:11.98,0:39:14.49,Default,,0000,0000,0000,,z equals little f\Nof xy, as a graph
Dialogue: 0,0:39:14.49,0:39:18.24,Default,,0000,0000,0000,,of the function of two variables\Nover a domain in plane.
Dialogue: 0,0:39:18.24,0:39:19.98,Default,,0000,0000,0000,,We computed the\Ngradient of that.
Dialogue: 0,0:39:19.98,0:39:22.86,Default,,0000,0000,0000,,But that's what we did all\Nthrough the [? meter ?].
Dialogue: 0,0:39:22.86,0:39:24.62,Default,,0000,0000,0000,,So that's no fun.
Dialogue: 0,0:39:24.62,0:39:27.14,Default,,0000,0000,0000,,We know that too well.
Dialogue: 0,0:39:27.14,0:39:33.68,Default,,0000,0000,0000,,On this problem, I\Ngave you some new piece
Dialogue: 0,0:39:33.68,0:39:34.83,Default,,0000,0000,0000,,of information last time.
Dialogue: 0,0:39:34.83,0:39:38.19,Default,,0000,0000,0000,,So I said, if you have\Na vector field that
Dialogue: 0,0:39:38.19,0:39:44.48,Default,,0000,0000,0000,,looks F 1i plus F\N2j plus F 3k, where
Dialogue: 0,0:39:44.48,0:39:50.44,Default,,0000,0000,0000,,Fi is C1, that means\Nthat the differentiable
Dialogue: 0,0:39:50.44,0:39:52.92,Default,,0000,0000,0000,,and the derivatives\Nare continuous,
Dialogue: 0,0:39:52.92,0:39:56.89,Default,,0000,0000,0000,,what was the divergence of it?
Dialogue: 0,0:39:56.89,0:39:59.37,Default,,0000,0000,0000,,Well, that was before\Nthe Easter break.
Dialogue: 0,0:39:59.37,0:40:01.36,Default,,0000,0000,0000,,And I know we had a long break.
Dialogue: 0,0:40:01.36,0:40:06.32,Default,,0000,0000,0000,,I cannot recover from this break\Nso easily, because it was long.
Dialogue: 0,0:40:06.32,0:40:08.30,Default,,0000,0000,0000,,And I also traveled last week.
Dialogue: 0,0:40:08.30,0:40:13.70,Default,,0000,0000,0000,,But before I traveled, I\Nremember that I gave you this.
Dialogue: 0,0:40:13.70,0:40:15.85,Default,,0000,0000,0000,,And you memorized it.
Dialogue: 0,0:40:15.85,0:40:17.67,Default,,0000,0000,0000,,Most of you memorised it.
Dialogue: 0,0:40:17.67,0:40:19.86,Default,,0000,0000,0000,,How was it?
Dialogue: 0,0:40:19.86,0:40:23.31,Default,,0000,0000,0000,,The first component\Ndifferentiated with respect
Dialogue: 0,0:40:23.31,0:40:28.73,Default,,0000,0000,0000,,to the first variable\Nplus the second component
Dialogue: 0,0:40:28.73,0:40:33.17,Default,,0000,0000,0000,,differentiated with respect\Nto the second variable.
Dialogue: 0,0:40:33.17,0:40:37.61,Default,,0000,0000,0000,,Plus the third component\Ndifferentiated with respect
Dialogue: 0,0:40:37.61,0:40:41.06,Default,,0000,0000,0000,,to the third variable.
Dialogue: 0,0:40:41.06,0:40:46.27,Default,,0000,0000,0000,,So I'm asking you, as\Nan exercise, like I
Dialogue: 0,0:40:46.27,0:40:49.39,Default,,0000,0000,0000,,did last time, the same thing.
Dialogue: 0,0:40:49.39,0:40:54.85,Default,,0000,0000,0000,,Exercise one for this section.
Dialogue: 0,0:40:54.85,0:40:57.32,Default,,0000,0000,0000,,Compute divergence\Nof the gradient
Dialogue: 0,0:40:57.32,0:41:04.52,Default,,0000,0000,0000,,of F, where F is a\NC1 function of xyz.
Dialogue: 0,0:41:04.52,0:41:07.00,Default,,0000,0000,0000,,That means F is\N[? like this ?] differentiable
Dialogue: 0,0:41:07.00,0:41:08.99,Default,,0000,0000,0000,,and with continuous derivatives.
Dialogue: 0,0:41:08.99,0:41:10.48,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:41:10.48,0:41:15.45,Default,,0000,0000,0000,,It means that you have to\Ncompute divergence of F sub xi
Dialogue: 0,0:41:15.45,0:41:20.42,Default,,0000,0000,0000,,plus F sub yj plus F sub zk.
Dialogue: 0,0:41:20.42,0:41:24.01,Default,,0000,0000,0000,,And you're thinking,\NI can do that!
Dialogue: 0,0:41:24.01,0:41:29.75,Default,,0000,0000,0000,,By definition, I take the\Nfirst component-- who was that?
Dialogue: 0,0:41:29.75,0:41:30.75,Default,,0000,0000,0000,,Hmm?
Dialogue: 0,0:41:30.75,0:41:32.24,Default,,0000,0000,0000,,STUDENT: Brian.
Dialogue: 0,0:41:32.24,0:41:33.24,Default,,0000,0000,0000,,PROFESSOR: Oh, right.
Dialogue: 0,0:41:33.24,0:41:35.23,Default,,0000,0000,0000,,I thought that somebody\Nwanted to come in
Dialogue: 0,0:41:35.23,0:41:37.72,Default,,0000,0000,0000,,and then he heard me\Nand changed his mind.
Dialogue: 0,0:41:37.72,0:41:39.21,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,0:41:39.21,0:41:41.02,Default,,0000,0000,0000,,PROFESSOR: F sub x\Nparentheses [INAUDIBLE]
Dialogue: 0,0:41:41.02,0:41:45.01,Default,,0000,0000,0000,,x plus F sub-- like when\Nyou go on a blind date
Dialogue: 0,0:41:45.01,0:41:47.29,Default,,0000,0000,0000,,and you see, change your mind.
Dialogue: 0,0:41:47.29,0:41:47.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:41:47.91,0:41:53.78,Default,,0000,0000,0000,,F sub y y plus F sub z z.
Dialogue: 0,0:41:53.78,0:41:57.57,Default,,0000,0000,0000,,Do you remember that\NI gave away 95 cents
Dialogue: 0,0:41:57.57,0:41:59.99,Default,,0000,0000,0000,,for this type of question?
Dialogue: 0,0:41:59.99,0:42:03.23,Default,,0000,0000,0000,,So what was this operator?
Dialogue: 0,0:42:03.23,0:42:05.45,Default,,0000,0000,0000,,We can write it better.
Dialogue: 0,0:42:05.45,0:42:09.34,Default,,0000,0000,0000,,We can write it using the\Nsecond partial derivatives
Dialogue: 0,0:42:09.34,0:42:12.36,Default,,0000,0000,0000,,with respect to z, y, and z.
Dialogue: 0,0:42:12.36,0:42:15.77,Default,,0000,0000,0000,,And we gave a name to this one.
Dialogue: 0,0:42:15.77,0:42:17.41,Default,,0000,0000,0000,,We called this names--
Dialogue: 0,0:42:17.41,0:42:18.20,Default,,0000,0000,0000,,STUDENT: Laplacian.
Dialogue: 0,0:42:18.20,0:42:19.18,Default,,0000,0000,0000,,PROFESSOR: Laplacian.
Dialogue: 0,0:42:19.18,0:42:21.12,Default,,0000,0000,0000,,Laplace operator.
Dialogue: 0,0:42:21.12,0:42:22.58,Default,,0000,0000,0000,,Laplace.
Dialogue: 0,0:42:22.58,0:42:25.51,Default,,0000,0000,0000,,Laplace.
Dialogue: 0,0:42:25.51,0:42:26.48,Default,,0000,0000,0000,,Laplacian.
Dialogue: 0,0:42:26.48,0:42:29.06,Default,,0000,0000,0000,,That's how you spell it.
Dialogue: 0,0:42:29.06,0:42:30.83,Default,,0000,0000,0000,,Laplac-ian.
Dialogue: 0,0:42:30.83,0:42:33.26,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:42:33.26,0:42:44.06,Default,,0000,0000,0000,,Of F. And then what do you have?
Dialogue: 0,0:42:44.06,0:42:48.91,Default,,0000,0000,0000,,
Dialogue: 0,0:42:48.91,0:42:51.86,Default,,0000,0000,0000,,You have to introduce\Na new notation in.
Dialogue: 0,0:42:51.86,0:42:54.02,Default,,0000,0000,0000,,When you see this\Ntriangle that looks
Dialogue: 0,0:42:54.02,0:42:56.38,Default,,0000,0000,0000,,like an equilateral\Ntriangle, this
Dialogue: 0,0:42:56.38,0:42:58.65,Default,,0000,0000,0000,,means Laplacian of something.
Dialogue: 0,0:42:58.65,0:43:01.08,Default,,0000,0000,0000,,So if you have a function\Nof two variables-- so
Dialogue: 0,0:43:01.08,0:43:03.38,Default,,0000,0000,0000,,let's say z equals F of xy.
Dialogue: 0,0:43:03.38,0:43:07.80,Default,,0000,0000,0000,,What is the Laplacian\Nof this little f?
Dialogue: 0,0:43:07.80,0:43:11.72,Default,,0000,0000,0000,,Little f x x plus little f y y.
Dialogue: 0,0:43:11.72,0:43:14.67,Default,,0000,0000,0000,,So we could be second\Npartial with respect
Dialogue: 0,0:43:14.67,0:43:17.46,Default,,0000,0000,0000,,to x plus the second\Npartial with respect to y.
Dialogue: 0,0:43:17.46,0:43:20.53,Default,,0000,0000,0000,,What if I have something else?
Dialogue: 0,0:43:20.53,0:43:26.20,Default,,0000,0000,0000,,Like let me give you a\Nmore general function.
Dialogue: 0,0:43:26.20,0:43:28.89,Default,,0000,0000,0000,,Let's say I have a\Ndifferentiable function
Dialogue: 0,0:43:28.89,0:43:31.57,Default,,0000,0000,0000,,of N variables with\Ncontinuous derivatives.
Dialogue: 0,0:43:31.57,0:43:33.52,Default,,0000,0000,0000,,And it looks like crazy.
Dialogue: 0,0:43:33.52,0:43:35.47,Default,,0000,0000,0000,,It looks like that.
Dialogue: 0,0:43:35.47,0:43:39.48,Default,,0000,0000,0000,,x1, x2, x n minus what?
Dialogue: 0,0:43:39.48,0:43:46.74,Default,,0000,0000,0000,,Well, the Laplace operator in\Nthis case will be F sub x1 x1
Dialogue: 0,0:43:46.74,0:43:48.43,Default,,0000,0000,0000,,plus [? A of ?] sub x2 x2.
Dialogue: 0,0:43:48.43,0:43:53.26,Default,,0000,0000,0000,,Which means the partial of\NF, the second derivative
Dialogue: 0,0:43:53.26,0:43:55.24,Default,,0000,0000,0000,,with respect to x2.
Dialogue: 0,0:43:55.24,0:44:03.16,Default,,0000,0000,0000,,And plus the last derivative\Nwith respect-- two [INAUDIBLE]
Dialogue: 0,0:44:03.16,0:44:04.64,Default,,0000,0000,0000,,with respect to\Nthe same variable.
Dialogue: 0,0:44:04.64,0:44:07.12,Default,,0000,0000,0000,,The last variable is xm minus 1.
Dialogue: 0,0:44:07.12,0:44:09.10,Default,,0000,0000,0000,,This could be one million and 1.
Dialogue: 0,0:44:09.10,0:44:10.62,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:44:10.62,0:44:14.27,Default,,0000,0000,0000,,You can have this as many\Nvariables as you want.
Dialogue: 0,0:44:14.27,0:44:17.48,Default,,0000,0000,0000,,Now, actually in\Nengineering, there
Dialogue: 0,0:44:17.48,0:44:20.44,Default,,0000,0000,0000,,are functions that\Nhave many parameters.
Dialogue: 0,0:44:20.44,0:44:22.09,Default,,0000,0000,0000,,You have three\Nspecial opponents.
Dialogue: 0,0:44:22.09,0:44:22.90,Default,,0000,0000,0000,,Then you have time.
Dialogue: 0,0:44:22.90,0:44:25.08,Default,,0000,0000,0000,,Then you have temperature,\Nthen you have pressure,
Dialogue: 0,0:44:25.08,0:44:27.02,Default,,0000,0000,0000,,then you have god knows what.
Dialogue: 0,0:44:27.02,0:44:29.78,Default,,0000,0000,0000,,The surface tension\Nof the membrane.
Dialogue: 0,0:44:29.78,0:44:32.07,Default,,0000,0000,0000,,Many things.
Dialogue: 0,0:44:32.07,0:44:34.35,Default,,0000,0000,0000,,You really have a\Nmillion parameters.
Dialogue: 0,0:44:34.35,0:44:35.70,Default,,0000,0000,0000,,Actually, it's impossible.
Dialogue: 0,0:44:35.70,0:44:38.56,Default,,0000,0000,0000,,It's even hard to work\Nwith 10 parameters.
Dialogue: 0,0:44:38.56,0:44:41.60,Default,,0000,0000,0000,,Imagine always\Nworking with equations
Dialogue: 0,0:44:41.60,0:44:47.96,Default,,0000,0000,0000,,that have lots of variables\Nand having do deal with that.
Dialogue: 0,0:44:47.96,0:44:52.69,Default,,0000,0000,0000,,In fluid flows,\Nhydrodynamical problems,
Dialogue: 0,0:44:52.69,0:44:56.18,Default,,0000,0000,0000,,most the time in\N3D turbulent flows,
Dialogue: 0,0:44:56.18,0:44:58.93,Default,,0000,0000,0000,,for example, then you have\Nxyz spatial coordinates
Dialogue: 0,0:44:58.93,0:45:02.53,Default,,0000,0000,0000,,and time T. So even\Nwith four variables,
Dialogue: 0,0:45:02.53,0:45:07.11,Default,,0000,0000,0000,,once you get those operators,\Nyou could have something like F
Dialogue: 0,0:45:07.11,0:45:14.25,Default,,0000,0000,0000,,sub x x x t plus g sub\Nx x t plus and so on.
Dialogue: 0,0:45:14.25,0:45:17.98,Default,,0000,0000,0000,,All sorts of ugly components.
Dialogue: 0,0:45:17.98,0:45:21.78,Default,,0000,0000,0000,,Sometimes you'll have\Nequations of fluid flows
Dialogue: 0,0:45:21.78,0:45:24.32,Default,,0000,0000,0000,,in dynamic software.
Dialogue: 0,0:45:24.32,0:45:26.17,Default,,0000,0000,0000,,Fluid flows with\Nturbulence are really
Dialogue: 0,0:45:26.17,0:45:30.68,Default,,0000,0000,0000,,an area of\Nmathematics in itself,
Dialogue: 0,0:45:30.68,0:45:36.57,Default,,0000,0000,0000,,of really complicated equations\Nwith most of the operators.
Dialogue: 0,0:45:36.57,0:45:38.55,Default,,0000,0000,0000,,I was looking at\Nthem in Georgia,
Dialogue: 0,0:45:38.55,0:45:40.04,Default,,0000,0000,0000,,where I went to this conference.
Dialogue: 0,0:45:40.04,0:45:43.03,Default,,0000,0000,0000,,Most of those\Nequations were order 4.
Dialogue: 0,0:45:43.03,0:45:47.00,Default,,0000,0000,0000,,Of course, most of them you\Ncannot even think about solving
Dialogue: 0,0:45:47.00,0:45:50.00,Default,,0000,0000,0000,,by hand, or with\Nany known methods.
Dialogue: 0,0:45:50.00,0:45:54.23,Default,,0000,0000,0000,,You can solve them numerically\Nwith computational software.
Dialogue: 0,0:45:54.23,0:45:58.10,Default,,0000,0000,0000,,That is the only [INAUDIBLE]\Nthat modern mathematics
Dialogue: 0,0:45:58.10,0:46:01.39,Default,,0000,0000,0000,,has in some areas right now.
Dialogue: 0,0:46:01.39,0:46:04.27,Default,,0000,0000,0000,,The right software, in\Norder to find solutions
Dialogue: 0,0:46:04.27,0:46:06.87,Default,,0000,0000,0000,,to a fluid flow with turbulence.
Dialogue: 0,0:46:06.87,0:46:09.31,Default,,0000,0000,0000,,That is the solution to\Nthis type of equation.
Dialogue: 0,0:46:09.31,0:46:13.70,Default,,0000,0000,0000,,Like [INAUDIBLE], for example.
Dialogue: 0,0:46:13.70,0:46:18.71,Default,,0000,0000,0000,,Now we are going to see--\Nwell, you are going to see.
Dialogue: 0,0:46:18.71,0:46:20.88,Default,,0000,0000,0000,,I'm too old and I saw\Nthat 20 years ago.
Dialogue: 0,0:46:20.88,0:46:23.65,Default,,0000,0000,0000,,When you're going\N3350 [INAUDIBLE]
Dialogue: 0,0:46:23.65,0:46:25.10,Default,,0000,0000,0000,,differential equations.
Dialogue: 0,0:46:25.10,0:46:29.91,Default,,0000,0000,0000,,And then, if you do PD\N3350 one in engineering,
Dialogue: 0,0:46:29.91,0:46:32.31,Default,,0000,0000,0000,,You're going to see\Nlots of equations
Dialogue: 0,0:46:32.31,0:46:34.43,Default,,0000,0000,0000,,that are hard to solve.
Dialogue: 0,0:46:34.43,0:46:37.05,Default,,0000,0000,0000,,But in many of them, you're\Ngoing to see partials,
Dialogue: 0,0:46:37.05,0:46:38.12,Default,,0000,0000,0000,,like that.
Dialogue: 0,0:46:38.12,0:46:40.14,Default,,0000,0000,0000,,And you're going to\Nsay, oh, thank god
Dialogue: 0,0:46:40.14,0:46:42.18,Default,,0000,0000,0000,,that I like partials\Nin Calc Three
Dialogue: 0,0:46:42.18,0:46:43.53,Default,,0000,0000,0000,,so they became my friends.
Dialogue: 0,0:46:43.53,0:46:47.35,Default,,0000,0000,0000,,And you'll never have\Nheadaches-- [? you know what ?]
Dialogue: 0,0:46:47.35,0:46:50.74,Default,,0000,0000,0000,,would be easy, if you understood\Nthat notion of differential
Dialogue: 0,0:46:50.74,0:46:55.38,Default,,0000,0000,0000,,well, the notion of partial\Nderivatives very well.
Dialogue: 0,0:46:55.38,0:46:59.38,Default,,0000,0000,0000,,So I'm going to erase this one.
Dialogue: 0,0:46:59.38,0:47:13.64,Default,,0000,0000,0000,,
Dialogue: 0,0:47:13.64,0:47:14.46,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:47:14.46,0:47:17.88,Default,,0000,0000,0000,,And then I'll say, I\Ndon't how many of you--
Dialogue: 0,0:47:17.88,0:47:21.69,Default,,0000,0000,0000,,I'll try to make this\Nformula more visible.
Dialogue: 0,0:47:21.69,0:47:25.01,Default,,0000,0000,0000,,Some of you maybe, who\Nare engineering majors
Dialogue: 0,0:47:25.01,0:47:27.99,Default,,0000,0000,0000,,know about curl.
Dialogue: 0,0:47:27.99,0:47:30.51,Default,,0000,0000,0000,,Have you heard about curl?
Dialogue: 0,0:47:30.51,0:47:32.83,Default,,0000,0000,0000,,Curl of a vector value function.
Dialogue: 0,0:47:32.83,0:47:33.33,Default,,0000,0000,0000,,No.
Dialogue: 0,0:47:33.33,0:47:34.82,Default,,0000,0000,0000,,You haven't.
Dialogue: 0,0:47:34.82,0:47:38.30,Default,,0000,0000,0000,,Suppose that you have a\Nvector value function.
Dialogue: 0,0:47:38.30,0:47:44.76,Default,,0000,0000,0000,,
Dialogue: 0,0:47:44.76,0:47:49.73,Default,,0000,0000,0000,,That is F of coordinates\Nx, y, z, the coordinates.
Dialogue: 0,0:47:49.73,0:47:53.47,Default,,0000,0000,0000,,The C1 of over\Nseven domain omega.
Dialogue: 0,0:47:53.47,0:47:57.53,Default,,0000,0000,0000,,Omega is the domain that your\Nspecial coordinates live in.
Dialogue: 0,0:47:57.53,0:47:59.88,Default,,0000,0000,0000,,Xyz living some potato.
Dialogue: 0,0:47:59.88,0:48:02.07,Default,,0000,0000,0000,,That's it.
Dialogue: 0,0:48:02.07,0:48:06.83,Default,,0000,0000,0000,,Whose solid body enclosed\Nby a closed surface.
Dialogue: 0,0:48:06.83,0:48:11.15,Default,,0000,0000,0000,,In that potato, F is a\Ndifferentiable function
Dialogue: 0,0:48:11.15,0:48:16.20,Default,,0000,0000,0000,,with respect to xyz, and the\Nderivatives are continuous.
Dialogue: 0,0:48:16.20,0:48:19.48,Default,,0000,0000,0000,,Now, in most cases, if\Nyou work with Laplacian,
Dialogue: 0,0:48:19.48,0:48:21.61,Default,,0000,0000,0000,,this is not enough C1.
Dialogue: 0,0:48:21.61,0:48:24.10,Default,,0000,0000,0000,,If you work with Laplacian,\Nwhat do you want?
Dialogue: 0,0:48:24.10,0:48:25.10,Default,,0000,0000,0000,,What do you need?
Dialogue: 0,0:48:25.10,0:48:28.10,Default,,0000,0000,0000,,You have F sub x\Nx plus F sub y1.
Dialogue: 0,0:48:28.10,0:48:29.59,Default,,0000,0000,0000,,So you need C2.
Dialogue: 0,0:48:29.59,0:48:32.59,Default,,0000,0000,0000,,You work with at least C2.
Dialogue: 0,0:48:32.59,0:48:35.08,Default,,0000,0000,0000,,Many examples have C infinity.
Dialogue: 0,0:48:35.08,0:48:37.83,Default,,0000,0000,0000,,That means you're having\Nreally beautiful functions that
Dialogue: 0,0:48:37.83,0:48:39.07,Default,,0000,0000,0000,,are elementary.
Dialogue: 0,0:48:39.07,0:48:41.07,Default,,0000,0000,0000,,Some of them even\Npolynomial approximations.
Dialogue: 0,0:48:41.07,0:48:43.93,Default,,0000,0000,0000,,And then you really\Ncan differentiate
Dialogue: 0,0:48:43.93,0:48:47.81,Default,,0000,0000,0000,,them ad infinitum and all\Nthe derivatives [INAUDIBLE],
Dialogue: 0,0:48:47.81,0:48:50.29,Default,,0000,0000,0000,,and then you can\Ncall yourself lucky.
Dialogue: 0,0:48:50.29,0:48:54.26,Default,,0000,0000,0000,,How do you introduce the\Nnotion of curl of it?
Dialogue: 0,0:48:54.26,0:48:57.73,Default,,0000,0000,0000,,And it sounds funny, and this\Nis why they made this fun.
Dialogue: 0,0:48:57.73,0:49:01.20,Default,,0000,0000,0000,,And my hair used to be\Ncurly, but I shaved my head
Dialogue: 0,0:49:01.20,0:49:04.34,Default,,0000,0000,0000,,over the holiday,\Nand now it's between.
Dialogue: 0,0:49:04.34,0:49:09.18,Default,,0000,0000,0000,,So curl of F is something\Nthat looks horrible
Dialogue: 0,0:49:09.18,0:49:12.14,Default,,0000,0000,0000,,when you try to memorize it.
Dialogue: 0,0:49:12.14,0:49:15.51,Default,,0000,0000,0000,,So you say, OK, if I'm going\Nto get this on the final,
Dialogue: 0,0:49:15.51,0:49:18.03,Default,,0000,0000,0000,,you better wear this T-shirt.
Dialogue: 0,0:49:18.03,0:49:21.51,Default,,0000,0000,0000,,No, there is something\Nbetter than that.
Dialogue: 0,0:49:21.51,0:49:24.89,Default,,0000,0000,0000,,One time I was the wearing-- OK.
Dialogue: 0,0:49:24.89,0:49:29.93,Default,,0000,0000,0000,,My students got no permission\Nfrom the [INAUDIBLE]
Dialogue: 0,0:49:29.93,0:49:33.37,Default,,0000,0000,0000,,to come in with a cheat sheet.
Dialogue: 0,0:49:33.37,0:49:36.26,Default,,0000,0000,0000,,But I was wearing a T-shirt\Nthat had Green's theorem.
Dialogue: 0,0:49:36.26,0:49:37.64,Default,,0000,0000,0000,,I don't know how\Nmany of you have
Dialogue: 0,0:49:37.64,0:49:39.06,Default,,0000,0000,0000,,heard about Green's theorem.
Dialogue: 0,0:49:39.06,0:49:41.05,Default,,0000,0000,0000,,We are going to learn\Nit in two weeks.
Dialogue: 0,0:49:41.05,0:49:44.26,Default,,0000,0000,0000,,And I was wearing that T-shirt.
Dialogue: 0,0:49:44.26,0:49:46.73,Default,,0000,0000,0000,,And it was by accident, OK?
Dialogue: 0,0:49:46.73,0:49:49.70,Default,,0000,0000,0000,,I didn't do it on purpose\Nto help my students cheat.
Dialogue: 0,0:49:49.70,0:49:53.32,Default,,0000,0000,0000,,So one student at some\Npoint goes like, well, I
Dialogue: 0,0:49:53.32,0:49:54.64,Default,,0000,0000,0000,,don't remember Green's theorem.
Dialogue: 0,0:49:54.64,0:49:56.03,Default,,0000,0000,0000,,And then he looked my T-shirt.
Dialogue: 0,0:49:56.03,0:49:56.61,Default,,0000,0000,0000,,Oh, all right.
Dialogue: 0,0:49:56.61,0:49:58.09,Default,,0000,0000,0000,,Never mind.
Dialogue: 0,0:49:58.09,0:50:01.78,Default,,0000,0000,0000,,So I had Green's theorem\Non my shirt, [INAUDIBLE].
Dialogue: 0,0:50:01.78,0:50:04.33,Default,,0000,0000,0000,,
Dialogue: 0,0:50:04.33,0:50:08.47,Default,,0000,0000,0000,,But it's hard to wear like\N10 T-shirts, one for the-- I
Dialogue: 0,0:50:08.47,0:50:12.13,Default,,0000,0000,0000,,have one for the formula of the\Ncurvature of a curve in space.
Dialogue: 0,0:50:12.13,0:50:14.56,Default,,0000,0000,0000,,Remember that one,\Nhow it is so nasty?
Dialogue: 0,0:50:14.56,0:50:16.02,Default,,0000,0000,0000,,OK, I have this one.
Dialogue: 0,0:50:16.02,0:50:16.100,Default,,0000,0000,0000,,I have Green's theorem.
Dialogue: 0,0:50:16.100,0:50:19.41,Default,,0000,0000,0000,,I have [INAUDIBLE], all the\Nimportant formulas actually.
Dialogue: 0,0:50:19.41,0:50:20.75,Default,,0000,0000,0000,,I have 10 T-shirts.
Dialogue: 0,0:50:20.75,0:50:23.22,Default,,0000,0000,0000,,And then I was\Nthinking, how will I
Dialogue: 0,0:50:23.22,0:50:27.16,Default,,0000,0000,0000,,be if I were like taking ten\NT-shirts on top of the other
Dialogue: 0,0:50:27.16,0:50:31.12,Default,,0000,0000,0000,,and taking them one off at\Na time during the final.
Dialogue: 0,0:50:31.12,0:50:32.78,Default,,0000,0000,0000,,There is no cheat sheet.
Dialogue: 0,0:50:32.78,0:50:35.51,Default,,0000,0000,0000,,There are no formula\Nsheets, no nothing.
Dialogue: 0,0:50:35.51,0:50:38.06,Default,,0000,0000,0000,,But I would look like\NJoey from "Friends."
Dialogue: 0,0:50:38.06,0:50:41.95,Default,,0000,0000,0000,,Remember Joey, when he was\Ndressed in many layers.
Dialogue: 0,0:50:41.95,0:50:47.41,Default,,0000,0000,0000,,So rather than\Nthat, I say ask me.
Dialogue: 0,0:50:47.41,0:50:49.97,Default,,0000,0000,0000,,Say oh, you know,\NI'm freaking out.
Dialogue: 0,0:50:49.97,0:50:55.10,Default,,0000,0000,0000,,I'm taking this final,\Nand I forgot curl.
Dialogue: 0,0:50:55.10,0:50:59.32,Default,,0000,0000,0000,,Rather than not attempting\Nthe complex problem at all,
Dialogue: 0,0:50:59.32,0:51:03.78,Default,,0000,0000,0000,,ask me before the exam,\Nand I will remind everybody
Dialogue: 0,0:51:03.78,0:51:07.70,Default,,0000,0000,0000,,how to set up the curl formula.
Dialogue: 0,0:51:07.70,0:51:11.38,Default,,0000,0000,0000,,So you simply have\Nto think in terms
Dialogue: 0,0:51:11.38,0:51:15.42,Default,,0000,0000,0000,,of operators-- ddx, ddy, ddz.
Dialogue: 0,0:51:15.42,0:51:16.39,Default,,0000,0000,0000,,What are these?
Dialogue: 0,0:51:16.39,0:51:22.09,Default,,0000,0000,0000,,These are derivative operators.
Dialogue: 0,0:51:22.09,0:51:29.51,Default,,0000,0000,0000,,So if you take this and\Nmultiply it by a function,
Dialogue: 0,0:51:29.51,0:51:34.26,Default,,0000,0000,0000,,that means df, ds-- [INAUDIBLE].
Dialogue: 0,0:51:34.26,0:51:39.71,Default,,0000,0000,0000,,All right, so in this\Ncase, if F is-- I'll
Dialogue: 0,0:51:39.71,0:51:54.05,Default,,0000,0000,0000,,go by my T-shirt-- PI plus QJ\Nplus RK, where PQ and R are all
Dialogue: 0,0:51:54.05,0:52:00.72,Default,,0000,0000,0000,,scalar functions of xyz.
Dialogue: 0,0:52:00.72,0:52:07.92,Default,,0000,0000,0000,,
Dialogue: 0,0:52:07.92,0:52:09.84,Default,,0000,0000,0000,,STUDENT: Then we\Nwill not forget it.
Dialogue: 0,0:52:09.84,0:52:11.76,Default,,0000,0000,0000,,PROFESSOR: Then we are\Nno longer forget it,
Dialogue: 0,0:52:11.76,0:52:15.49,Default,,0000,0000,0000,,and you'll no longer\Nneed my T-shirt.
Dialogue: 0,0:52:15.49,0:52:18.58,Default,,0000,0000,0000,,All right, so how\Ndo you do that?
Dialogue: 0,0:52:18.58,0:52:22.43,Default,,0000,0000,0000,,You go expand along\Nyour first row,
Dialogue: 0,0:52:22.43,0:52:28.78,Default,,0000,0000,0000,,I times whoever the minor\Nwill be, which is this guy.
Dialogue: 0,0:52:28.78,0:52:31.49,Default,,0000,0000,0000,,How do you do the\N[? cowboy ?] problem?
Dialogue: 0,0:52:31.49,0:52:34.17,Default,,0000,0000,0000,,These guys multiply each other.
Dialogue: 0,0:52:34.17,0:52:38.53,Default,,0000,0000,0000,,So you go dr, dy.
Dialogue: 0,0:52:38.53,0:52:39.22,Default,,0000,0000,0000,,Plus or minus?
Dialogue: 0,0:52:39.22,0:52:45.41,Default,,0000,0000,0000,,Minus dq, dz.
Dialogue: 0,0:52:45.41,0:52:55.50,Default,,0000,0000,0000,,Close times I. So the I is\Nthe corresponding element
Dialogue: 0,0:52:55.50,0:52:58.40,Default,,0000,0000,0000,,to the minor that\NI just completed.
Dialogue: 0,0:52:58.40,0:53:03.65,Default,,0000,0000,0000,,This minor is the determinant,\Nwhich is exactly this guy.
Dialogue: 0,0:53:03.65,0:53:07.03,Default,,0000,0000,0000,,And this is exactly\Nwhat my T-shirt says.
Dialogue: 0,0:53:07.03,0:53:08.13,Default,,0000,0000,0000,,Right, precisely.
Dialogue: 0,0:53:08.13,0:53:09.38,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:53:09.38,0:53:12.90,Default,,0000,0000,0000,,The second term, if\Nwe put the minus-- no,
Dialogue: 0,0:53:12.90,0:53:14.84,Default,,0000,0000,0000,,they changed the signs.
Dialogue: 0,0:53:14.84,0:53:15.78,Default,,0000,0000,0000,,That's the thing.
Dialogue: 0,0:53:15.78,0:53:23.97,Default,,0000,0000,0000,,I would put minus, because I am\Nexpanding along the first row.
Dialogue: 0,0:53:23.97,0:53:28.01,Default,,0000,0000,0000,,And the second that I'm in\Nminus something minor times
Dialogue: 0,0:53:28.01,0:53:29.79,Default,,0000,0000,0000,,J. Which minor?
Dialogue: 0,0:53:29.79,0:53:34.16,Default,,0000,0000,0000,,Let me make in the lime.
Dialogue: 0,0:53:34.16,0:53:35.52,Default,,0000,0000,0000,,Lime is a nice color.
Dialogue: 0,0:53:35.52,0:53:50.99,Default,,0000,0000,0000,,And then I'll take this,\Nthis, this, and that-- dr,
Dialogue: 0,0:53:50.99,0:53:58.21,Default,,0000,0000,0000,,dx shooting [? cowboys ?]\Nthere-- minus dq, dz.
Dialogue: 0,0:53:58.21,0:54:10.07,Default,,0000,0000,0000,,And of course they wrote\Ndq, dz minus dr, dx.
Dialogue: 0,0:54:10.07,0:54:12.68,Default,,0000,0000,0000,,So I would leave it like that.
Dialogue: 0,0:54:12.68,0:54:13.68,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:54:13.68,0:54:16.16,Default,,0000,0000,0000,,You can put the\Nminus in if you want.
Dialogue: 0,0:54:16.16,0:54:19.62,Default,,0000,0000,0000,,Plus the k dot.
Dialogue: 0,0:54:19.62,0:54:22.31,Default,,0000,0000,0000,,k goes at the end.
Dialogue: 0,0:54:22.31,0:54:24.29,Default,,0000,0000,0000,,All right, now k\Ngoes at the end.
Dialogue: 0,0:54:24.29,0:54:27.28,Default,,0000,0000,0000,,
Dialogue: 0,0:54:27.28,0:54:38.54,Default,,0000,0000,0000,,And then k multiplies this\Ndeterminant-- dq, dx minus dp,
Dialogue: 0,0:54:38.54,0:54:39.04,Default,,0000,0000,0000,,dy.
Dialogue: 0,0:54:39.04,0:54:43.97,Default,,0000,0000,0000,,
Dialogue: 0,0:54:43.97,0:54:45.62,Default,,0000,0000,0000,,dq, dx minus dp, dy.
Dialogue: 0,0:54:45.62,0:54:48.43,Default,,0000,0000,0000,,
Dialogue: 0,0:54:48.43,0:54:49.34,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:54:49.34,0:54:49.84,Default,,0000,0000,0000,,No.
Dialogue: 0,0:54:49.84,0:54:51.70,Default,,0000,0000,0000,,It is not going to\Nbe hard to memorize.
Dialogue: 0,0:54:51.70,0:54:54.01,Default,,0000,0000,0000,,So then how did we do that?
Dialogue: 0,0:54:54.01,0:54:57.59,Default,,0000,0000,0000,,We set up the first row to\Nbe I, J, K, the second row
Dialogue: 0,0:54:57.59,0:54:59.87,Default,,0000,0000,0000,,to be ddx, ddy, and ddz.
Dialogue: 0,0:54:59.87,0:55:04.26,Default,,0000,0000,0000,,And then all in order the\Ncomponents of your vector value
Dialogue: 0,0:55:04.26,0:55:07.19,Default,,0000,0000,0000,,function in the exact\Norder they are with respect
Dialogue: 0,0:55:07.19,0:55:11.12,Default,,0000,0000,0000,,to the standard basis i j k.
Dialogue: 0,0:55:11.12,0:55:13.86,Default,,0000,0000,0000,,All right, now there\Nare other names
Dialogue: 0,0:55:13.86,0:55:18.49,Default,,0000,0000,0000,,and other symbols for\Ncurl of F. They use
Dialogue: 0,0:55:18.49,0:55:21.77,Default,,0000,0000,0000,,curl because it's in English.
Dialogue: 0,0:55:21.77,0:55:24.11,Default,,0000,0000,0000,,Well actually, in\NGreat Britain I
Dialogue: 0,0:55:24.11,0:55:30.29,Default,,0000,0000,0000,,saw that they used [INAUDIBLE],\Nor else they use both.
Dialogue: 0,0:55:30.29,0:55:34.45,Default,,0000,0000,0000,,In my language, in Romanian,\Nwe call it [? rotore. ?]
Dialogue: 0,0:55:34.45,0:55:38.39,Default,,0000,0000,0000,,And I saw that in French\Nit's very similar.
Dialogue: 0,0:55:38.39,0:55:40.07,Default,,0000,0000,0000,,They use the same.
Dialogue: 0,0:55:40.07,0:55:45.20,Default,,0000,0000,0000,,Now in the mechanical\Nengineering notation
Dialogue: 0,0:55:45.20,0:55:46.62,Default,,0000,0000,0000,,it's funny.
Dialogue: 0,0:55:46.62,0:55:53.64,Default,,0000,0000,0000,,They use another symbol and a\Ncross [? broad dot ?] symbol F.
Dialogue: 0,0:55:53.64,0:56:00.39,Default,,0000,0000,0000,,And by that they mean\Ncurl F. So if you
Dialogue: 0,0:56:00.39,0:56:03.69,Default,,0000,0000,0000,,talk to a professor who's\Nin mechanical engineering,
Dialogue: 0,0:56:03.69,0:56:06.26,Default,,0000,0000,0000,,or fluid mechanics,\Nor something,
Dialogue: 0,0:56:06.26,0:56:10.68,Default,,0000,0000,0000,,when they talk about curl,\Nthey will use this notation.
Dialogue: 0,0:56:10.68,0:56:13.53,Default,,0000,0000,0000,,When they use this\Nother notation,
Dialogue: 0,0:56:13.53,0:56:16.56,Default,,0000,0000,0000,,what do you think this is again?
Dialogue: 0,0:56:16.56,0:56:17.74,Default,,0000,0000,0000,,Divergence, yes.
Dialogue: 0,0:56:17.74,0:56:21.30,Default,,0000,0000,0000,,I told you last time\Nthat is divergence of F.
Dialogue: 0,0:56:21.30,0:56:26.09,Default,,0000,0000,0000,,So make the distinction\Nbetween-- again,
Dialogue: 0,0:56:26.09,0:56:28.85,Default,,0000,0000,0000,,when are you leaving?
Dialogue: 0,0:56:28.85,0:56:29.82,Default,,0000,0000,0000,,Huh?
Dialogue: 0,0:56:29.82,0:56:34.05,Default,,0000,0000,0000,,OK, so you have\Nbeen in [INAUDIBLE].
Dialogue: 0,0:56:34.05,0:56:38.62,Default,,0000,0000,0000,,And then we have\Nthis distinction
Dialogue: 0,0:56:38.62,0:56:40.27,Default,,0000,0000,0000,,we use here, like\Nfor dot product
Dialogue: 0,0:56:40.27,0:56:42.86,Default,,0000,0000,0000,,and you use here\Nas a cross product.
Dialogue: 0,0:56:42.86,0:56:46.99,Default,,0000,0000,0000,,Now you have to understand\Nthe conceptual difference is
Dialogue: 0,0:56:46.99,0:56:49.28,Default,,0000,0000,0000,,huge between these guys.
Dialogue: 0,0:56:49.28,0:56:52.21,Default,,0000,0000,0000,,This is a scalar function.
Dialogue: 0,0:56:52.21,0:56:55.76,Default,,0000,0000,0000,,This is a vector function--\Nvector, scalar-- vector,
Dialogue: 0,0:56:55.76,0:56:58.22,Default,,0000,0000,0000,,scalar, vector scalar.
Dialogue: 0,0:56:58.22,0:57:00.19,Default,,0000,0000,0000,,Because I've had to do\Nit on so [INAUDIBLE].
Dialogue: 0,0:57:00.19,0:57:01.66,Default,,0000,0000,0000,,It makes [INAUDIBLE].
Dialogue: 0,0:57:01.66,0:57:06.58,Default,,0000,0000,0000,,And I heard of\Ncolleagues complaining
Dialogue: 0,0:57:06.58,0:57:10.03,Default,,0000,0000,0000,,while grading the final\Nthat the students did not
Dialogue: 0,0:57:10.03,0:57:14.99,Default,,0000,0000,0000,,understand that this is a\Nvector, and this is a scalar.
Dialogue: 0,0:57:14.99,0:57:18.80,Default,,0000,0000,0000,,OK, a few simple exercises--\NI'm going to go ahead and do
Dialogue: 0,0:57:18.80,0:57:20.51,Default,,0000,0000,0000,,some of them.
Dialogue: 0,0:57:20.51,0:57:24.45,Default,,0000,0000,0000,,We tried to make the\Ndata on the final exam
Dialogue: 0,0:57:24.45,0:57:30.31,Default,,0000,0000,0000,,very accessible and very\Neasy to apply in problems.
Dialogue: 0,0:57:30.31,0:57:36.74,Default,,0000,0000,0000,,And one of the problems that--\Nwe'll start with example 2--
Dialogue: 0,0:57:36.74,0:57:40.70,Default,,0000,0000,0000,,would be this one.
Dialogue: 0,0:57:40.70,0:57:43.25,Default,,0000,0000,0000,,And you may think, why?
Dialogue: 0,0:57:43.25,0:57:45.74,Default,,0000,0000,0000,,Sometimes we put it in disguise.
Dialogue: 0,0:57:45.74,0:57:50.46,Default,,0000,0000,0000,,And we said assume you\Nhave a sphere-- that's
Dialogue: 0,0:57:50.46,0:58:07.59,Default,,0000,0000,0000,,the unit sphere-- of\Norigin O. And say compute.
Dialogue: 0,0:58:07.59,0:58:10.50,Default,,0000,0000,0000,,
Dialogue: 0,0:58:10.50,0:58:13.84,Default,,0000,0000,0000,,What is the equation of\Nthe unit sphere, guys?
Dialogue: 0,0:58:13.84,0:58:17.72,Default,,0000,0000,0000,,X squared plus y squared plus\Nz squared equals one, right?
Dialogue: 0,0:58:17.72,0:58:23.75,Default,,0000,0000,0000,,From [INAUDIBLE], F\Nequals normal-- external
Dialogue: 0,0:58:23.75,0:58:35.27,Default,,0000,0000,0000,,normal-- to the unit sphere\Npointing out, [? through ?]
Dialogue: 0,0:58:35.27,0:58:49.77,Default,,0000,0000,0000,,than N is the same at a\Ndifferent point as the position
Dialogue: 0,0:58:49.77,0:58:50.27,Default,,0000,0000,0000,,vector.
Dialogue: 0,0:58:50.27,0:58:55.21,Default,,0000,0000,0000,,
Dialogue: 0,0:58:55.21,0:58:57.08,Default,,0000,0000,0000,,Then compute.
Dialogue: 0,0:58:57.08,0:59:00.96,Default,,0000,0000,0000,,
Dialogue: 0,0:59:00.96,0:59:11.79,Default,,0000,0000,0000,,[? Now follow. ?] Gradient\Nof F, divergence of F,
Dialogue: 0,0:59:11.79,0:59:15.78,Default,,0000,0000,0000,,and curl of F. Now that\Nshould be a piece of cake.
Dialogue: 0,0:59:15.78,0:59:19.28,Default,,0000,0000,0000,,Now one is not [INAUDIBLE]\Nso much of a piece of cake
Dialogue: 0,0:59:19.28,0:59:23.22,Default,,0000,0000,0000,,if you don't understand what\Nthe problem wants from you.
Dialogue: 0,0:59:23.22,0:59:28.24,Default,,0000,0000,0000,,It is to actually graph\Nthe expression of this one.
Dialogue: 0,0:59:28.24,0:59:32.46,Default,,0000,0000,0000,,So you're going to\Nsay what is the normal
Dialogue: 0,0:59:32.46,0:59:34.91,Default,,0000,0000,0000,,to a function like that?
Dialogue: 0,0:59:34.91,0:59:37.36,Default,,0000,0000,0000,,First of all, we just\Ntalked today about it.
Dialogue: 0,0:59:37.36,0:59:42.22,Default,,0000,0000,0000,,If you have a function, even\Nif it's implicitly as F of x,
Dialogue: 0,0:59:42.22,0:59:52.53,Default,,0000,0000,0000,,y, z equals c, in that case N\Nis your friend from the past.
Dialogue: 0,0:59:52.53,1:00:00.83,Default,,0000,0000,0000,,If it's a unit normal,\Nunit normal to a surface
Dialogue: 0,1:00:00.83,1:00:03.80,Default,,0000,0000,0000,,happens all the\Ntime in engineering.
Dialogue: 0,1:00:03.80,1:00:06.76,Default,,0000,0000,0000,,Whether you do solid\Nmechanics or fluid mechanics,
Dialogue: 0,1:00:06.76,1:00:09.97,Default,,0000,0000,0000,,you always have to\Ncomplete these things.
Dialogue: 0,1:00:09.97,1:00:11.76,Default,,0000,0000,0000,,This is going to be hard.
Dialogue: 0,1:00:11.76,1:00:24.18,Default,,0000,0000,0000,,The gradient of F\Ndivided by the length
Dialogue: 0,1:00:24.18,1:00:27.10,Default,,0000,0000,0000,,of-- but here I have a problem.
Dialogue: 0,1:00:27.10,1:00:32.94,Default,,0000,0000,0000,,I have to put G here, because\NG will be my position vector.
Dialogue: 0,1:00:32.94,1:00:35.89,Default,,0000,0000,0000,,This is the point x,y,z.
Dialogue: 0,1:00:35.89,1:00:38.89,Default,,0000,0000,0000,,Or you prefer big R. But\NI think I prefer big G,
Dialogue: 0,1:00:38.89,1:00:41.90,Default,,0000,0000,0000,,because big R looks\Nlike a scalar radius,
Dialogue: 0,1:00:41.90,1:00:43.29,Default,,0000,0000,0000,,and I don't like that.
Dialogue: 0,1:00:43.29,1:00:47.26,Default,,0000,0000,0000,,So the position vector\Nwill be the circle middle
Dialogue: 0,1:00:47.26,1:00:52.18,Default,,0000,0000,0000,,that starts at the origin and\Nwhose N is on the surface,
Dialogue: 0,1:00:52.18,1:00:53.16,Default,,0000,0000,0000,,right?
Dialogue: 0,1:00:53.16,1:00:57.08,Default,,0000,0000,0000,,And this is the equation,\Nxy equals yj plus [? ek1. ?]
Dialogue: 0,1:00:57.08,1:01:02.43,Default,,0000,0000,0000,,Because my point x,y,z has a\Ncorresponding vector xi plus yj
Dialogue: 0,1:01:02.43,1:01:04.55,Default,,0000,0000,0000,,plus zk-- big deal.
Dialogue: 0,1:01:04.55,1:01:08.13,Default,,0000,0000,0000,,Now I'm trying to convince\Nyou that, for the unit
Dialogue: 0,1:01:08.13,1:01:12.57,Default,,0000,0000,0000,,normal for the sphere, I\Nhave the same kind of thing.
Dialogue: 0,1:01:12.57,1:01:17.33,Default,,0000,0000,0000,,So how do we compute\Nthis normally?
Dialogue: 0,1:01:17.33,1:01:23.09,Default,,0000,0000,0000,,I take the function F that\Nimplicitly defines the surface.
Dialogue: 0,1:01:23.09,1:01:27.02,Default,,0000,0000,0000,,All right, so in my case\NF is something else.
Dialogue: 0,1:01:27.02,1:01:29.97,Default,,0000,0000,0000,,What is it? x squared plus\Ny squared plus z squared.
Dialogue: 0,1:01:29.97,1:01:34.38,Default,,0000,0000,0000,,
Dialogue: 0,1:01:34.38,1:01:37.33,Default,,0000,0000,0000,,Let's compute it.
Dialogue: 0,1:01:37.33,1:01:40.44,Default,,0000,0000,0000,,N is going to be [INAUDIBLE].
Dialogue: 0,1:01:40.44,1:01:41.68,Default,,0000,0000,0000,,It's very nice.
Dialogue: 0,1:01:41.68,1:01:50.74,Default,,0000,0000,0000,,2x comma 2y comma 2z divided\Nby the square root of the sums.
Dialogue: 0,1:01:50.74,1:01:52.40,Default,,0000,0000,0000,,Do I like this?
Dialogue: 0,1:01:52.40,1:01:56.28,Default,,0000,0000,0000,,Uh, no, but I'll have to do\Nit whether I like it or not.
Dialogue: 0,1:01:56.28,1:02:02.59,Default,,0000,0000,0000,,
Dialogue: 0,1:02:02.59,1:02:05.71,Default,,0000,0000,0000,,I want to simplify\Nup and down via 2.
Dialogue: 0,1:02:05.71,1:02:07.26,Default,,0000,0000,0000,,Can I do that?
Dialogue: 0,1:02:07.26,1:02:08.08,Default,,0000,0000,0000,,Of course I can.
Dialogue: 0,1:02:08.08,1:02:15.95,Default,,0000,0000,0000,,I'm going to get x,y,z divided\Nby square root of x squared
Dialogue: 0,1:02:15.95,1:02:18.11,Default,,0000,0000,0000,,plus y squared plus z squared.
Dialogue: 0,1:02:18.11,1:02:22.05,Default,,0000,0000,0000,,
Dialogue: 0,1:02:22.05,1:02:23.04,Default,,0000,0000,0000,,And this was 1.
Dialogue: 0,1:02:23.04,1:02:28.20,Default,,0000,0000,0000,,
Dialogue: 0,1:02:28.20,1:02:29.70,Default,,0000,0000,0000,,STUDENT: Wouldn't\Nthere still be a 2
Dialogue: 0,1:02:29.70,1:02:32.41,Default,,0000,0000,0000,,there, because it's 2\Nsquared [INAUDIBLE]?
Dialogue: 0,1:02:32.41,1:02:33.89,Default,,0000,0000,0000,,PROFESSOR: No, I pulled it out.
Dialogue: 0,1:02:33.89,1:02:35.65,Default,,0000,0000,0000,,That's exactly what I said.
Dialogue: 0,1:02:35.65,1:02:37.24,Default,,0000,0000,0000,,There was a 4 inside.
Dialogue: 0,1:02:37.24,1:02:39.18,Default,,0000,0000,0000,,I pulled out with the forceps.
Dialogue: 0,1:02:39.18,1:02:41.13,Default,,0000,0000,0000,,I put it up here,\Nsquare root of 4.
Dialogue: 0,1:02:41.13,1:02:45.01,Default,,0000,0000,0000,,And I have a 2 here,\Nand that cancels out.
Dialogue: 0,1:02:45.01,1:02:48.20,Default,,0000,0000,0000,,So I got something much\Nsimpler than you guys
Dialogue: 0,1:02:48.20,1:02:50.37,Default,,0000,0000,0000,,expected at first.
Dialogue: 0,1:02:50.37,1:02:59.32,Default,,0000,0000,0000,,I got xi plus yj plus\Nzk as being the normal.
Dialogue: 0,1:02:59.32,1:03:01.27,Default,,0000,0000,0000,,Did you expect this?
Dialogue: 0,1:03:01.27,1:03:03.71,Default,,0000,0000,0000,,And you were supposed to\Nexpect that this is y,
Dialogue: 0,1:03:03.71,1:03:07.12,Default,,0000,0000,0000,,because this is the position\Nvector that has one length.
Dialogue: 0,1:03:07.12,1:03:09.99,Default,,0000,0000,0000,,The length of a\Nroot vector is 1,
Dialogue: 0,1:03:09.99,1:03:11.44,Default,,0000,0000,0000,,and the point is on the sphere.
Dialogue: 0,1:03:11.44,1:03:14.70,Default,,0000,0000,0000,,The normal will be\Nexactly the continuation.
Dialogue: 0,1:03:14.70,1:03:16.90,Default,,0000,0000,0000,,Take your root\Nvector, and continue
Dialogue: 0,1:03:16.90,1:03:20.08,Default,,0000,0000,0000,,in the same direction--\Nthis is the beauty
Dialogue: 0,1:03:20.08,1:03:23.84,Default,,0000,0000,0000,,of the normal to a surface,\Nthat it continues the radius.
Dialogue: 0,1:03:23.84,1:03:27.75,Default,,0000,0000,0000,,It continues the radius of the\Nsphere in the same direction.
Dialogue: 0,1:03:27.75,1:03:30.16,Default,,0000,0000,0000,,So you copy and paste\Nyour vector here.
Dialogue: 0,1:03:30.16,1:03:34.87,Default,,0000,0000,0000,,Position vector G will be\Nthe same as the normal N.
Dialogue: 0,1:03:34.87,1:03:38.06,Default,,0000,0000,0000,,All you do is you shift,\Nbut it's the same vector
Dialogue: 0,1:03:38.06,1:03:40.35,Default,,0000,0000,0000,,at the different point.
Dialogue: 0,1:03:40.35,1:03:43.53,Default,,0000,0000,0000,,Instead of starting at\NO, it starts at P. So
Dialogue: 0,1:03:43.53,1:03:45.55,Default,,0000,0000,0000,,[? that ?] is the same vector.
Dialogue: 0,1:03:45.55,1:03:48.58,Default,,0000,0000,0000,,So you take the radius vector\Nfrom inside the sphere--
Dialogue: 0,1:03:48.58,1:03:51.32,Default,,0000,0000,0000,,the position vector--\Nand you shift it out,
Dialogue: 0,1:03:51.32,1:03:54.79,Default,,0000,0000,0000,,and that's the\Nnormal to the sphere.
Dialogue: 0,1:03:54.79,1:03:59.29,Default,,0000,0000,0000,,So the equation is still\Nxi plus yj plus zk.
Dialogue: 0,1:03:59.29,1:03:59.79,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:03:59.79,1:04:02.22,Default,,0000,0000,0000,,STUDENT: Does it remain the\Nsame for any other functions,
Dialogue: 0,1:04:02.22,1:04:03.20,Default,,0000,0000,0000,,like [INAUDIBLE]?
Dialogue: 0,1:04:03.20,1:04:07.81,Default,,0000,0000,0000,,
Dialogue: 0,1:04:07.81,1:04:09.80,Default,,0000,0000,0000,,PROFESSOR: For the\Nunit sphere, yes it is.
Dialogue: 0,1:04:09.80,1:04:12.80,Default,,0000,0000,0000,,But for a general sphere, no.
Dialogue: 0,1:04:12.80,1:04:15.76,Default,,0000,0000,0000,,For example, what\Nif my sphere will be
Dialogue: 0,1:04:15.76,1:04:19.36,Default,,0000,0000,0000,,of center origin and radius R?
Dialogue: 0,1:04:19.36,1:04:22.61,Default,,0000,0000,0000,,
Dialogue: 0,1:04:22.61,1:04:28.75,Default,,0000,0000,0000,,And its position vector\Nv is x,y,z-- like that.
Dialogue: 0,1:04:28.75,1:04:31.64,Default,,0000,0000,0000,,
Dialogue: 0,1:04:31.64,1:04:33.38,Default,,0000,0000,0000,,[INAUDIBLE] I don't know.
Dialogue: 0,1:04:33.38,1:04:34.09,Default,,0000,0000,0000,,G, right?
Dialogue: 0,1:04:34.09,1:04:35.61,Default,,0000,0000,0000,,That's the position normal.
Dialogue: 0,1:04:35.61,1:04:40.56,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] just\Ndivide them by the R.
Dialogue: 0,1:04:40.56,1:04:44.35,Default,,0000,0000,0000,,PROFESSOR: You just\Ndivide by the R.
Dialogue: 0,1:04:44.35,1:04:47.96,Default,,0000,0000,0000,,So instead of\Nradius being big R,
Dialogue: 0,1:04:47.96,1:04:50.58,Default,,0000,0000,0000,,your unit vector\Nwill be this one.
Dialogue: 0,1:04:50.58,1:04:52.66,Default,,0000,0000,0000,,And you take this one\Nand shift it here,
Dialogue: 0,1:04:52.66,1:04:53.94,Default,,0000,0000,0000,,and that's all you have.
Dialogue: 0,1:04:53.94,1:04:55.38,Default,,0000,0000,0000,,For the sphere, it's beautiful.
Dialogue: 0,1:04:55.38,1:04:58.08,Default,,0000,0000,0000,,For any surface in general, no.
Dialogue: 0,1:04:58.08,1:04:59.98,Default,,0000,0000,0000,,Let me show you.
Dialogue: 0,1:04:59.98,1:05:04.28,Default,,0000,0000,0000,,You have a bunch of [INAUDIBLE],\Nand your position vectors
Dialogue: 0,1:05:04.28,1:05:06.66,Default,,0000,0000,0000,,look like crazies like that.
Dialogue: 0,1:05:06.66,1:05:11.99,Default,,0000,0000,0000,,And the normals could\Nbe-- they don't have
Dialogue: 0,1:05:11.99,1:05:13.24,Default,,0000,0000,0000,,to continue their position.
Dialogue: 0,1:05:13.24,1:05:23.22,Default,,0000,0000,0000,,They could be-- it depends how\Nthe tangent planes look like.
Dialogue: 0,1:05:23.22,1:05:31.20,Default,,0000,0000,0000,,And the tangent\Nplane at the point
Dialogue: 0,1:05:31.20,1:05:33.70,Default,,0000,0000,0000,,has to be perpendicular\Nto the normal.
Dialogue: 0,1:05:33.70,1:05:37.81,Default,,0000,0000,0000,,So the normal field is the\NN of [INAUDIBLE] vectors.
Dialogue: 0,1:05:37.81,1:05:41.20,Default,,0000,0000,0000,,But the little thingies\Nthat look like rectangles
Dialogue: 0,1:05:41.20,1:05:44.02,Default,,0000,0000,0000,,or whatever they are--\Nthose are the tangent planes
Dialogue: 0,1:05:44.02,1:05:45.68,Default,,0000,0000,0000,,of those points.
Dialogue: 0,1:05:45.68,1:05:48.58,Default,,0000,0000,0000,,So in general there is\Nno obvious relationship
Dialogue: 0,1:05:48.58,1:05:53.24,Default,,0000,0000,0000,,between the position and\Nthe normal for the surface.
Dialogue: 0,1:05:53.24,1:05:55.43,Default,,0000,0000,0000,,You are really lucky\Nfor this [? field. ?]
Dialogue: 0,1:05:55.43,1:06:00.17,Default,,0000,0000,0000,,And for many reasons, like\Nhow beautiful the sphere is,
Dialogue: 0,1:06:00.17,1:06:04.12,Default,,0000,0000,0000,,these functions will\Nbe easy to compute.
Dialogue: 0,1:06:04.12,1:06:06.59,Default,,0000,0000,0000,,Can you tell me what they\Nare without computing?
Dialogue: 0,1:06:06.59,1:06:11.03,Default,,0000,0000,0000,,Because that should\Nbe a piece of cake.
Dialogue: 0,1:06:11.03,1:06:14.49,Default,,0000,0000,0000,,What is the gradient field?
Dialogue: 0,1:06:14.49,1:06:19.43,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nto that one?
Dialogue: 0,1:06:19.43,1:06:22.24,Default,,0000,0000,0000,,That's the x, y, and z.
Dialogue: 0,1:06:22.24,1:06:24.89,Default,,0000,0000,0000,,PROFESSOR: For the sphere.
Dialogue: 0,1:06:24.89,1:06:26.71,Default,,0000,0000,0000,,STUDENT: 2x, 2y--
Dialogue: 0,1:06:26.71,1:06:36.80,Default,,0000,0000,0000,,PROFESSOR: Actually, let's do it\Nfor both divergence G and curl
Dialogue: 0,1:06:36.80,1:06:46.84,Default,,0000,0000,0000,,G. And you say wait, they\Nwill be-- so gradient-- no,
Dialogue: 0,1:06:46.84,1:06:47.63,Default,,0000,0000,0000,,I meant here.
Dialogue: 0,1:06:47.63,1:06:50.41,Default,,0000,0000,0000,,You don't have gradient.
Dialogue: 0,1:06:50.41,1:06:54.84,Default,,0000,0000,0000,,When F is a scalar function,\Nthen you have gradient.
Dialogue: 0,1:06:54.84,1:07:01.27,Default,,0000,0000,0000,,Then for that gradient you're\Ngoing to have divergence.
Dialogue: 0,1:07:01.27,1:07:05.33,Default,,0000,0000,0000,,And for that-- I changed\Nnotations, that's shy
Dialogue: 0,1:07:05.33,1:07:06.86,Default,,0000,0000,0000,,I have to fix it.
Dialogue: 0,1:07:06.86,1:07:10.82,Default,,0000,0000,0000,,Because F used to be that,\Nand it's not a vector anymore.
Dialogue: 0,1:07:10.82,1:07:12.80,Default,,0000,0000,0000,,So big F is not\Na vector anymore.
Dialogue: 0,1:07:12.80,1:07:16.76,Default,,0000,0000,0000,,It's a scalar function, and now\NI have to change the problem.
Dialogue: 0,1:07:16.76,1:07:18.24,Default,,0000,0000,0000,,What is the gradient there?
Dialogue: 0,1:07:18.24,1:07:19.73,Default,,0000,0000,0000,,What's divergence\Nof the gradient?
Dialogue: 0,1:07:19.73,1:07:24.18,Default,,0000,0000,0000,,[INAUDIBLE] gradient of F.\NAnd for the G that I gave you,
Dialogue: 0,1:07:24.18,1:07:27.66,Default,,0000,0000,0000,,I want the divergence\Nin the curve?
Dialogue: 0,1:07:27.66,1:07:31.32,Default,,0000,0000,0000,,So I made the problem\Nfluffier that it was before.
Dialogue: 0,1:07:31.32,1:07:34.59,Default,,0000,0000,0000,,More things to\Nconfuse for practice.
Dialogue: 0,1:07:34.59,1:07:35.48,Default,,0000,0000,0000,,What's the gradient?
Dialogue: 0,1:07:35.48,1:07:36.60,Default,,0000,0000,0000,,We did it before.
Dialogue: 0,1:07:36.60,1:07:37.10,Default,,0000,0000,0000,,2x--
Dialogue: 0,1:07:37.10,1:07:38.26,Default,,0000,0000,0000,,STUDENT: 2xi, 2--
Dialogue: 0,1:07:38.26,1:07:42.47,Default,,0000,0000,0000,,PROFESSOR: 2y, 2z-- we are at a\N[? 93 ?] point p on the sphere.
Dialogue: 0,1:07:42.47,1:07:46.20,Default,,0000,0000,0000,,It could be anywhere--\Nanywhere in space.
Dialogue: 0,1:07:46.20,1:07:49.16,Default,,0000,0000,0000,,What's the divergence\Nof this individual?
Dialogue: 0,1:07:49.16,1:07:51.64,Default,,0000,0000,0000,,So remember guys,\Nwhat I told you?
Dialogue: 0,1:07:51.64,1:07:54.61,Default,,0000,0000,0000,,First component differentiated\Nwith a straight 2x
Dialogue: 0,1:07:54.61,1:07:57.33,Default,,0000,0000,0000,,plus second component\Ndifferentiated with respect
Dialogue: 0,1:07:57.33,1:08:01.04,Default,,0000,0000,0000,,to y plus third\Ncomponent differentiated
Dialogue: 0,1:08:01.04,1:08:04.02,Default,,0000,0000,0000,,with respect to z.
Dialogue: 0,1:08:04.02,1:08:11.95,Default,,0000,0000,0000,,2 plus 2 plus 2 equals\N6-- piece of cake.
Dialogue: 0,1:08:11.95,1:08:18.02,Default,,0000,0000,0000,,And curl of the gradient\Nof F-- is that hard?
Dialogue: 0,1:08:18.02,1:08:18.86,Default,,0000,0000,0000,,[? STUDENT: Yeah. ?]
Dialogue: 0,1:08:18.86,1:08:22.31,Default,,0000,0000,0000,,PROFESSOR: No, but we have\Nto know the definition.
Dialogue: 0,1:08:22.31,1:08:26.74,Default,,0000,0000,0000,,And without looking at the\NT-shirt, how do we do that?
Dialogue: 0,1:08:26.74,1:08:30.19,Default,,0000,0000,0000,,
Dialogue: 0,1:08:30.19,1:08:36.90,Default,,0000,0000,0000,,The determinant-- I, J, K.\NOperators-- ddx, ddy, and ddz.
Dialogue: 0,1:08:36.90,1:08:40.88,Default,,0000,0000,0000,,
Dialogue: 0,1:08:40.88,1:08:44.85,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\N2x, 2y, 2z, correct?
Dialogue: 0,1:08:44.85,1:08:48.83,Default,,0000,0000,0000,,PROFESSOR: And we copy and\Npaste the three components.
Dialogue: 0,1:08:48.83,1:08:50.81,Default,,0000,0000,0000,,[INAUDIBLE] in the trash.
Dialogue: 0,1:08:50.81,1:08:53.83,Default,,0000,0000,0000,,I'll take the blue.
Dialogue: 0,1:08:53.83,1:08:57.96,Default,,0000,0000,0000,,So we put 2x, 2y, 2z.
Dialogue: 0,1:08:57.96,1:09:00.79,Default,,0000,0000,0000,,Do you think it's going\Nto be easy or hard?
Dialogue: 0,1:09:00.79,1:09:02.13,Default,,0000,0000,0000,,Do you see the answer?
Dialogue: 0,1:09:02.13,1:09:06.19,Default,,0000,0000,0000,,Some of are very sharp,\Nand you may see the answer.
Dialogue: 0,1:09:06.19,1:09:10.25,Default,,0000,0000,0000,,For example, when the\Ncowboys shoot at each other
Dialogue: 0,1:09:10.25,1:09:14.41,Default,,0000,0000,0000,,like this, dz, dy is here.
Dialogue: 0,1:09:14.41,1:09:16.34,Default,,0000,0000,0000,,dy, dz is here.
Dialogue: 0,1:09:16.34,1:09:23.97,Default,,0000,0000,0000,,So this, as a minor, is\N0-- 0I, an eye for an eye.
Dialogue: 0,1:09:23.97,1:09:25.76,Default,,0000,0000,0000,,And what else?
Dialogue: 0,1:09:25.76,1:09:29.66,Default,,0000,0000,0000,,dz, dx-- dx dz, same\Nthing, minus 0j.
Dialogue: 0,1:09:29.66,1:09:32.40,Default,,0000,0000,0000,,Is this meant to say minus 0j?
Dialogue: 0,1:09:32.40,1:09:33.21,Default,,0000,0000,0000,,Yes it is.
Dialogue: 0,1:09:33.21,1:09:37.72,Default,,0000,0000,0000,,But I did it because I want\Nyou to have the good habit
Dialogue: 0,1:09:37.72,1:09:40.19,Default,,0000,0000,0000,,of saying plus minus plus.
Dialogue: 0,1:09:40.19,1:09:42.91,Default,,0000,0000,0000,,And that's finally\Nthe same kind of thing
Dialogue: 0,1:09:42.91,1:09:46.88,Default,,0000,0000,0000,,that'll give you 0k\Nif you think that when
Dialogue: 0,1:09:46.88,1:09:49.31,Default,,0000,0000,0000,,you do partial derivative of\Ny with respect to [? f ?],
Dialogue: 0,1:09:49.31,1:09:50.77,Default,,0000,0000,0000,,you get 0.
Dialogue: 0,1:09:50.77,1:09:52.23,Default,,0000,0000,0000,,You have 0.
Dialogue: 0,1:09:52.23,1:09:57.58,Default,,0000,0000,0000,,So some student of mine asked,\Nso this is the 0 vector,
Dialogue: 0,1:09:57.58,1:10:01.82,Default,,0000,0000,0000,,how in the world do I\Nwrite a 0 vector on short?
Dialogue: 0,1:10:01.82,1:10:03.01,Default,,0000,0000,0000,,Let me show you how.
Dialogue: 0,1:10:03.01,1:10:04.18,Default,,0000,0000,0000,,You're going to laugh at me.
Dialogue: 0,1:10:04.18,1:10:08.99,Default,,0000,0000,0000,,Some people write 0 bar,\Nwhich means the 0 vector.
Dialogue: 0,1:10:08.99,1:10:11.56,Default,,0000,0000,0000,,Some other people don't\Nlike it, it's silly.
Dialogue: 0,1:10:11.56,1:10:17.64,Default,,0000,0000,0000,,Some people write O with\Ndouble like that, meaning that,
Dialogue: 0,1:10:17.64,1:10:21.50,Default,,0000,0000,0000,,hey, this is a vector element,\Nthe vector with its components
Dialogue: 0,1:10:21.50,1:10:25.84,Default,,0000,0000,0000,,of 0, 0, 0-- to distinguish\Nthat vector from the number 0,
Dialogue: 0,1:10:25.84,1:10:33.62,Default,,0000,0000,0000,,which is not in bold-- So the\Nnotations for the vector are 0.
Dialogue: 0,1:10:33.62,1:10:37.55,Default,,0000,0000,0000,,So I'm going to\Nwrite here 0, 0, 0.
Dialogue: 0,1:10:37.55,1:10:39.51,Default,,0000,0000,0000,,How about Mr. G?
Dialogue: 0,1:10:39.51,1:10:42.46,Default,,0000,0000,0000,,Mr. G will act similarly.
Dialogue: 0,1:10:42.46,1:10:47.25,Default,,0000,0000,0000,,When you do the divergence\Nit's going to be-- 1
Dialogue: 0,1:10:47.25,1:10:50.32,Default,,0000,0000,0000,,plus 1 plus 1 equals 3.
Dialogue: 0,1:10:50.32,1:10:53.52,Default,,0000,0000,0000,,
Dialogue: 0,1:10:53.52,1:10:55.98,Default,,0000,0000,0000,,You should remember this thing.
Dialogue: 0,1:10:55.98,1:10:59.18,Default,,0000,0000,0000,,We are going to do\Nthe divergence 3,
Dialogue: 0,1:10:59.18,1:11:02.87,Default,,0000,0000,0000,,and they will ask you to do a\Ntriple integral of a divergence
Dialogue: 0,1:11:02.87,1:11:04.35,Default,,0000,0000,0000,,of a vector field.
Dialogue: 0,1:11:04.35,1:11:06.64,Default,,0000,0000,0000,,And when you do\Nthat, you are going
Dialogue: 0,1:11:06.64,1:11:08.78,Default,,0000,0000,0000,,to get a triple\Ninteger of something
Dialogue: 0,1:11:08.78,1:11:13.33,Default,,0000,0000,0000,,like 3, which is a custom, which\Nwill make your life very easy.
Dialogue: 0,1:11:13.33,1:11:16.43,Default,,0000,0000,0000,,So you will very easily\Ncompute those triple integrals
Dialogue: 0,1:11:16.43,1:11:18.15,Default,,0000,0000,0000,,of constants.
Dialogue: 0,1:11:18.15,1:11:22.64,Default,,0000,0000,0000,,Curl of G, G being\Nof [? a. ?] OK?
Dialogue: 0,1:11:22.64,1:11:26.27,Default,,0000,0000,0000,,I should make the distinction\Nbetween a scalar function
Dialogue: 0,1:11:26.27,1:11:30.60,Default,,0000,0000,0000,,and a vector function by putting\Na G bar on the vector function.
Dialogue: 0,1:11:30.60,1:11:34.93,Default,,0000,0000,0000,,How about this?
Dialogue: 0,1:11:34.93,1:11:35.75,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:11:35.75,1:11:38.16,Default,,0000,0000,0000,,No, because it's\Nthe same fellow.
Dialogue: 0,1:11:38.16,1:11:40.92,Default,,0000,0000,0000,,Instead of that, I\Nhave just x, y, z.
Dialogue: 0,1:11:40.92,1:11:42.77,Default,,0000,0000,0000,,The answer will be the same.
Dialogue: 0,1:11:42.77,1:11:48.83,Default,,0000,0000,0000,,So I still want to get\N0, 0, 0-- the vector 0.
Dialogue: 0,1:11:48.83,1:11:52.67,Default,,0000,0000,0000,,So the point was that\Nwe will give you enough.
Dialogue: 0,1:11:52.67,1:11:54.85,Default,,0000,0000,0000,,You may expect them\Nto be very hard,
Dialogue: 0,1:11:54.85,1:11:57.84,Default,,0000,0000,0000,,but they are not\Ngoing to be very hard.
Dialogue: 0,1:11:57.84,1:12:02.33,Default,,0000,0000,0000,,Let's do one more like the\Nones we have in the book.
Dialogue: 0,1:12:02.33,1:12:06.32,Default,,0000,0000,0000,,What do you think\Nthis one will be?
Dialogue: 0,1:12:06.32,1:12:10.82,Default,,0000,0000,0000,,I'm making you a new\Nvector value function.
Dialogue: 0,1:12:10.82,1:12:16.30,Default,,0000,0000,0000,,That's maybe two\Nlittle exercises
Dialogue: 0,1:12:16.30,1:12:19.92,Default,,0000,0000,0000,,we can do just working\Nexercise three, four,
Dialogue: 0,1:12:19.92,1:12:21.30,Default,,0000,0000,0000,,I don't know what they are.
Dialogue: 0,1:12:21.30,1:12:23.90,Default,,0000,0000,0000,,
Dialogue: 0,1:12:23.90,1:12:28.28,Default,,0000,0000,0000,,Let me give you R\Nvector of x, y, z
Dialogue: 0,1:12:28.28,1:12:38.47,Default,,0000,0000,0000,,equals yzI plus xzj plus xyk.
Dialogue: 0,1:12:38.47,1:12:41.17,Default,,0000,0000,0000,,Compute the curl.
Dialogue: 0,1:12:41.17,1:12:47.38,Default,,0000,0000,0000,,Let me write it like engineers\Ndo just for fun-- [INAUDIBLE]
Dialogue: 0,1:12:47.38,1:12:48.37,Default,,0000,0000,0000,,cross.
Dialogue: 0,1:12:48.37,1:12:54.36,Default,,0000,0000,0000,,R is the same as\Ncurl R, which is I,
Dialogue: 0,1:12:54.36,1:12:59.84,Default,,0000,0000,0000,,J, K-- oh my god--\Nddx, ddy, ddz.
Dialogue: 0,1:12:59.84,1:13:02.50,Default,,0000,0000,0000,,
Dialogue: 0,1:13:02.50,1:13:07.82,Default,,0000,0000,0000,,Why is z-- xz-- xy.
Dialogue: 0,1:13:07.82,1:13:13.49,Default,,0000,0000,0000,,Are you saying oh, that's\Nnot so easy anymore.
Dialogue: 0,1:13:13.49,1:13:17.36,Default,,0000,0000,0000,,You-- you will see\Nthat it becomes easy,
Dialogue: 0,1:13:17.36,1:13:22.16,Default,,0000,0000,0000,,OK? i times what is the minor?
Dialogue: 0,1:13:22.16,1:13:24.88,Default,,0000,0000,0000,,This times-- x, right?
Dialogue: 0,1:13:24.88,1:13:31.88,Default,,0000,0000,0000,,Minus x plus minus j\Ntimes 1 minus what?
Dialogue: 0,1:13:31.88,1:13:34.81,Default,,0000,0000,0000,,Minor will be the red thingie.
Dialogue: 0,1:13:34.81,1:13:39.21,Default,,0000,0000,0000,,And the red thingie\Nis beautiful,
Dialogue: 0,1:13:39.21,1:13:45.43,Default,,0000,0000,0000,,because it's gonna be y\Nminus y plus k times--
Dialogue: 0,1:13:45.43,1:13:49.81,Default,,0000,0000,0000,,who do you think it's\Ngonna be? z z minus.
Dialogue: 0,1:13:49.81,1:13:53.22,Default,,0000,0000,0000,,So it's still 0.
Dialogue: 0,1:13:53.22,1:13:57.46,Default,,0000,0000,0000,,Do we expect something\Nlike that on the final?
Dialogue: 0,1:13:57.46,1:13:58.90,Default,,0000,0000,0000,,An easy computation.
Dialogue: 0,1:13:58.90,1:14:03.49,Default,,0000,0000,0000,,Somebody says, find me the\Ncurve of this function.
Dialogue: 0,1:14:03.49,1:14:04.87,Default,,0000,0000,0000,,And the functions\Nusually we give
Dialogue: 0,1:14:04.87,1:14:06.46,Default,,0000,0000,0000,,you are nice and significant.
Dialogue: 0,1:14:06.46,1:14:11.88,Default,,0000,0000,0000,,Something where the\Nresult will be pretty.
Dialogue: 0,1:14:11.88,1:14:12.38,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:14:12.38,1:14:15.84,Default,,0000,0000,0000,,
Dialogue: 0,1:14:15.84,1:14:18.95,Default,,0000,0000,0000,,Let me see what else I wanted.
Dialogue: 0,1:14:18.95,1:14:25.90,Default,,0000,0000,0000,,I'm gonna-- I have space here.
Dialogue: 0,1:14:25.90,1:14:45.74,Default,,0000,0000,0000,,So compute the curl and\NLaplace operator of f of xyz
Dialogue: 0,1:14:45.74,1:14:56.24,Default,,0000,0000,0000,,equals x squared yzi plus x y\Nsquared zj plus xy z squared k.
Dialogue: 0,1:14:56.24,1:15:02.53,Default,,0000,0000,0000,,
Dialogue: 0,1:15:02.53,1:15:04.47,Default,,0000,0000,0000,,Of divergence.
Dialogue: 0,1:15:04.47,1:15:05.93,Default,,0000,0000,0000,,Sorry, guys.
Dialogue: 0,1:15:05.93,1:15:09.33,Default,,0000,0000,0000,,This is not a-- it's\Nnot a scalar function.
Dialogue: 0,1:15:09.33,1:15:11.24,Default,,0000,0000,0000,,I want the divergence\Nand the curl.
Dialogue: 0,1:15:11.24,1:15:13.15,Default,,0000,0000,0000,,The curl will be a vector.
Dialogue: 0,1:15:13.15,1:15:15.55,Default,,0000,0000,0000,,The divergence will\Nbe a scalar function.
Dialogue: 0,1:15:15.55,1:15:18.19,Default,,0000,0000,0000,,Later on I'll give you a\Nnice function where you can
Dialogue: 0,1:15:18.19,1:15:20.30,Default,,0000,0000,0000,,compute the Laplace operator.
Dialogue: 0,1:15:20.30,1:15:23.80,Default,,0000,0000,0000,,That's gonna have to\Nbe a scalar function.
Dialogue: 0,1:15:23.80,1:15:27.30,Default,,0000,0000,0000,,And although the\NLaplace operator
Dialogue: 0,1:15:27.30,1:15:29.79,Default,,0000,0000,0000,,can be generalized\Nto vector functions,
Dialogue: 0,1:15:29.79,1:15:31.78,Default,,0000,0000,0000,,and I'll tell you later\Nhow-- what that is.
Dialogue: 0,1:15:31.78,1:15:32.77,Default,,0000,0000,0000,,It's very easy.
Dialogue: 0,1:15:32.77,1:15:37.62,Default,,0000,0000,0000,,It's practically the Laplace\Noperators in every direction.
Dialogue: 0,1:15:37.62,1:15:38.45,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:15:38.45,1:15:42.28,Default,,0000,0000,0000,,So let's see the curl.
Dialogue: 0,1:15:42.28,1:15:48.50,Default,,0000,0000,0000,,
Dialogue: 0,1:15:48.50,1:15:56.39,Default,,0000,0000,0000,,i j k, d dx, d dy, d dz.
Dialogue: 0,1:15:56.39,1:15:58.74,Default,,0000,0000,0000,,Today I'm gonna cook\Nup the homework.
Dialogue: 0,1:15:58.74,1:16:01.12,Default,,0000,0000,0000,,And with all the practice\Nthat we are doing now,
Dialogue: 0,1:16:01.12,1:16:04.88,Default,,0000,0000,0000,,you should have absolutely\Nno problem doing the homework
Dialogue: 0,1:16:04.88,1:16:06.84,Default,,0000,0000,0000,,for the first two sections.
Dialogue: 0,1:16:06.84,1:16:10.33,Default,,0000,0000,0000,,At least for the section--\Ntoday's section, 13.1.
Dialogue: 0,1:16:10.33,1:16:15.34,Default,,0000,0000,0000,,x squared yz, x y\Nsquared z, xy z squared.
Dialogue: 0,1:16:15.34,1:16:17.39,Default,,0000,0000,0000,,You see there is some\Nsort of symmetry.
Dialogue: 0,1:16:17.39,1:16:18.80,Default,,0000,0000,0000,,I'm playing a game here.
Dialogue: 0,1:16:18.80,1:16:23.94,Default,,0000,0000,0000,,
Dialogue: 0,1:16:23.94,1:16:29.84,Default,,0000,0000,0000,,So I have i.
Dialogue: 0,1:16:29.84,1:16:33.93,Default,,0000,0000,0000,,I want you to tell me\N[INAUDIBLE], see now,
Dialogue: 0,1:16:33.93,1:16:35.62,Default,,0000,0000,0000,,I don't work much in groups.
Dialogue: 0,1:16:35.62,1:16:38.12,Default,,0000,0000,0000,,I don't make you\Nwork in groups, but I
Dialogue: 0,1:16:38.12,1:16:40.69,Default,,0000,0000,0000,,want you to answer my question.
Dialogue: 0,1:16:40.69,1:16:46.92,Default,,0000,0000,0000,,So what is going to be this\Nminor that I-- the first thing
Dialogue: 0,1:16:46.92,1:16:47.42,Default,,0000,0000,0000,,is gonna be?
Dialogue: 0,1:16:47.42,1:16:48.91,Default,,0000,0000,0000,,STUDENT: x z squared.
Dialogue: 0,1:16:48.91,1:16:49.90,Default,,0000,0000,0000,,PROFESSOR: Very good.
Dialogue: 0,1:16:49.90,1:16:52.38,Default,,0000,0000,0000,,STUDENT: Minus x y squared.
Dialogue: 0,1:16:52.38,1:16:57.32,Default,,0000,0000,0000,,
Dialogue: 0,1:16:57.32,1:16:58.32,Default,,0000,0000,0000,,PROFESSOR: Minus j.
Dialogue: 0,1:16:58.32,1:17:00.30,Default,,0000,0000,0000,,Potential [? plus or ?] minus.
Dialogue: 0,1:17:00.30,1:17:01.78,Default,,0000,0000,0000,,OK, what is the next guy?
Dialogue: 0,1:17:01.78,1:17:03.07,Default,,0000,0000,0000,,STUDENT: y z squared.
Dialogue: 0,1:17:03.07,1:17:07.97,Default,,0000,0000,0000,,PROFESSOR: y z\Nsquared, thank you.
Dialogue: 0,1:17:07.97,1:17:08.95,Default,,0000,0000,0000,,STUDENT: x squared.
Dialogue: 0,1:17:08.95,1:17:12.87,Default,,0000,0000,0000,,PROFESSOR: x squared, right?
Dialogue: 0,1:17:12.87,1:17:16.79,Default,,0000,0000,0000,,Plus k times--
Dialogue: 0,1:17:16.79,1:17:20.71,Default,,0000,0000,0000,,STUDENT: y squared z.
Dialogue: 0,1:17:20.71,1:17:23.16,Default,,0000,0000,0000,,PROFESSOR: So, you\Nsee what I'm doing?
Dialogue: 0,1:17:23.16,1:17:27.50,Default,,0000,0000,0000,,I'm doing this from\Nrespect to x. y squared z.
Dialogue: 0,1:17:27.50,1:17:28.97,Default,,0000,0000,0000,,You said it, right.
Dialogue: 0,1:17:28.97,1:17:32.88,Default,,0000,0000,0000,,Minus this guy, x squared z.
Dialogue: 0,1:17:32.88,1:17:37.63,Default,,0000,0000,0000,,Can I write it more-- I don't\Nreally like the way I wrote it.
Dialogue: 0,1:17:37.63,1:17:38.75,Default,,0000,0000,0000,,But I'll write like that.
Dialogue: 0,1:17:38.75,1:17:45.40,Default,,0000,0000,0000,,How about x times z squared\Nminus y squared i minus j.
Dialogue: 0,1:17:45.40,1:17:47.39,Default,,0000,0000,0000,,Or maybe better plus j.
Dialogue: 0,1:17:47.39,1:17:49.38,Default,,0000,0000,0000,,I'll change this up.
Dialogue: 0,1:17:49.38,1:17:50.87,Default,,0000,0000,0000,,Plus j at the end.
Dialogue: 0,1:17:50.87,1:17:55.34,Default,,0000,0000,0000,,Because it's the\Nvector. y times x
Dialogue: 0,1:17:55.34,1:18:02.80,Default,,0000,0000,0000,,squared minus z squared\Nj plus-- who gets out?
Dialogue: 0,1:18:02.80,1:18:09.26,Default,,0000,0000,0000,,z-- z times y squared\Nminus x squared k.
Dialogue: 0,1:18:09.26,1:18:12.74,Default,,0000,0000,0000,,
Dialogue: 0,1:18:12.74,1:18:13.73,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:18:13.73,1:18:16.71,Default,,0000,0000,0000,,Good There is some\Nsymmetry in there.
Dialogue: 0,1:18:16.71,1:18:19.70,Default,,0000,0000,0000,,The break in the symmetry\Nis in the middle.
Dialogue: 0,1:18:19.70,1:18:23.17,Default,,0000,0000,0000,,Because, as you\Nsee, x is separate,
Dialogue: 0,1:18:23.17,1:18:26.62,Default,,0000,0000,0000,,and then z is followed\Nby y, and then
Dialogue: 0,1:18:26.62,1:18:31.77,Default,,0000,0000,0000,,x squared-- x is followed\Nby z and y is followed by x.
Dialogue: 0,1:18:31.77,1:18:37.10,Default,,0000,0000,0000,,So I have some-- some\Nsymmetry of some sort.
Dialogue: 0,1:18:37.10,1:18:40.08,Default,,0000,0000,0000,,
Dialogue: 0,1:18:40.08,1:18:42.56,Default,,0000,0000,0000,,What else did I want?
Dialogue: 0,1:18:42.56,1:18:44.06,Default,,0000,0000,0000,,Divergence operator.
Dialogue: 0,1:18:44.06,1:18:47.54,Default,,0000,0000,0000,,And that will be the\Nlast example of the kind.
Dialogue: 0,1:18:47.54,1:18:50.02,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:18:50.02,1:18:51.96,Default,,0000,0000,0000,,How do you write the divergence?
Dialogue: 0,1:18:51.96,1:18:52.50,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,1:18:52.50,1:18:53.00,Default,,0000,0000,0000,,Very easy?
Dialogue: 0,1:18:53.00,1:18:55.51,Default,,0000,0000,0000,,
Dialogue: 0,1:18:55.51,1:18:58.77,Default,,0000,0000,0000,,I'm going go ask you to simplify\Nbecause I don't like it,
Dialogue: 0,1:18:58.77,1:19:01.23,Default,,0000,0000,0000,,like, as a sum.
Dialogue: 0,1:19:01.23,1:19:02.21,Default,,0000,0000,0000,,2xyz--
Dialogue: 0,1:19:02.21,1:19:10.56,Default,,0000,0000,0000,,STUDENT: 2xyz plus\N2xyz plus 2xyz.
Dialogue: 0,1:19:10.56,1:19:13.41,Default,,0000,0000,0000,,PROFESSOR: And now you see\Nwhy I don't like it as a sum.
Dialogue: 0,1:19:13.41,1:19:17.39,Default,,0000,0000,0000,,Because it's 6xyz and it's\Nvery pretty like that.
Dialogue: 0,1:19:17.39,1:19:20.12,Default,,0000,0000,0000,,I'd like you-- on\Nthe exam, I'd like
Dialogue: 0,1:19:20.12,1:19:24.34,Default,,0000,0000,0000,,you to take the function\Nand box the answer,
Dialogue: 0,1:19:24.34,1:19:28.82,Default,,0000,0000,0000,,and that's all I want you to do.
Dialogue: 0,1:19:28.82,1:19:29.32,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:19:29.32,1:19:31.30,Default,,0000,0000,0000,,I'm gonna go ahead and erase.
Dialogue: 0,1:19:31.30,1:19:49.69,Default,,0000,0000,0000,,
Dialogue: 0,1:19:49.69,1:19:53.17,Default,,0000,0000,0000,,I'm going to move\Non to 13.2, but I'd
Dialogue: 0,1:19:53.17,1:19:59.85,Default,,0000,0000,0000,,like to review some physics\Na little bit with you
Dialogue: 0,1:19:59.85,1:20:02.80,Default,,0000,0000,0000,,and see what you\Nremember from physics.
Dialogue: 0,1:20:02.80,1:20:21.50,Default,,0000,0000,0000,,
Dialogue: 0,1:20:21.50,1:20:23.46,Default,,0000,0000,0000,,It's a little bit messy.
Dialogue: 0,1:20:23.46,1:20:27.89,Default,,0000,0000,0000,,I'll use this instead because\NI like the board to be clean.
Dialogue: 0,1:20:27.89,1:20:34.26,Default,,0000,0000,0000,,If I were to ask you to\Nremember work in physics,
Dialogue: 0,1:20:34.26,1:20:41.71,Default,,0000,0000,0000,,I would say-- I'm changing a\Nlittle bit the order in 13.2.
Dialogue: 0,1:20:41.71,1:20:46.69,Default,,0000,0000,0000,,I'd like you to go\Nback in time and see
Dialogue: 0,1:20:46.69,1:20:48.68,Default,,0000,0000,0000,,what work was in physics class.
Dialogue: 0,1:20:48.68,1:20:53.66,Default,,0000,0000,0000,,
Dialogue: 0,1:20:53.66,1:20:56.15,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] force dx.
Dialogue: 0,1:20:56.15,1:20:59.67,Default,,0000,0000,0000,,
Dialogue: 0,1:20:59.67,1:21:02.09,Default,,0000,0000,0000,,PROFESSOR: What if you\Ndidn't know any calculus?
Dialogue: 0,1:21:02.09,1:21:04.32,Default,,0000,0000,0000,,Let's go a long time.
Dialogue: 0,1:21:04.32,1:21:10.78,Default,,0000,0000,0000,,The section is\N13.2 preliminaries.
Dialogue: 0,1:21:10.78,1:21:13.47,Default,,0000,0000,0000,,STUDENT: Force\Nmultiplied distance.
Dialogue: 0,1:21:13.47,1:21:15.42,Default,,0000,0000,0000,,PROFESSOR: Very good.
Dialogue: 0,1:21:15.42,1:21:17.38,Default,,0000,0000,0000,,Preliminary work.
Dialogue: 0,1:21:17.38,1:21:25.50,Default,,0000,0000,0000,,[INAUDIBLE] The notion of work\Nfrom physics-- hey, come on.
Dialogue: 0,1:21:25.50,1:21:34.96,Default,,0000,0000,0000,,
Dialogue: 0,1:21:34.96,1:21:39.13,Default,,0000,0000,0000,,Physics, or engineering,\Nmechanics, whatever you study,
Dialogue: 0,1:21:39.13,1:21:39.80,Default,,0000,0000,0000,,work.
Dialogue: 0,1:21:39.80,1:21:43.92,Default,,0000,0000,0000,,Imagine that you're\Ntaking a-- this
Dialogue: 0,1:21:43.92,1:21:47.39,Default,,0000,0000,0000,,is your body that you're playing\Nwith-- not your own body,
Dialogue: 0,1:21:47.39,1:21:50.47,Default,,0000,0000,0000,,but the body you are\Nacting on in physics.
Dialogue: 0,1:21:50.47,1:21:58.48,Default,,0000,0000,0000,,And you are dragging this\Nobject from a place A
Dialogue: 0,1:21:58.48,1:22:00.89,Default,,0000,0000,0000,,to a place B, another position.
Dialogue: 0,1:22:00.89,1:22:04.26,Default,,0000,0000,0000,,A B is the distance.
Dialogue: 0,1:22:04.26,1:22:09.19,Default,,0000,0000,0000,,And the force is parallel\Nto the trajectory.
Dialogue: 0,1:22:09.19,1:22:10.40,Default,,0000,0000,0000,,This is very important.
Dialogue: 0,1:22:10.40,1:22:12.35,Default,,0000,0000,0000,,This is a simpler case.
Dialogue: 0,1:22:12.35,1:22:14.22,Default,,0000,0000,0000,,In general, it's not so simple.
Dialogue: 0,1:22:14.22,1:22:17.84,Default,,0000,0000,0000,,So this force is acting,\Nand it's a constant force.
Dialogue: 0,1:22:17.84,1:22:21.91,Default,,0000,0000,0000,,And you pull the object\Nfrom one place to another.
Dialogue: 0,1:22:21.91,1:22:24.04,Default,,0000,0000,0000,,That's case one.
Dialogue: 0,1:22:24.04,1:22:28.01,Default,,0000,0000,0000,,In case two, life is harder.
Dialogue: 0,1:22:28.01,1:22:35.82,Default,,0000,0000,0000,,You actually pull\Nthe poor object
Dialogue: 0,1:22:35.82,1:22:38.27,Default,,0000,0000,0000,,with the force in\Nthis direction.
Dialogue: 0,1:22:38.27,1:22:40.68,Default,,0000,0000,0000,,Actually, most of\Nus do that, right?
Dialogue: 0,1:22:40.68,1:22:45.09,Default,,0000,0000,0000,,If I were to have a gliding\Nobject on the surface,
Dialogue: 0,1:22:45.09,1:22:48.50,Default,,0000,0000,0000,,I would actually act on\Nthat object in the direction
Dialogue: 0,1:22:48.50,1:22:51.10,Default,,0000,0000,0000,,of my arm by pulling it.
Dialogue: 0,1:22:51.10,1:22:56.52,Default,,0000,0000,0000,,So when I displace this body\Nfrom point a to point b,
Dialogue: 0,1:22:56.52,1:23:01.14,Default,,0000,0000,0000,,I still travel the\Ndistance d, but-- so d
Dialogue: 0,1:23:01.14,1:23:04.23,Default,,0000,0000,0000,,is a displacement vector\Nthat can be written like--
Dialogue: 0,1:23:04.23,1:23:07.20,Default,,0000,0000,0000,,or it can be drawn like that.
Dialogue: 0,1:23:07.20,1:23:09.55,Default,,0000,0000,0000,,I have to be smart\Nin both cases,
Dialogue: 0,1:23:09.55,1:23:11.32,Default,,0000,0000,0000,,figure out what\NI want from life,
Dialogue: 0,1:23:11.32,1:23:13.53,Default,,0000,0000,0000,,because it's not so clear.
Dialogue: 0,1:23:13.53,1:23:15.61,Default,,0000,0000,0000,,When they taught me I\Nthink the first time
Dialogue: 0,1:23:15.61,1:23:18.46,Default,,0000,0000,0000,,I was in-- oh my\NGod-- eighth grade,
Dialogue: 0,1:23:18.46,1:23:20.77,Default,,0000,0000,0000,,and they-- that was\Na long time ago.
Dialogue: 0,1:23:20.77,1:23:26.64,Default,,0000,0000,0000,,In this case, w is\Ngonna be a scalar
Dialogue: 0,1:23:26.64,1:23:32.26,Default,,0000,0000,0000,,and I'm gonna have the magnitude\Nof the force F. F is a vector,
Dialogue: 0,1:23:32.26,1:23:36.90,Default,,0000,0000,0000,,but to indicate it in Newtons\Nor whatever I measure it in,
Dialogue: 0,1:23:36.90,1:23:40.85,Default,,0000,0000,0000,,it's gonna be in magnitude.
Dialogue: 0,1:23:40.85,1:23:44.55,Default,,0000,0000,0000,,Times the little d,\Nbut instead of little d
Dialogue: 0,1:23:44.55,1:23:48.18,Default,,0000,0000,0000,,I should be a little\Nsmarter and say,
Dialogue: 0,1:23:48.18,1:23:51.86,Default,,0000,0000,0000,,Magdalena, this is the magnitude\Nof the vector A B, which
Dialogue: 0,1:23:51.86,1:23:52.90,Default,,0000,0000,0000,,is a displacement vector.
Dialogue: 0,1:23:52.90,1:23:55.52,Default,,0000,0000,0000,,
Dialogue: 0,1:23:55.52,1:23:58.59,Default,,0000,0000,0000,,So this is called work.
Dialogue: 0,1:23:58.59,1:24:03.15,Default,,0000,0000,0000,,And if the force is 10\NNewtons, and the distance
Dialogue: 0,1:24:03.15,1:24:07.48,Default,,0000,0000,0000,,is 10 meters, because we\Nwant to go international.
Dialogue: 0,1:24:07.48,1:24:10.97,Default,,0000,0000,0000,,We want to be global, right,\Nat Texas Tech, so good.
Dialogue: 0,1:24:10.97,1:24:12.18,Default,,0000,0000,0000,,So we have 100 Newton-meters.
Dialogue: 0,1:24:12.18,1:24:17.45,Default,,0000,0000,0000,,
Dialogue: 0,1:24:17.45,1:24:23.03,Default,,0000,0000,0000,,Now you can measure-- well,\Nyou can have another example.
Dialogue: 0,1:24:23.03,1:24:27.91,Default,,0000,0000,0000,,I'm thinking gravity and then\Nyou can say it in pounds,
Dialogue: 0,1:24:27.91,1:24:29.87,Default,,0000,0000,0000,,and that measures force.
Dialogue: 0,1:24:29.87,1:24:32.31,Default,,0000,0000,0000,,And you have other units\Nthat are not international.
Dialogue: 0,1:24:32.31,1:24:34.26,Default,,0000,0000,0000,,I'm not gonna mess up.
Dialogue: 0,1:24:34.26,1:24:38.31,Default,,0000,0000,0000,,When you have the work\Nin this case, though,
Dialogue: 0,1:24:38.31,1:24:41.43,Default,,0000,0000,0000,,it's more complicated.
Dialogue: 0,1:24:41.43,1:24:44.10,Default,,0000,0000,0000,,And I'm not gonna be mad\Nat you. [INAUDIBLE] is
Dialogue: 0,1:24:44.10,1:24:45.80,Default,,0000,0000,0000,,trying to tell me what it is.
Dialogue: 0,1:24:45.80,1:24:48.94,Default,,0000,0000,0000,,I'm not going to be mad at\Nthe people who don't know
Dialogue: 0,1:24:48.94,1:24:51.34,Default,,0000,0000,0000,,what the work is in this case.
Dialogue: 0,1:24:51.34,1:24:56.64,Default,,0000,0000,0000,,Although, I was looking\Nat-- I am the person
Dialogue: 0,1:24:56.64,1:25:01.56,Default,,0000,0000,0000,,who has run a\Ncommittee to oversee
Dialogue: 0,1:25:01.56,1:25:04.26,Default,,0000,0000,0000,,the finals for different\Nmath classes, all the math
Dialogue: 0,1:25:04.26,1:25:05.52,Default,,0000,0000,0000,,classes we offer here.
Dialogue: 0,1:25:05.52,1:25:09.95,Default,,0000,0000,0000,,And every semester I see the\N[? streak ?] pre-calculus,
Dialogue: 0,1:25:09.95,1:25:11.98,Default,,0000,0000,0000,,calc 1, calc 2, calc 3.
Dialogue: 0,1:25:11.98,1:25:13.35,Default,,0000,0000,0000,,In trig and\Npre-calculus, they'll
Dialogue: 0,1:25:13.35,1:25:15.88,Default,,0000,0000,0000,,always have a work there.
Dialogue: 0,1:25:15.88,1:25:19.93,Default,,0000,0000,0000,,And I was wondering how many\Nof you took pre-calculus,
Dialogue: 0,1:25:19.93,1:25:22.52,Default,,0000,0000,0000,,and how many of you\Nremember that you
Dialogue: 0,1:25:22.52,1:25:26.16,Default,,0000,0000,0000,,studied this in pre-calculus.
Dialogue: 0,1:25:26.16,1:25:27.82,Default,,0000,0000,0000,,It's a little bit awkward.
Dialogue: 0,1:25:27.82,1:25:31.40,Default,,0000,0000,0000,,I'm thinking, how do they do\Nit, but I gave you the formula.
Dialogue: 0,1:25:31.40,1:25:35.77,Default,,0000,0000,0000,,And they say the force in\Nitself as a vector dot product
Dialogue: 0,1:25:35.77,1:25:38.68,Default,,0000,0000,0000,,the displacement vector.
Dialogue: 0,1:25:38.68,1:25:41.67,Default,,0000,0000,0000,,So they are both\Nforces in dot product.
Dialogue: 0,1:25:41.67,1:25:45.48,Default,,0000,0000,0000,,And I was surprised to see\Nthat they gave you [INAUDIBLE]
Dialogue: 0,1:25:45.48,1:25:51.42,Default,,0000,0000,0000,,If I were to express it,\Nhow would I express it?
Dialogue: 0,1:25:51.42,1:25:55.34,Default,,0000,0000,0000,,I'll say the magnitude\Nof F, of course,
Dialogue: 0,1:25:55.34,1:25:57.79,Default,,0000,0000,0000,,in Newtons, whatever it\Nis, times the magnitude
Dialogue: 0,1:25:57.79,1:26:00.24,Default,,0000,0000,0000,,of the displacement vector--
Dialogue: 0,1:26:00.24,1:26:01.72,Default,,0000,0000,0000,,STUDENT: Multiply those cosines.
Dialogue: 0,1:26:01.72,1:26:04.58,Default,,0000,0000,0000,,PROFESSOR: Cosine of\Nthe angle between.
Dialogue: 0,1:26:04.58,1:26:06.58,Default,,0000,0000,0000,,And I'm too lazy, I\Ndon't know, theta.
Dialogue: 0,1:26:06.58,1:26:08.98,Default,,0000,0000,0000,,Let's call it angle theta.
Dialogue: 0,1:26:08.98,1:26:16.20,Default,,0000,0000,0000,,Because I don't want to\Ninclude that in locations, OK?
Dialogue: 0,1:26:16.20,1:26:18.46,Default,,0000,0000,0000,,It really doesn't matter\Nin which direction
Dialogue: 0,1:26:18.46,1:26:23.34,Default,,0000,0000,0000,,I'm going, because cosine theta,\Nthank God, is an even function.
Dialogue: 0,1:26:23.34,1:26:25.79,Default,,0000,0000,0000,,It's equal to cosine\Nof minus theta.
Dialogue: 0,1:26:25.79,1:26:29.71,Default,,0000,0000,0000,,So whether I go this way or\Nthat way, it's the same cosine.
Dialogue: 0,1:26:29.71,1:26:30.69,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:26:30.69,1:26:33.63,Default,,0000,0000,0000,,So the cosine of the\Nangle between the two.
Dialogue: 0,1:26:33.63,1:26:37.06,Default,,0000,0000,0000,,It's very easy when you\Ndon't need calculus.
Dialogue: 0,1:26:37.06,1:26:41.96,Default,,0000,0000,0000,,But when you use calculus,\Nbecause your trajectory is
Dialogue: 0,1:26:41.96,1:26:45.88,Default,,0000,0000,0000,,no longer a line, life is\Nbecoming more complicated.
Dialogue: 0,1:26:45.88,1:26:49.73,Default,,0000,0000,0000,,So we have to come up\Nwith a different formula,
Dialogue: 0,1:26:49.73,1:26:53.58,Default,,0000,0000,0000,,with a different notion of work.
Dialogue: 0,1:26:53.58,1:26:56.51,Default,,0000,0000,0000,,I'm gonna erase-- are\Nyou guys done with that?
Dialogue: 0,1:26:56.51,1:26:58.20,Default,,0000,0000,0000,,Is it visible?
Dialogue: 0,1:26:58.20,1:26:59.19,Default,,0000,0000,0000,,You're done.
Dialogue: 0,1:26:59.19,1:27:08.12,Default,,0000,0000,0000,,
Dialogue: 0,1:27:08.12,1:27:08.62,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:27:08.62,1:27:14.10,Default,,0000,0000,0000,,
Dialogue: 0,1:27:14.10,1:27:18.68,Default,,0000,0000,0000,,So again, life is not so\Neasy in reality anymore.
Dialogue: 0,1:27:18.68,1:27:28.50,Default,,0000,0000,0000,,I have a particle in physics-- a\Nphoton enters a four-star hotel
Dialogue: 0,1:27:28.50,1:27:32.84,Default,,0000,0000,0000,,and says-- talks to the\Nbellboy, and the bellboy,
Dialogue: 0,1:27:32.84,1:27:34.78,Default,,0000,0000,0000,,can I help you\Nwith your luggage.
Dialogue: 0,1:27:34.78,1:27:36.73,Default,,0000,0000,0000,,No, I'm traveling light.
Dialogue: 0,1:27:36.73,1:27:37.69,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,1:27:37.69,1:27:41.39,Default,,0000,0000,0000,,So the particle, the\Nphoton-- whatever.
Dialogue: 0,1:27:41.39,1:27:45.19,Default,,0000,0000,0000,,A particle is moving-- is\Nmoving on a trajectory.
Dialogue: 0,1:27:45.19,1:27:47.79,Default,,0000,0000,0000,,Suppose that\Ntrajectory is planar,
Dialogue: 0,1:27:47.79,1:27:50.69,Default,,0000,0000,0000,,just to make your\Nlife easier at first.
Dialogue: 0,1:27:50.69,1:27:53.43,Default,,0000,0000,0000,,It's in the plane x y.
Dialogue: 0,1:27:53.43,1:27:56.84,Default,,0000,0000,0000,,And this is the little\Nparticle that's moving.
Dialogue: 0,1:27:56.84,1:28:04.16,Default,,0000,0000,0000,,And this is R. And\Nthat is x i plus yj.
Dialogue: 0,1:28:04.16,1:28:08.55,Default,,0000,0000,0000,,
Dialogue: 0,1:28:08.55,1:28:11.48,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:28:11.48,1:28:13.90,Default,,0000,0000,0000,,And this is the point x, y.
Dialogue: 0,1:28:13.90,1:28:16.35,Default,,0000,0000,0000,,And that's the position,\Nthe current position
Dialogue: 0,1:28:16.35,1:28:18.32,Default,,0000,0000,0000,,of my particle, right now.
Dialogue: 0,1:28:18.32,1:28:20.65,Default,,0000,0000,0000,,Not in the past,\Nnot in the future.
Dialogue: 0,1:28:20.65,1:28:23.70,Default,,0000,0000,0000,,My particle is moving,\Nand this is now.
Dialogue: 0,1:28:23.70,1:28:26.13,Default,,0000,0000,0000,,Suppose time doesn't even exist.
Dialogue: 0,1:28:26.13,1:28:30.22,Default,,0000,0000,0000,,We think of the movies\Nthat we saw lately,
Dialogue: 0,1:28:30.22,1:28:33.67,Default,,0000,0000,0000,,in The Theory of Everything.
Dialogue: 0,1:28:33.67,1:28:38.74,Default,,0000,0000,0000,,So then, they say OK, we only\Ncare about now. x, y is now
Dialogue: 0,1:28:38.74,1:28:43.74,Default,,0000,0000,0000,,and that is the current\Nposition vector.
Dialogue: 0,1:28:43.74,1:28:49.60,Default,,0000,0000,0000,,Well, what would be\Nthe work between now--
Dialogue: 0,1:28:49.60,1:28:55.48,Default,,0000,0000,0000,,whatever now-- and the next,\Nlet's say, this is gonna be x1,
Dialogue: 0,1:28:55.48,1:28:57.25,Default,,0000,0000,0000,,y1.
Dialogue: 0,1:28:57.25,1:28:59.22,Default,,0000,0000,0000,,And this is x0, y0.
Dialogue: 0,1:28:59.22,1:29:04.63,Default,,0000,0000,0000,,
Dialogue: 0,1:29:04.63,1:29:14.00,Default,,0000,0000,0000,,That's the general formula,\Nwill be x i plus yj.
Dialogue: 0,1:29:14.00,1:29:19.15,Default,,0000,0000,0000,,So I actually cannot\Nforget about time.
Dialogue: 0,1:29:19.15,1:29:21.07,Default,,0000,0000,0000,,Not as much as I want.
Dialogue: 0,1:29:21.07,1:29:27.50,Default,,0000,0000,0000,,So x and y-- x and y are\Nboth changing in time.
Dialogue: 0,1:29:27.50,1:29:30.96,Default,,0000,0000,0000,,We're gonna have x equals\Nx sub t, y equals y sub t.
Dialogue: 0,1:29:30.96,1:29:34.92,Default,,0000,0000,0000,,Do you guys remember what we\Ncall that kind of equation
Dialogue: 0,1:29:34.92,1:29:39.35,Default,,0000,0000,0000,,for a curve from here to here?
Dialogue: 0,1:29:39.35,1:29:40.28,Default,,0000,0000,0000,,Para--
Dialogue: 0,1:29:40.28,1:29:41.32,Default,,0000,0000,0000,,STUDENT: Parametrization.
Dialogue: 0,1:29:41.32,1:29:45.74,Default,,0000,0000,0000,,PROFESSOR: Parametrization,\Nor parametric equations.
Dialogue: 0,1:29:45.74,1:29:50.15,Default,,0000,0000,0000,,Parametric equations.
Dialogue: 0,1:29:50.15,1:29:57.03,Default,,0000,0000,0000,,
Dialogue: 0,1:29:57.03,1:29:59.02,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:29:59.02,1:30:04.33,Default,,0000,0000,0000,,So I have may the\Nforce be with you.
Dialogue: 0,1:30:04.33,1:30:06.78,Default,,0000,0000,0000,,I have a force.
Dialogue: 0,1:30:06.78,1:30:10.89,Default,,0000,0000,0000,,I have a force, and I have\Nsome sort of displacement.
Dialogue: 0,1:30:10.89,1:30:15.33,Default,,0000,0000,0000,,But I cannot express that\Ndisplacement linearly anymore.
Dialogue: 0,1:30:15.33,1:30:18.63,Default,,0000,0000,0000,,I'm moving along a\N[INAUDIBLE] of a curve.
Dialogue: 0,1:30:18.63,1:30:22.57,Default,,0000,0000,0000,,So I have to think differently.
Dialogue: 0,1:30:22.57,1:30:27.81,Default,,0000,0000,0000,,And the work will be defined,\Nwhether you like it or not,
Dialogue: 0,1:30:27.81,1:30:35.76,Default,,0000,0000,0000,,as F vector field, dot\Nproduct with dR over c.
Dialogue: 0,1:30:35.76,1:30:40.08,Default,,0000,0000,0000,,And you say, what in\Nthe world is that?
Dialogue: 0,1:30:40.08,1:30:43.54,Default,,0000,0000,0000,,What would c be?
Dialogue: 0,1:30:43.54,1:30:46.43,Default,,0000,0000,0000,,How do I integrate along a path?
Dialogue: 0,1:30:46.43,1:30:50.34,Default,,0000,0000,0000,,And I will tell you in a\Nsecond what we mean by that.
Dialogue: 0,1:30:50.34,1:30:53.23,Default,,0000,0000,0000,,
Dialogue: 0,1:30:53.23,1:30:58.60,Default,,0000,0000,0000,,Meaning that-- this is by\Ndefinition if you want.
Dialogue: 0,1:30:58.60,1:31:03.49,Default,,0000,0000,0000,,This is like an\Napplication of calculus 1.
Dialogue: 0,1:31:03.49,1:31:07.70,Default,,0000,0000,0000,,It can be proved, but we\Ndon't-- we do a rigorous job
Dialogue: 0,1:31:07.70,1:31:12.32,Default,,0000,0000,0000,,in the book about introducing\Nand proving that along
Dialogue: 0,1:31:12.32,1:31:13.77,Default,,0000,0000,0000,,a curvilinear path.
Dialogue: 0,1:31:13.77,1:31:17.66,Default,,0000,0000,0000,,This is gonna be-- where\Nam I here, at time t0?
Dialogue: 0,1:31:17.66,1:31:19.71,Default,,0000,0000,0000,,And this is at time t1.
Dialogue: 0,1:31:19.71,1:31:23.37,Default,,0000,0000,0000,,That means x0 is x of t0.
Dialogue: 0,1:31:23.37,1:31:26.58,Default,,0000,0000,0000,,y0 is y of t0.
Dialogue: 0,1:31:26.58,1:31:31.53,Default,,0000,0000,0000,,And at the finish point,\NI'm at x1, which is x of t1,
Dialogue: 0,1:31:31.53,1:31:34.88,Default,,0000,0000,0000,,and y1 equals y of t1.
Dialogue: 0,1:31:34.88,1:31:40.17,Default,,0000,0000,0000,,So between t0 and\Nt1, I have traveled
Dialogue: 0,1:31:40.17,1:31:46.13,Default,,0000,0000,0000,,and I have F where\Nmeasure at x of t y of t,
Dialogue: 0,1:31:46.13,1:31:50.26,Default,,0000,0000,0000,,where t is between--\Nmoving between t0 and t1.
Dialogue: 0,1:31:50.26,1:31:53.04,Default,,0000,0000,0000,,I'm done with this\Nis the F part.
Dialogue: 0,1:31:53.04,1:31:54.98,Default,,0000,0000,0000,,What is the dR?
Dialogue: 0,1:31:54.98,1:31:57.87,Default,,0000,0000,0000,,Now you guys know\Nabout differential.
Dialogue: 0,1:31:57.87,1:32:00.36,Default,,0000,0000,0000,,Thank God you know\Nabout differential,
Dialogue: 0,1:32:00.36,1:32:06.85,Default,,0000,0000,0000,,because this is gonna\Nhelp you very much.
Dialogue: 0,1:32:06.85,1:32:07.85,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:32:07.85,1:32:18.10,Default,,0000,0000,0000,,So instead of dR, I'm going\Nto write dot, and let's
Dialogue: 0,1:32:18.10,1:32:24.30,Default,,0000,0000,0000,,see how I write-- what I write\Nin terms of dR. You may say,
Dialogue: 0,1:32:24.30,1:32:28.02,Default,,0000,0000,0000,,well, what does she mean?
Dialogue: 0,1:32:28.02,1:32:34.62,Default,,0000,0000,0000,,dR was dxi plus dyj.
Dialogue: 0,1:32:34.62,1:32:36.28,Default,,0000,0000,0000,,And you say, why is that?
Dialogue: 0,1:32:36.28,1:32:37.07,Default,,0000,0000,0000,,I don't understand.
Dialogue: 0,1:32:37.07,1:32:41.49,Default,,0000,0000,0000,,Because R itself is x i plus yj.
Dialogue: 0,1:32:41.49,1:32:46.64,Default,,0000,0000,0000,,And x is a function of t,\Nand y is a function of t.
Dialogue: 0,1:32:46.64,1:32:49.91,Default,,0000,0000,0000,,That means that when you\Napply the differential,
Dialogue: 0,1:32:49.91,1:32:53.57,Default,,0000,0000,0000,,you are gonna apply the\Ndifferentials to dx and dy,
Dialogue: 0,1:32:53.57,1:32:57.00,Default,,0000,0000,0000,,and these are gonna be\Ninfinitesimal displacement.
Dialogue: 0,1:32:57.00,1:33:00.44,Default,,0000,0000,0000,,Infinitesimal displacement.
Dialogue: 0,1:33:00.44,1:33:07.80,Default,,0000,0000,0000,,
Dialogue: 0,1:33:07.80,1:33:09.77,Default,,0000,0000,0000,,Infinitesimal displacement.
Dialogue: 0,1:33:09.77,1:33:14.19,Default,,0000,0000,0000,,What is an infinitesimal\Ndisplacement in terms of time?
Dialogue: 0,1:33:14.19,1:33:16.59,Default,,0000,0000,0000,,Well, we have our\Nparametric equations.
Dialogue: 0,1:33:16.59,1:33:21.28,Default,,0000,0000,0000,,So Mr. dx as a differential\Nis just x prime dt.
Dialogue: 0,1:33:21.28,1:33:24.06,Default,,0000,0000,0000,,It's like in the\N[INAUDIBLE] substitution.
Dialogue: 0,1:33:24.06,1:33:26.16,Default,,0000,0000,0000,,dy is just y prime dt.
Dialogue: 0,1:33:26.16,1:33:28.62,Default,,0000,0000,0000,,So let me write this down again.
Dialogue: 0,1:33:28.62,1:33:44.55,Default,,0000,0000,0000,,This is x prime of t i plus\Ny prime of t j times dt.
Dialogue: 0,1:33:44.55,1:33:48.03,Default,,0000,0000,0000,,So Mr. dt is like\Na common factor.
Dialogue: 0,1:33:48.03,1:33:50.02,Default,,0000,0000,0000,,If he wants to go\Nout for a walk,
Dialogue: 0,1:33:50.02,1:33:52.18,Default,,0000,0000,0000,,he says, I'm gonna\Ngo out for a walk.
Dialogue: 0,1:33:52.18,1:33:53.01,Default,,0000,0000,0000,,I go out for a walk.
Dialogue: 0,1:33:53.01,1:33:58.42,Default,,0000,0000,0000,,So dR is actually x\Nprime of t times i
Dialogue: 0,1:33:58.42,1:34:02.97,Default,,0000,0000,0000,,plus y prime of t times j dt.
Dialogue: 0,1:34:02.97,1:34:06.85,Default,,0000,0000,0000,,
Dialogue: 0,1:34:06.85,1:34:14.65,Default,,0000,0000,0000,,And this will represent\Nthe derivative of R
Dialogue: 0,1:34:14.65,1:34:16.05,Default,,0000,0000,0000,,with respect to pi.
Dialogue: 0,1:34:16.05,1:34:18.03,Default,,0000,0000,0000,,So that will be what?
Dialogue: 0,1:34:18.03,1:34:19.51,Default,,0000,0000,0000,,The differential.
Dialogue: 0,1:34:19.51,1:34:26.43,Default,,0000,0000,0000,,Differential of R\Nwith respect to pi.
Dialogue: 0,1:34:26.43,1:34:31.37,Default,,0000,0000,0000,,
Dialogue: 0,1:34:31.37,1:34:34.55,Default,,0000,0000,0000,,This is the same as\Nwriting dx i plus dy j.
Dialogue: 0,1:34:34.55,1:34:37.69,Default,,0000,0000,0000,,
Dialogue: 0,1:34:37.69,1:34:44.11,Default,,0000,0000,0000,,And it's the same as writing\NdR. Why is this happening?
Dialogue: 0,1:34:44.11,1:34:47.33,Default,,0000,0000,0000,,Because it's [INAUDIBLE]\NBecause x and y
Dialogue: 0,1:34:47.33,1:34:52.54,Default,,0000,0000,0000,,themselves are functions of one\Nvariable only, which is time.
Dialogue: 0,1:34:52.54,1:34:56.49,Default,,0000,0000,0000,,This is why it happens.
Dialogue: 0,1:34:56.49,1:34:59.95,Default,,0000,0000,0000,,Oh, so we will\Nsimply have to do--
Dialogue: 0,1:34:59.95,1:35:02.91,Default,,0000,0000,0000,,to learn new things, right?
Dialogue: 0,1:35:02.91,1:35:05.38,Default,,0000,0000,0000,,We are gonna have\Nto learn new things,
Dialogue: 0,1:35:05.38,1:35:10.08,Default,,0000,0000,0000,,like integral from a time--\Nfixed time 0 to t1, which
Dialogue: 0,1:35:10.08,1:35:15.46,Default,,0000,0000,0000,,is 10 seconds, of a dot product\Nbetween a certain vector that
Dialogue: 0,1:35:15.46,1:35:18.61,Default,,0000,0000,0000,,depends on time and another\Nvector that depends on time,
Dialogue: 0,1:35:18.61,1:35:20.55,Default,,0000,0000,0000,,and dt.
Dialogue: 0,1:35:20.55,1:35:23.47,Default,,0000,0000,0000,,So we are gonna\Nhave to learn how
Dialogue: 0,1:35:23.47,1:35:28.00,Default,,0000,0000,0000,,to compute the work through this\Ntype of curvilinear integral.
Dialogue: 0,1:35:28.00,1:35:32.49,Default,,0000,0000,0000,,And this is-- this is called\Neither path integral-- path
Dialogue: 0,1:35:32.49,1:35:50.91,Default,,0000,0000,0000,,integral along the curve c, or\Ncurvilinear integral along c.
Dialogue: 0,1:35:50.91,1:35:53.85,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:35:53.85,1:35:56.80,Default,,0000,0000,0000,,STUDENT: Let's say if I\Nmove the force this is
Dialogue: 0,1:35:56.80,1:35:58.76,Default,,0000,0000,0000,,[INAUDIBLE] function, correct?
Dialogue: 0,1:35:58.76,1:36:01.95,Default,,0000,0000,0000,,So if I can find\N[? arc length ?] that
Dialogue: 0,1:36:01.95,1:36:03.72,Default,,0000,0000,0000,,is between the x--
Dialogue: 0,1:36:03.72,1:36:04.57,Default,,0000,0000,0000,,PROFESSOR: Yeah--
Dialogue: 0,1:36:04.57,1:36:05.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] points.
Dialogue: 0,1:36:05.74,1:36:08.87,Default,,0000,0000,0000,,PROFESSOR: Yeah, we will do\Nthe one with that length next.
Dialogue: 0,1:36:08.87,1:36:10.91,Default,,0000,0000,0000,,The [? reason ?] so--
Dialogue: 0,1:36:10.91,1:36:11.83,Default,,0000,0000,0000,,STUDENT: Is it harder?
Dialogue: 0,1:36:11.83,1:36:12.79,Default,,0000,0000,0000,,PROFESSOR: No.
Dialogue: 0,1:36:12.79,1:36:15.06,Default,,0000,0000,0000,,No, you can pass\Nthrough a plane.
Dialogue: 0,1:36:15.06,1:36:17.50,Default,,0000,0000,0000,,And you can-- we'll\Ndo that next time.
Dialogue: 0,1:36:17.50,1:36:20.91,Default,,0000,0000,0000,,You will have an integral\Nwith respect to S.
Dialogue: 0,1:36:20.91,1:36:23.34,Default,,0000,0000,0000,,So the integration will\Nbe with respect to dS,
Dialogue: 0,1:36:23.34,1:36:24.32,Default,,0000,0000,0000,,they are correct.
Dialogue: 0,1:36:24.32,1:36:27.73,Default,,0000,0000,0000,,And then you will have a\Nfunction that depends on S
Dialogue: 0,1:36:27.73,1:36:31.34,Default,,0000,0000,0000,,[INAUDIBLE] So I'll-- for\Ntoday, I'll only teach you that.
Dialogue: 0,1:36:31.34,1:36:33.36,Default,,0000,0000,0000,,Next time I'll teach\Nyou that with respect
Dialogue: 0,1:36:33.36,1:36:36.77,Default,,0000,0000,0000,,to arc length, which is also\Nvery-- it's not hard at all.
Dialogue: 0,1:36:36.77,1:36:37.27,Default,,0000,0000,0000,,STUDENT: OK.
Dialogue: 0,1:36:37.27,1:36:41.53,Default,,0000,0000,0000,,PROFESSOR: So I will work\Nwith you on [INAUDIBLE]
Dialogue: 0,1:36:41.53,1:36:48.77,Default,,0000,0000,0000,,Now assume that we have-- I\Nwill spray all this thing.
Dialogue: 0,1:36:48.77,1:36:55.16,Default,,0000,0000,0000,,
Dialogue: 0,1:36:55.16,1:36:58.10,Default,,0000,0000,0000,,Assume that I have a problem.
Dialogue: 0,1:36:58.10,1:37:02.67,Default,,0000,0000,0000,,I have a parabola-- arc\Nof a parabola, all right?
Dialogue: 0,1:37:02.67,1:37:08.50,Default,,0000,0000,0000,,Between-- let's\Nsay the parabola is
Dialogue: 0,1:37:08.50,1:37:16.44,Default,,0000,0000,0000,,y equals x squared\Nbetween two points.
Dialogue: 0,1:37:16.44,1:37:19.42,Default,,0000,0000,0000,,
Dialogue: 0,1:37:19.42,1:37:21.90,Default,,0000,0000,0000,,And I'll ask you to\Ncompute some work,
Dialogue: 0,1:37:21.90,1:37:25.37,Default,,0000,0000,0000,,and I'll tell you in a second\Nwhat [INAUDIBLE] to do.
Dialogue: 0,1:37:25.37,1:37:38.26,Default,,0000,0000,0000,,
Dialogue: 0,1:37:38.26,1:37:52.65,Default,,0000,0000,0000,,So exercise-- assume\Nthe parabola y
Dialogue: 0,1:37:52.65,1:38:03.34,Default,,0000,0000,0000,,equals x squared between\Npoints A of coordinates 0, 0
Dialogue: 0,1:38:03.34,1:38:07.04,Default,,0000,0000,0000,,and point B of coordinates 1, 1.
Dialogue: 0,1:38:07.04,1:38:15.83,Default,,0000,0000,0000,,a, Parametrized this parabola\Nin the simplest way you can.
Dialogue: 0,1:38:15.83,1:38:31.07,Default,,0000,0000,0000,,
Dialogue: 0,1:38:31.07,1:38:41.36,Default,,0000,0000,0000,,And b, compute the work\Nalong this arc of a parabola,
Dialogue: 0,1:38:41.36,1:38:44.32,Default,,0000,0000,0000,,arc AB of this parabola.
Dialogue: 0,1:38:44.32,1:38:50.27,Default,,0000,0000,0000,,
Dialogue: 0,1:38:50.27,1:39:04.25,Default,,0000,0000,0000,,For [INAUDIBLE] the\Nfunction big F of t,
Dialogue: 0,1:39:04.25,1:39:09.92,Default,,0000,0000,0000,,you see that-- I'm going to say,\Nno, big F of the point x, y,
Dialogue: 0,1:39:09.92,1:39:15.13,Default,,0000,0000,0000,,because you haven't parametrized\Nthat yet, big F of x,
Dialogue: 0,1:39:15.13,1:39:22.31,Default,,0000,0000,0000,,y being xi plus yg.
Dialogue: 0,1:39:22.31,1:39:27.72,Default,,0000,0000,0000,,
Dialogue: 0,1:39:27.72,1:39:30.18,Default,,0000,0000,0000,,So you say, OK, wait a minute.
Dialogue: 0,1:39:30.18,1:39:34.61,Default,,0000,0000,0000,,W will be integral over\Nthe arc of a parabola.
Dialogue: 0,1:39:34.61,1:39:37.04,Default,,0000,0000,0000,,Do you want to draw that first?
Dialogue: 0,1:39:37.04,1:39:39.16,Default,,0000,0000,0000,,Yes, I need to draw that first.
Dialogue: 0,1:39:39.16,1:39:43.100,Default,,0000,0000,0000,,So I have this parabola from A\Nto B. A is of coordinates 0, 0.
Dialogue: 0,1:39:43.100,1:39:45.96,Default,,0000,0000,0000,,B is of coordinates 1, 1.
Dialogue: 0,1:39:45.96,1:39:48.91,Default,,0000,0000,0000,,And this is y equals x squared.
Dialogue: 0,1:39:48.91,1:39:52.10,Default,,0000,0000,0000,,So what kind of\Nparametrization is
Dialogue: 0,1:39:52.10,1:39:55.29,Default,,0000,0000,0000,,the simplest one, the\Nregular one that people take?
Dialogue: 0,1:39:55.29,1:39:57.26,Default,,0000,0000,0000,,Take x to be t?
Dialogue: 0,1:39:57.26,1:40:00.39,Default,,0000,0000,0000,,And of course, take y in that\Ncase. y will be t squared.
Dialogue: 0,1:40:00.39,1:40:01.55,Default,,0000,0000,0000,,And for 1 you have 1.
Dialogue: 0,1:40:01.55,1:40:04.02,Default,,0000,0000,0000,,For 0 you have 0.
Dialogue: 0,1:40:04.02,1:40:07.47,Default,,0000,0000,0000,,When you have that work by\Ndefinition, what was that?
Dialogue: 0,1:40:07.47,1:40:12.25,Default,,0000,0000,0000,,It was written as integral\Nor on the graph C. Let's call
Dialogue: 0,1:40:12.25,1:40:14.23,Default,,0000,0000,0000,,this path C a curvilinear path.
Dialogue: 0,1:40:14.23,1:40:16.46,Default,,0000,0000,0000,,Look, script C-- so beautiful.
Dialogue: 0,1:40:16.46,1:40:25.36,Default,,0000,0000,0000,,Let me [INAUDIBLE] red and\Ndraw the C of what is work?
Dialogue: 0,1:40:25.36,1:40:36.62,Default,,0000,0000,0000,,F force-- may the force be with\Nus-- dot dR. All righty, that's
Dialogue: 0,1:40:36.62,1:40:39.05,Default,,0000,0000,0000,,a little bit of a headache.
Dialogue: 0,1:40:39.05,1:40:45.91,Default,,0000,0000,0000,,This F is going to be-- can I\Nwrite an alternative formula
Dialogue: 0,1:40:45.91,1:40:50.81,Default,,0000,0000,0000,,that I have not written yet\Nbut I will write in a second?
Dialogue: 0,1:40:50.81,1:40:56.06,Default,,0000,0000,0000,,dR will be dxi plus dyj.
Dialogue: 0,1:40:56.06,1:41:01.63,Default,,0000,0000,0000,,So I can also\Nwrite that F dot dR
Dialogue: 0,1:41:01.63,1:41:04.56,Default,,0000,0000,0000,,as the dot product will seem to\Nbe-- what was the dot product
Dialogue: 0,1:41:04.56,1:41:06.02,Default,,0000,0000,0000,,guys, do you remember?
Dialogue: 0,1:41:06.02,1:41:10.43,Default,,0000,0000,0000,,First component times\Nfirst component, F1dx
Dialogue: 0,1:41:10.43,1:41:14.93,Default,,0000,0000,0000,,plus second scalar component\Ntimes second scalar component,
Dialogue: 0,1:41:14.93,1:41:15.43,Default,,0000,0000,0000,,F2dy.
Dialogue: 0,1:41:15.43,1:41:20.17,Default,,0000,0000,0000,,
Dialogue: 0,1:41:20.17,1:41:21.51,Default,,0000,0000,0000,,I'll write it down.
Dialogue: 0,1:41:21.51,1:41:26.38,Default,,0000,0000,0000,,Along the path C I'll\Nhave F1dx plus F2dy.
Dialogue: 0,1:41:26.38,1:41:29.89,Default,,0000,0000,0000,,But god knows what it's\Ngoing to be in terms of time.
Dialogue: 0,1:41:29.89,1:41:33.14,Default,,0000,0000,0000,,So I have to change\Nvariable thinking.
Dialogue: 0,1:41:33.14,1:41:36.69,Default,,0000,0000,0000,,Okey-dokey, Mr.\Ndx by substitution
Dialogue: 0,1:41:36.69,1:41:41.26,Default,,0000,0000,0000,,was x prime to T. Mr. dy by\Nsubstitution was y prime dt.
Dialogue: 0,1:41:41.26,1:41:46.01,Default,,0000,0000,0000,,So I'd rather write\Nthis in a simpler way.
Dialogue: 0,1:41:46.01,1:41:50.21,Default,,0000,0000,0000,,This is a new object,\Npath integral.
Dialogue: 0,1:41:50.21,1:41:52.98,Default,,0000,0000,0000,,But we know this\Nobject from Calc I
Dialogue: 0,1:41:52.98,1:41:56.83,Default,,0000,0000,0000,,as being a simple\Nintegral from time t0--
Dialogue: 0,1:41:56.83,1:42:04.07,Default,,0000,0000,0000,,I'll write it down-- time\Nt1, F1x prime of t plus F2y
Dialogue: 0,1:42:04.07,1:42:05.91,Default,,0000,0000,0000,,prime of t.
Dialogue: 0,1:42:05.91,1:42:09.76,Default,,0000,0000,0000,,This is the integral dt.
Dialogue: 0,1:42:09.76,1:42:12.93,Default,,0000,0000,0000,,This would be a\Npiece of cake for us
Dialogue: 0,1:42:12.93,1:42:16.07,Default,,0000,0000,0000,,to apply in this problem.
Dialogue: 0,1:42:16.07,1:42:19.93,Default,,0000,0000,0000,,Equals-- now you tell me\Nwhat I'm supposed to write.
Dialogue: 0,1:42:19.93,1:42:23.26,Default,,0000,0000,0000,,Because if you don't, I'm\Ngoing to not write anything.
Dialogue: 0,1:42:23.26,1:42:26.40,Default,,0000,0000,0000,,t0 for me is what time?
Dialogue: 0,1:42:26.40,1:42:28.37,Default,,0000,0000,0000,,When did we leave this?
Dialogue: 0,1:42:28.37,1:42:29.35,Default,,0000,0000,0000,,0.
Dialogue: 0,1:42:29.35,1:42:31.25,Default,,0000,0000,0000,,And when did we arrive?
Dialogue: 0,1:42:31.25,1:42:32.60,Default,,0000,0000,0000,,At 1 o'clock.
Dialogue: 0,1:42:32.60,1:42:38.15,Default,,0000,0000,0000,,We arrived when t is 1, or\Nevery one second or whatever
Dialogue: 0,1:42:38.15,1:42:40.29,Default,,0000,0000,0000,,depending on [INAUDIBLE].
Dialogue: 0,1:42:40.29,1:42:45.78,Default,,0000,0000,0000,,OK, from 0 to 1, now who is F1?
Dialogue: 0,1:42:45.78,1:42:48.36,Default,,0000,0000,0000,,F1 is this.
Dialogue: 0,1:42:48.36,1:42:49.35,Default,,0000,0000,0000,,But it drives me crazy.
Dialogue: 0,1:42:49.35,1:42:52.80,Default,,0000,0000,0000,,Because I need this\Nto be expressed in t.
Dialogue: 0,1:42:52.80,1:42:55.81,Default,,0000,0000,0000,,So I think of x and\Ny as functions of t.
Dialogue: 0,1:42:55.81,1:42:59.52,Default,,0000,0000,0000,,So if 1 is not x,\Nnot [INAUDIBLE]
Dialogue: 0,1:42:59.52,1:43:04.48,Default,,0000,0000,0000,,right here, but t, which is the\Nsame thing in parametrization--
Dialogue: 0,1:43:04.48,1:43:07.96,Default,,0000,0000,0000,,this is t, t times.
Dialogue: 0,1:43:07.96,1:43:10.44,Default,,0000,0000,0000,,Who is x prime?
Dialogue: 0,1:43:10.44,1:43:11.92,Default,,0000,0000,0000,,1, thank god.
Dialogue: 0,1:43:11.92,1:43:17.38,Default,,0000,0000,0000,,That is easy, times 1, plus F2.
Dialogue: 0,1:43:17.38,1:43:19.86,Default,,0000,0000,0000,,Who is F2?
Dialogue: 0,1:43:19.86,1:43:20.36,Default,,0000,0000,0000,,t squared.
Dialogue: 0,1:43:20.36,1:43:22.64,Default,,0000,0000,0000,,I'll have to write it down.
Dialogue: 0,1:43:22.64,1:43:25.42,Default,,0000,0000,0000,,Times who is y prime?
Dialogue: 0,1:43:25.42,1:43:26.23,Default,,0000,0000,0000,,2t.
Dialogue: 0,1:43:26.23,1:43:27.92,Default,,0000,0000,0000,,y prime is t2.
Dialogue: 0,1:43:27.92,1:43:34.14,Default,,0000,0000,0000,,So I write it down-- 2t, dt.
Dialogue: 0,1:43:34.14,1:43:37.98,Default,,0000,0000,0000,,
Dialogue: 0,1:43:37.98,1:43:40.97,Default,,0000,0000,0000,,So that's how I compute\Nthis integral back.
Dialogue: 0,1:43:40.97,1:43:41.82,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:43:41.82,1:43:47.37,Default,,0000,0000,0000,,No, because it's just a simple\Nintegral from Calculus I.
Dialogue: 0,1:43:47.37,1:43:49.90,Default,,0000,0000,0000,,So I have to integrate\Nwhat function?
Dialogue: 0,1:43:49.90,1:43:55.69,Default,,0000,0000,0000,,A polynomial, 2t cubed\Nplus t with respect
Dialogue: 0,1:43:55.69,1:43:59.72,Default,,0000,0000,0000,,to t between 0,\Ntime 0 and time 1.
Dialogue: 0,1:43:59.72,1:44:02.63,Default,,0000,0000,0000,,
Dialogue: 0,1:44:02.63,1:44:05.06,Default,,0000,0000,0000,,Good, let's do it.
Dialogue: 0,1:44:05.06,1:44:09.62,Default,,0000,0000,0000,,Because that's a piece of\Ncake-- 2 times t to the 4 over 4
Dialogue: 0,1:44:09.62,1:44:12.39,Default,,0000,0000,0000,,plus t squared over 2.
Dialogue: 0,1:44:12.39,1:44:16.78,Default,,0000,0000,0000,,I take the whole thing between,\NI apply the fundamental theorem
Dialogue: 0,1:44:16.78,1:44:21.59,Default,,0000,0000,0000,,of calculus, and I have between\Nt equals 1 up and t equals 0
Dialogue: 0,1:44:21.59,1:44:22.36,Default,,0000,0000,0000,,down.
Dialogue: 0,1:44:22.36,1:44:26.11,Default,,0000,0000,0000,,What's the final answer?
Dialogue: 0,1:44:26.11,1:44:27.60,Default,,0000,0000,0000,,It's a single final answer.
Dialogue: 0,1:44:27.60,1:44:30.43,Default,,0000,0000,0000,,And again, on the\Nexam, on the final,
Dialogue: 0,1:44:30.43,1:44:32.58,Default,,0000,0000,0000,,do not expect a\Nheadache computation.
Dialogue: 0,1:44:32.58,1:44:34.73,Default,,0000,0000,0000,,Do expect something\Nsimple like that
Dialogue: 0,1:44:34.73,1:44:36.71,Default,,0000,0000,0000,,where you don't\Nneed a calculator.
Dialogue: 0,1:44:36.71,1:44:40.51,Default,,0000,0000,0000,,You just have either integers\Nonly or simple fractions
Dialogue: 0,1:44:40.51,1:44:42.13,Default,,0000,0000,0000,,to add, and you\Nshould get the answer.
Dialogue: 0,1:44:42.13,1:44:44.34,Default,,0000,0000,0000,,What is the answer, guys?
Dialogue: 0,1:44:44.34,1:44:47.76,Default,,0000,0000,0000,,1-- 1/2 plus 1/2 equals 1.
Dialogue: 0,1:44:47.76,1:44:52.39,Default,,0000,0000,0000,,So 1 is the value\Nof the work in what?
Dialogue: 0,1:44:52.39,1:44:54.94,Default,,0000,0000,0000,,Measured in newtons\Ntimes meters,
Dialogue: 0,1:44:54.94,1:44:57.82,Default,,0000,0000,0000,,whatever your units are.
Dialogue: 0,1:44:57.82,1:45:01.63,Default,,0000,0000,0000,,When you drag the\Nobject from this point
Dialogue: 0,1:45:01.63,1:45:06.05,Default,,0000,0000,0000,,to this point, on which the\Nacting force is the only
Dialogue: 0,1:45:06.05,1:45:08.03,Default,,0000,0000,0000,,acting force-- it\Ncould be the result
Dialogue: 0,1:45:08.03,1:45:09.88,Default,,0000,0000,0000,,that there are several forces.
Dialogue: 0,1:45:09.88,1:45:13.32,Default,,0000,0000,0000,,That is that force\Nthat you have here.
Dialogue: 0,1:45:13.32,1:45:15.27,Default,,0000,0000,0000,,Is it useful?
Dialogue: 0,1:45:15.27,1:45:17.23,Default,,0000,0000,0000,,It's very useful for engineers.
Dialogue: 0,1:45:17.23,1:45:18.70,Default,,0000,0000,0000,,It's very useful for physicists.
Dialogue: 0,1:45:18.70,1:45:21.30,Default,,0000,0000,0000,,It's very useful for\Nanybody who works
Dialogue: 0,1:45:21.30,1:45:27.23,Default,,0000,0000,0000,,in applied mathematics, this\Nnotion of work given like that.
Dialogue: 0,1:45:27.23,1:45:29.62,Default,,0000,0000,0000,,I'm going to go ahead and erase.
Dialogue: 0,1:45:29.62,1:45:34.86,Default,,0000,0000,0000,,And I'll ask you one\Nthing here that is not
Dialogue: 0,1:45:34.86,1:45:38.81,Default,,0000,0000,0000,,in the book I think\Nas far as I remember.
Dialogue: 0,1:45:38.81,1:45:45.35,Default,,0000,0000,0000,,Can you guys prove that this\Nsophisticated formula becomes
Dialogue: 0,1:45:45.35,1:45:48.03,Default,,0000,0000,0000,,your formula of the\None you claimed,
Dialogue: 0,1:45:48.03,1:45:52.06,Default,,0000,0000,0000,,the first formula you gave me?
Dialogue: 0,1:45:52.06,1:45:53.05,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:45:53.05,1:45:58.01,Default,,0000,0000,0000,,Do you think it's\Nhard to prove this?
Dialogue: 0,1:45:58.01,1:46:01.48,Default,,0000,0000,0000,,OK, what if we have the\Nsimplest possible case.
Dialogue: 0,1:46:01.48,1:46:03.47,Default,,0000,0000,0000,,Let's think of--
Dialogue: 0,1:46:03.47,1:46:06.94,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:46:06.94,1:46:09.92,Default,,0000,0000,0000,,
Dialogue: 0,1:46:09.92,1:46:11.65,Default,,0000,0000,0000,,PROFESSOR: Yeah,\NI'm thinking maybe I
Dialogue: 0,1:46:11.65,1:46:22.32,Default,,0000,0000,0000,,should do, well, A to B, right?
Dialogue: 0,1:46:22.32,1:46:33.23,Default,,0000,0000,0000,,AB, what kind of expression\Ndo I have [INAUDIBLE]?
Dialogue: 0,1:46:33.23,1:46:38.19,Default,,0000,0000,0000,,If I take this to be-- I\Ncould have any line, right?
Dialogue: 0,1:46:38.19,1:46:39.85,Default,,0000,0000,0000,,I could have any line.
Dialogue: 0,1:46:39.85,1:46:43.50,Default,,0000,0000,0000,,But if I have any line,\NI can pick my frame
Dialogue: 0,1:46:43.50,1:46:46.93,Default,,0000,0000,0000,,according to my preference.
Dialogue: 0,1:46:46.93,1:46:49.42,Default,,0000,0000,0000,,Nobody's going to\Ntell me, well, you
Dialogue: 0,1:46:49.42,1:46:51.43,Default,,0000,0000,0000,,have to take the\Nframe like that,
Dialogue: 0,1:46:51.43,1:46:55.99,Default,,0000,0000,0000,,and then your line will be of\Nthe form ax plus by equals.
Dialogue: 0,1:46:55.99,1:47:04.76,Default,,0000,0000,0000,,No, I'll just take the\Nframe to be this one, where
Dialogue: 0,1:47:04.76,1:47:09.85,Default,,0000,0000,0000,,AB will be x axis, and A\Nwill be of coordinates 0, 0
Dialogue: 0,1:47:09.85,1:47:14.54,Default,,0000,0000,0000,,and B will be of\Ncoordinates B and 0.
Dialogue: 0,1:47:14.54,1:47:18.55,Default,,0000,0000,0000,,And this is just my line.
Dialogue: 0,1:47:18.55,1:47:28.10,Default,,0000,0000,0000,,So x will be moving between\N0 and B. And y is 0, right?
Dialogue: 0,1:47:28.10,1:47:31.39,Default,,0000,0000,0000,,It should be, at least.
Dialogue: 0,1:47:31.39,1:47:43.73,Default,,0000,0000,0000,,And then F, let's say, should\Nbe this function, this.
Dialogue: 0,1:47:43.73,1:47:48.53,Default,,0000,0000,0000,,
Dialogue: 0,1:47:48.53,1:47:50.57,Default,,0000,0000,0000,,I'll assume the\Nangle is constant,
Dialogue: 0,1:47:50.57,1:47:53.40,Default,,0000,0000,0000,,just like I had it with theta.
Dialogue: 0,1:47:53.40,1:47:57.01,Default,,0000,0000,0000,,And then it's acting all\Nthe way on your object.
Dialogue: 0,1:47:57.01,1:47:58.96,Default,,0000,0000,0000,,You have the same\Nangle here always.
Dialogue: 0,1:47:58.96,1:48:02.88,Default,,0000,0000,0000,,
Dialogue: 0,1:48:02.88,1:48:07.82,Default,,0000,0000,0000,,So F is F1i plus F2j.
Dialogue: 0,1:48:07.82,1:48:12.82,Default,,0000,0000,0000,,
Dialogue: 0,1:48:12.82,1:48:22.56,Default,,0000,0000,0000,,dR will be dxi plus dyj.
Dialogue: 0,1:48:22.56,1:48:29.77,Default,,0000,0000,0000,,
Dialogue: 0,1:48:29.77,1:48:32.39,Default,,0000,0000,0000,,But then you say, wait a minute,\Nbut didn't you say, Magdalena,
Dialogue: 0,1:48:32.39,1:48:34.40,Default,,0000,0000,0000,,that you are along this line?
Dialogue: 0,1:48:34.40,1:48:36.74,Default,,0000,0000,0000,,Didn't you say that y is 0?
Dialogue: 0,1:48:36.74,1:48:37.63,Default,,0000,0000,0000,,So which y?
Dialogue: 0,1:48:37.63,1:48:38.45,Default,,0000,0000,0000,,So there is no y.
Dialogue: 0,1:48:38.45,1:48:41.69,Default,,0000,0000,0000,,So this is 0, right?
Dialogue: 0,1:48:41.69,1:48:50.07,Default,,0000,0000,0000,,OK, Mr. x, I want to\Nparametrize my trajectory.
Dialogue: 0,1:48:50.07,1:48:53.39,Default,,0000,0000,0000,,How do I parametrize\Nit the simplest way?
Dialogue: 0,1:48:53.39,1:48:55.86,Default,,0000,0000,0000,,I'll take x to be t.
Dialogue: 0,1:48:55.86,1:49:00.79,Default,,0000,0000,0000,,And time will be\Nexactly between 0 and d.
Dialogue: 0,1:49:00.79,1:49:02.27,Default,,0000,0000,0000,,And y will be 0.
Dialogue: 0,1:49:02.27,1:49:06.21,Default,,0000,0000,0000,,And thank you god,\Nbecause that's easy.
Dialogue: 0,1:49:06.21,1:49:10.18,Default,,0000,0000,0000,,And so all you\Nneed to give me is
Dialogue: 0,1:49:10.18,1:49:17.02,Default,,0000,0000,0000,,W is integral of F\NdR C in that case.
Dialogue: 0,1:49:17.02,1:49:20.40,Default,,0000,0000,0000,,So what am I going\Nto have in that case?
Dialogue: 0,1:49:20.40,1:49:21.84,Default,,0000,0000,0000,,I'll have this formula.
Dialogue: 0,1:49:21.84,1:49:26.44,Default,,0000,0000,0000,,I'll skip a step, and\NI'll have that formula.
Dialogue: 0,1:49:26.44,1:49:30.28,Default,,0000,0000,0000,,And that means I have integral\Nfrom t0 equals 0 to t1
Dialogue: 0,1:49:30.28,1:49:35.74,Default,,0000,0000,0000,,equals B.
Dialogue: 0,1:49:35.74,1:49:39.21,Default,,0000,0000,0000,,F1-- now you have to tell\Nme what F1 will be. x prime
Dialogue: 0,1:49:39.21,1:49:40.70,Default,,0000,0000,0000,,[? noted ?] is 1.
Dialogue: 0,1:49:40.70,1:49:45.66,Default,,0000,0000,0000,,The second guy is 0, thank you\Nvery much, and [INAUDIBLE].
Dialogue: 0,1:49:45.66,1:49:49.13,Default,,0000,0000,0000,,
Dialogue: 0,1:49:49.13,1:49:52.97,Default,,0000,0000,0000,,F1 will be what?
Dialogue: 0,1:49:52.97,1:49:56.23,Default,,0000,0000,0000,,Well, life is nice.
Dialogue: 0,1:49:56.23,1:50:01.48,Default,,0000,0000,0000,,F1 will be the projection of\Nthe vector F on my x-axis.
Dialogue: 0,1:50:01.48,1:50:06.41,Default,,0000,0000,0000,,So F1 is the length of\Nthis blue vector, I'll say.
Dialogue: 0,1:50:06.41,1:50:09.33,Default,,0000,0000,0000,,So F1 is a scalar.
Dialogue: 0,1:50:09.33,1:50:12.25,Default,,0000,0000,0000,,Let's say F1 is a\Nscalar component.
Dialogue: 0,1:50:12.25,1:50:16.65,Default,,0000,0000,0000,,That means it's F\Nlength cosine theta.
Dialogue: 0,1:50:16.65,1:50:19.71,Default,,0000,0000,0000,,Because it's hypotenuse\Ntimes cosine theta.
Dialogue: 0,1:50:19.71,1:50:20.61,Default,,0000,0000,0000,,So it's easy.
Dialogue: 0,1:50:20.61,1:50:25.73,Default,,0000,0000,0000,,So you have length\Nof F, how much it is,
Dialogue: 0,1:50:25.73,1:50:29.89,Default,,0000,0000,0000,,how big this vector is, times\Ncosine theta, times what
Dialogue: 0,1:50:29.89,1:50:31.06,Default,,0000,0000,0000,,when you integrate it, guys?
Dialogue: 0,1:50:31.06,1:50:34.68,Default,,0000,0000,0000,,When you integrate 1 with\Nrespect to t, what do you get?
Dialogue: 0,1:50:34.68,1:50:36.94,Default,,0000,0000,0000,,t between d and 0.
Dialogue: 0,1:50:36.94,1:50:41.17,Default,,0000,0000,0000,,So you have t between d and 0.
Dialogue: 0,1:50:41.17,1:50:42.92,Default,,0000,0000,0000,,We got the formula.
Dialogue: 0,1:50:42.92,1:50:48.04,Default,,0000,0000,0000,,So we got that F length\Ntimes [INAUDIBLE]
Dialogue: 0,1:50:48.04,1:50:50.55,Default,,0000,0000,0000,,times cosine theta times d,\Nthis is the displacement.
Dialogue: 0,1:50:50.55,1:50:52.48,Default,,0000,0000,0000,,This is the cosine.
Dialogue: 0,1:50:52.48,1:50:57.42,Default,,0000,0000,0000,,This is the magnitude of\Nthe force that I'm-- look,
Dialogue: 0,1:50:57.42,1:50:59.39,Default,,0000,0000,0000,,this is the force.
Dialogue: 0,1:50:59.39,1:51:01.86,Default,,0000,0000,0000,,My force is along my arm.
Dialogue: 0,1:51:01.86,1:51:04.83,Default,,0000,0000,0000,,I'm just dragging\Nthis poor object.
Dialogue: 0,1:51:04.83,1:51:07.30,Default,,0000,0000,0000,,The force I'm\Nacting with, suppose
Dialogue: 0,1:51:07.30,1:51:11.74,Default,,0000,0000,0000,,it's always the same parallel\Nto that that I can feel.
Dialogue: 0,1:51:11.74,1:51:16.17,Default,,0000,0000,0000,,So that's what I have, F\Ncosine theta, and it was easy.
Dialogue: 0,1:51:16.17,1:51:21.10,Default,,0000,0000,0000,,So as a particular case\Nof this nasty integral,
Dialogue: 0,1:51:21.10,1:51:26.94,Default,,0000,0000,0000,,I have my old work from\Nschool that I had to believe.
Dialogue: 0,1:51:26.94,1:51:30.23,Default,,0000,0000,0000,,I tell you guys, I did\Nnot believe a word.
Dialogue: 0,1:51:30.23,1:51:32.98,Default,,0000,0000,0000,,Because my teacher\Nin eighth grade
Dialogue: 0,1:51:32.98,1:51:35.66,Default,,0000,0000,0000,,came up with this\Nout of nothing,
Dialogue: 0,1:51:35.66,1:51:38.81,Default,,0000,0000,0000,,and we were supposed to\Nbe good students preparing
Dialogue: 0,1:51:38.81,1:51:42.97,Default,,0000,0000,0000,,for a high school like this\Nkind of scientific-- back home,
Dialogue: 0,1:51:42.97,1:51:45.13,Default,,0000,0000,0000,,there are different\Nkinds of high school.
Dialogue: 0,1:51:45.13,1:51:47.40,Default,,0000,0000,0000,,There is scientific high\Nschool with emphasis
Dialogue: 0,1:51:47.40,1:51:48.23,Default,,0000,0000,0000,,in math and physics.
Dialogue: 0,1:51:48.23,1:51:49.95,Default,,0000,0000,0000,,There is one for\Nchemistry/biology.
Dialogue: 0,1:51:49.95,1:51:54.78,Default,,0000,0000,0000,,There is one for language,\Nlinguistics, [INAUDIBLE],
Dialogue: 0,1:51:54.78,1:51:55.75,Default,,0000,0000,0000,,and so on.
Dialogue: 0,1:51:55.75,1:51:59.61,Default,,0000,0000,0000,,And I was for the\Nmath and physics one.
Dialogue: 0,1:51:59.61,1:52:03.98,Default,,0000,0000,0000,,And I had to solve this formula\Nwithout understanding it.
Dialogue: 0,1:52:03.98,1:52:07.98,Default,,0000,0000,0000,,And it took me many\Nother years to understand
Dialogue: 0,1:52:07.98,1:52:10.93,Default,,0000,0000,0000,,that it's just a little\Npiece of a big picture,
Dialogue: 0,1:52:10.93,1:52:16.27,Default,,0000,0000,0000,,and that there's\Nsomething bigger than what
Dialogue: 0,1:52:16.27,1:52:18.35,Default,,0000,0000,0000,,we were taught in eighth grade.
Dialogue: 0,1:52:18.35,1:52:28.44,Default,,0000,0000,0000,,
Dialogue: 0,1:52:28.44,1:52:31.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:52:31.74,1:52:39.67,Default,,0000,0000,0000,,
Dialogue: 0,1:52:39.67,1:52:43.64,Default,,0000,0000,0000,,PROFESSOR: Yeah,\Nyeah, it's true.
Dialogue: 0,1:52:43.64,1:52:47.11,Default,,0000,0000,0000,,Now I want to ask\Nyou a question.
Dialogue: 0,1:52:47.11,1:52:58.72,Default,,0000,0000,0000,,So do you think that I would get\Nany kind of conservation laws
Dialogue: 0,1:52:58.72,1:53:03.11,Default,,0000,0000,0000,,in physics that\Napply to calculus?
Dialogue: 0,1:53:03.11,1:53:10.36,Default,,0000,0000,0000,,I mean, how hard is it\Nreally to compute the work?
Dialogue: 0,1:53:10.36,1:53:15.01,Default,,0000,0000,0000,,
Dialogue: 0,1:53:15.01,1:53:18.18,Default,,0000,0000,0000,,And I'm making an\Nannouncement now.
Dialogue: 0,1:53:18.18,1:53:22.57,Default,,0000,0000,0000,,Since I have not\Ngiven you a break,
Dialogue: 0,1:53:22.57,1:53:26.96,Default,,0000,0000,0000,,I have to let you\Ngo in a few minutes.
Dialogue: 0,1:53:26.96,1:53:29.91,Default,,0000,0000,0000,,
Dialogue: 0,1:53:29.91,1:53:32.39,Default,,0000,0000,0000,,But I'm making a\Nbig announcement
Dialogue: 0,1:53:32.39,1:53:33.84,Default,,0000,0000,0000,,without proving it.
Dialogue: 0,1:53:33.84,1:53:42.10,Default,,0000,0000,0000,,
Dialogue: 0,1:53:42.10,1:53:46.12,Default,,0000,0000,0000,,So we will, in about\None week at the maximum,
Dialogue: 0,1:53:46.12,1:54:11.53,Default,,0000,0000,0000,,in maximum one week, study the\Nindependence of path of work
Dialogue: 0,1:54:11.53,1:54:24.35,Default,,0000,0000,0000,,if that work is performed\Nby a conservative force.
Dialogue: 0,1:54:24.35,1:54:31.74,Default,,0000,0000,0000,,
Dialogue: 0,1:54:31.74,1:54:33.73,Default,,0000,0000,0000,,And you're going to\Nsay, wait a minute,
Dialogue: 0,1:54:33.73,1:54:37.62,Default,,0000,0000,0000,,what the heck is a conservative\Nforce and what does she mean?
Dialogue: 0,1:54:37.62,1:54:42.48,Default,,0000,0000,0000,,Well, I just showed you that\Nthe work is a path integral.
Dialogue: 0,1:54:42.48,1:54:44.91,Default,,0000,0000,0000,,We don't know what that is.
Dialogue: 0,1:54:44.91,1:54:46.36,Default,,0000,0000,0000,,I'll introduce more.
Dialogue: 0,1:54:46.36,1:54:50.47,Default,,0000,0000,0000,,I just introduced the definition\Nof a path integral with respect
Dialogue: 0,1:54:50.47,1:54:54.24,Default,,0000,0000,0000,,to parametrization, general\Nparametrization with respect
Dialogue: 0,1:54:54.24,1:54:54.74,Default,,0000,0000,0000,,to t.
Dialogue: 0,1:54:54.74,1:54:57.70,Default,,0000,0000,0000,,So that becomes an integral\Nwith respect to dt,
Dialogue: 0,1:54:57.70,1:55:01.02,Default,,0000,0000,0000,,like the one in\NCalc I. This is how
Dialogue: 0,1:55:01.02,1:55:02.66,Default,,0000,0000,0000,,you have to view it at first.
Dialogue: 0,1:55:02.66,1:55:08.18,Default,,0000,0000,0000,,But guys, if this force\Nis not just any force,
Dialogue: 0,1:55:08.18,1:55:26.84,Default,,0000,0000,0000,,it's something magic, if F comes\Nfrom a scalar potential that
Dialogue: 0,1:55:26.84,1:55:33.14,Default,,0000,0000,0000,,is F represents the gradient\Nof a scalar function F--
Dialogue: 0,1:55:33.14,1:55:44.35,Default,,0000,0000,0000,,this is called scalar\Npotential-- then
Dialogue: 0,1:55:44.35,1:55:51.95,Default,,0000,0000,0000,,F is called-- now let's\Nsee how much money I
Dialogue: 0,1:55:51.95,1:55:56.91,Default,,0000,0000,0000,,have for just the last two or\Nthree minutes that I have left.
Dialogue: 0,1:55:56.91,1:55:59.39,Default,,0000,0000,0000,,I don't have money\Nor I have money?
Dialogue: 0,1:55:59.39,1:56:00.90,Default,,0000,0000,0000,,Come on, big money.
Dialogue: 0,1:56:00.90,1:56:03.60,Default,,0000,0000,0000,,
Dialogue: 0,1:56:03.60,1:56:07.03,Default,,0000,0000,0000,,No, I have $5.
Dialogue: 0,1:56:07.03,1:56:10.77,Default,,0000,0000,0000,,I was looking for $1.
Dialogue: 0,1:56:10.77,1:56:14.68,Default,,0000,0000,0000,,Here, I'll give you $5\Nif you give me $4 back
Dialogue: 0,1:56:14.68,1:56:18.17,Default,,0000,0000,0000,,if you guess-- I don't know.
Dialogue: 0,1:56:18.17,1:56:21.16,Default,,0000,0000,0000,,So maybe in your\Nengineering courses-- maybe
Dialogue: 0,1:56:21.16,1:56:23.15,Default,,0000,0000,0000,,I give you some candy instead.
Dialogue: 0,1:56:23.15,1:56:30.64,Default,,0000,0000,0000,,
Dialogue: 0,1:56:30.64,1:56:38.54,Default,,0000,0000,0000,,So if there is a scalar function\Nlittle f of coordinates x,
Dialogue: 0,1:56:38.54,1:56:41.18,Default,,0000,0000,0000,,y, whatever you\Nhave in the problem,
Dialogue: 0,1:56:41.18,1:56:44.43,Default,,0000,0000,0000,,so that big F will be the nabla.
Dialogue: 0,1:56:44.43,1:56:47.07,Default,,0000,0000,0000,,F nabla means the gradient.
Dialogue: 0,1:56:47.07,1:56:50.03,Default,,0000,0000,0000,,We say that F comes\Nfrom a scalar potential.
Dialogue: 0,1:56:50.03,1:57:00.63,Default,,0000,0000,0000,,But it has also a name,\Nwhich is called-- god.
Dialogue: 0,1:57:00.63,1:57:06.11,Default,,0000,0000,0000,,It starts with a C, ends\Nwith an E. In that case,
Dialogue: 0,1:57:06.11,1:57:12.10,Default,,0000,0000,0000,,if this is going to be equal\Nto nabla F, in that case,
Dialogue: 0,1:57:12.10,1:57:14.84,Default,,0000,0000,0000,,there is a magic theorem\Nthat I'm anticipating.
Dialogue: 0,1:57:14.84,1:57:16.66,Default,,0000,0000,0000,,I'm not proving.
Dialogue: 0,1:57:16.66,1:57:18.55,Default,,0000,0000,0000,,I'm doing exercises right now.
Dialogue: 0,1:57:18.55,1:57:22.24,Default,,0000,0000,0000,,We'll see it in two sections,\Nthat the work does not depend
Dialogue: 0,1:57:22.24,1:57:23.94,Default,,0000,0000,0000,,on the path you are taking.
Dialogue: 0,1:57:23.94,1:57:27.24,Default,,0000,0000,0000,,So you can go from A to B like\Nthat, or you can go like this.
Dialogue: 0,1:57:27.24,1:57:28.12,Default,,0000,0000,0000,,You can go like this.
Dialogue: 0,1:57:28.12,1:57:28.61,Default,,0000,0000,0000,,You can go like this.
Dialogue: 0,1:57:28.61,1:57:29.49,Default,,0000,0000,0000,,You can go like that.
Dialogue: 0,1:57:29.49,1:57:33.03,Default,,0000,0000,0000,,You can go on a parabola,\Non a line, on anything.
Dialogue: 0,1:57:33.03,1:57:34.75,Default,,0000,0000,0000,,The result is always the same.
Dialogue: 0,1:57:34.75,1:57:36.29,Default,,0000,0000,0000,,And it's like the\Nfundamental theorem
Dialogue: 0,1:57:36.29,1:57:39.02,Default,,0000,0000,0000,,of Calc III in\Nplane for the work.
Dialogue: 0,1:57:39.02,1:57:42.85,Default,,0000,0000,0000,,So you have little f endpoint.
Dialogue: 0,1:57:42.85,1:57:45.94,Default,,0000,0000,0000,,STUDENT: Is that [INAUDIBLE].
Dialogue: 0,1:57:45.94,1:57:47.98,Default,,0000,0000,0000,,PROFESSOR: Little\Nf of [INAUDIBLE].
Dialogue: 0,1:57:47.98,1:57:51.66,Default,,0000,0000,0000,,So all that matters is\Ncomputing this scalar potential
Dialogue: 0,1:57:51.66,1:57:53.52,Default,,0000,0000,0000,,here and here, making\Nthe difference,
Dialogue: 0,1:57:53.52,1:57:55.01,Default,,0000,0000,0000,,and that will be your work.
Dialogue: 0,1:57:55.01,1:57:56.50,Default,,0000,0000,0000,,It's a magic thing.
Dialogue: 0,1:57:56.50,1:57:58.98,Default,,0000,0000,0000,,In mechanical\Nengineering maybe you
Dialogue: 0,1:57:58.98,1:58:03.44,Default,,0000,0000,0000,,met it, in physics-- in\Nmechanical engineering,
Dialogue: 0,1:58:03.44,1:58:06.66,Default,,0000,0000,0000,,because that's where you guys\Ndrag all sorts of objects
Dialogue: 0,1:58:06.66,1:58:09.39,Default,,0000,0000,0000,,around.
Dialogue: 0,1:58:09.39,1:58:10.88,Default,,0000,0000,0000,,STUDENT: Conservative.
Dialogue: 0,1:58:10.88,1:58:12.86,Default,,0000,0000,0000,,PROFESSOR: Ah, thank god.
Dialogue: 0,1:58:12.86,1:58:16.34,Default,,0000,0000,0000,,Rachel, you're a\Nmath major I think.
Dialogue: 0,1:58:16.34,1:58:18.32,Default,,0000,0000,0000,,You're an engineering major.
Dialogue: 0,1:58:18.32,1:58:20.80,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:58:20.80,1:58:24.27,Default,,0000,0000,0000,,PROFESSOR: Wow, OK, and\Nwho else said conservative?
Dialogue: 0,1:58:24.27,1:58:28.16,Default,,0000,0000,0000,,And were there other people\Nwho said conservative?
Dialogue: 0,1:58:28.16,1:58:29.93,Default,,0000,0000,0000,,I'm sorry I don't have.
Dialogue: 0,1:58:29.93,1:58:32.48,Default,,0000,0000,0000,,Well, next time I'll\Nbring a bunch of dollars,
Dialogue: 0,1:58:32.48,1:58:35.74,Default,,0000,0000,0000,,and I'll start giving\Nprizes as dollar bills
Dialogue: 0,1:58:35.74,1:58:37.96,Default,,0000,0000,0000,,like I used to give in\Ndifferential equations.
Dialogue: 0,1:58:37.96,1:58:39.86,Default,,0000,0000,0000,,Everybody was so\Nhappy in my class.
Dialogue: 0,1:58:39.86,1:58:42.25,Default,,0000,0000,0000,,Because for everything that\Nthey got quickly and right,
Dialogue: 0,1:58:42.25,1:58:44.16,Default,,0000,0000,0000,,they got $1.
Dialogue: 0,1:58:44.16,1:58:47.03,Default,,0000,0000,0000,,So conservative-- very good.
Dialogue: 0,1:58:47.03,1:58:52.16,Default,,0000,0000,0000,,Remember that for\Nthe next few lessons.
Dialogue: 0,1:58:52.16,1:58:57.48,Default,,0000,0000,0000,,We will show that when this f is\Nmagical, that is conservative,
Dialogue: 0,1:58:57.48,1:59:02.73,Default,,0000,0000,0000,,you guys don't have to\Ncompute the integral at all.
Dialogue: 0,1:59:02.73,1:59:05.02,Default,,0000,0000,0000,,There's no parametrization,\Nno nothing.
Dialogue: 0,1:59:05.02,1:59:07.19,Default,,0000,0000,0000,,It really doesn't depend\Non what path you take.
Dialogue: 0,1:59:07.19,1:59:11.13,Default,,0000,0000,0000,,All you would need is to figure\Nwho this little f will be,
Dialogue: 0,1:59:11.13,1:59:12.60,Default,,0000,0000,0000,,this scalar potential.
Dialogue: 0,1:59:12.60,1:59:14.08,Default,,0000,0000,0000,,Our future work can do that.
Dialogue: 0,1:59:14.08,1:59:18.37,Default,,0000,0000,0000,,And then you compute the values\Nof that scalar potential here
Dialogue: 0,1:59:18.37,1:59:19.89,Default,,0000,0000,0000,,and here, make the difference.
Dialogue: 0,1:59:19.89,1:59:23.60,Default,,0000,0000,0000,,And for sure you'll have\Nsuch a problem in the final.
Dialogue: 0,1:59:23.60,1:59:26.19,Default,,0000,0000,0000,,So I'm just anticipating\Nit, because I
Dialogue: 0,1:59:26.19,1:59:32.60,Default,,0000,0000,0000,,want this to be absorbed\Nin time into your system.
Dialogue: 0,1:59:32.60,1:59:34.78,Default,,0000,0000,0000,,When we will do the\Nfinal exam review,
Dialogue: 0,1:59:34.78,1:59:36.81,Default,,0000,0000,0000,,you should be baptized\Nin this kind of problem
Dialogue: 0,1:59:36.81,1:59:41.77,Default,,0000,0000,0000,,so that everybody will get\N100% on that for the final.
Dialogue: 0,1:59:41.77,1:59:43.74,Default,,0000,0000,0000,,OK, now I'll let you go.
Dialogue: 0,1:59:43.74,1:59:45.24,Default,,0000,0000,0000,,Sorry I didn't give you a break.
Dialogue: 0,1:59:45.24,1:59:47.85,Default,,0000,0000,0000,,But now I give you more time.
Dialogue: 0,1:59:47.85,1:59:49.84,Default,,0000,0000,0000,,And enjoy the day.
Dialogue: 0,1:59:49.84,1:59:51.34,Default,,0000,0000,0000,,I'll see you Thursday.
Dialogue: 0,1:59:51.34,1:59:58.82,Default,,0000,0000,0000,,
Dialogue: 0,1:59:58.82,1:59:59.82,Default,,0000,0000,0000,,I'm moving to my office.
Dialogue: 0,1:59:59.82,2:00:02.82,Default,,0000,0000,0000,,If you have questions,\Nyou can come to my office.
Dialogue: 0,2:00:02.82,2:00:06.31,Default,,0000,0000,0000,,
Dialogue: 0,2:00:06.31,2:00:09.30,Default,,0000,0000,0000,,Maybe you were getting close.
Dialogue: 0,2:00:09.30,2:00:13.80,Default,,0000,0000,0000,,How did-- did you know,\Nor it just came to you?
Dialogue: 0,2:00:13.80,2:00:17.79,Default,,0000,0000,0000,,[BACKGROUND CHATTER]
Dialogue: 0,2:00:17.79,2:01:03.52,Default,,0000,0000,0000,,
Dialogue: 0,2:01:03.52,2:01:05.44,Default,,0000,0000,0000,,STUDENT: Do you know\Nwhat section it would be?
Dialogue: 0,2:01:05.44,2:01:09.47,Default,,0000,0000,0000,,Because I don't even think\Nhe's listed or anything.
Dialogue: 0,2:01:09.47,2:01:12.37,Default,,0000,0000,0000,,PROFESSOR: Send me an email\Nif you don't figure it out.
Dialogue: 0,2:01:12.37,2:01:15.35,Default,,0000,0000,0000,,But for sure [INAUDIBLE].
Dialogue: 0,2:01:15.35,2:01:17.83,Default,,0000,0000,0000,,STUDENT: OK, because I was\Ngoing to do the honors,
Dialogue: 0,2:01:17.83,2:01:19.32,Default,,0000,0000,0000,,but it was with [INAUDIBLE].
Dialogue: 0,2:01:19.32,2:01:20.31,Default,,0000,0000,0000,,I don't know if he's\Ngood, or she's good.
Dialogue: 0,2:01:20.31,2:01:21.23,Default,,0000,0000,0000,,PROFESSOR: She's good.
Dialogue: 0,2:01:21.23,2:01:25.37,Default,,0000,0000,0000,,But he's fantastic in the\Nsense that he will help you
Dialogue: 0,2:01:25.37,2:01:27.16,Default,,0000,0000,0000,,whenever you stumble.
Dialogue: 0,2:01:27.16,2:01:29.86,Default,,0000,0000,0000,,He's an extremely good teacher.
Dialogue: 0,2:01:29.86,2:01:31.66,Default,,0000,0000,0000,,He explains really well.
Dialogue: 0,2:01:31.66,2:01:32.86,Default,,0000,0000,0000,,He has a talent.
Dialogue: 0,2:01:32.86,2:01:36.24,Default,,0000,0000,0000,,
Dialogue: 0,2:01:36.24,2:01:37.36,Default,,0000,0000,0000,,STUDENT: I'll look for him.
Dialogue: 0,2:01:37.36,2:01:38.26,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,2:01:38.26,2:01:40.36,Default,,0000,0000,0000,,PROFESSOR: And if\Nyou don't get him,
Dialogue: 0,2:01:40.36,2:01:43.36,Default,,0000,0000,0000,,she is good as well-- not\Nexceptional like he is.
Dialogue: 0,2:01:43.36,2:01:44.86,Default,,0000,0000,0000,,He's an exceptional teacher.
Dialogue: 0,2:01:44.86,2:01:48.76,Default,,0000,0000,0000,,
Dialogue: 0,2:01:48.76,2:01:50.56,Default,,0000,0000,0000,,STUDENT: I'll go to the office.
Dialogue: 0,2:01:50.56,2:01:53.31,Default,,0000,0000,0000,,PROFESSOR: Yes,\Nyes, [INAUDIBLE].
Dialogue: 0,2:01:53.31,2:01:54.91,Default,,0000,0000,0000,,