[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.79,Default,,0000,0000,0000,,
Dialogue: 0,0:00:01.79,0:00:05.02,Default,,0000,0000,0000,,MAGDALENA TODA: Welcome\Nto our review of 13.1.
Dialogue: 0,0:00:05.02,0:00:07.41,Default,,0000,0000,0000,,How many of you didn't\Nget your exams back?
Dialogue: 0,0:00:07.41,0:00:10.38,Default,,0000,0000,0000,,I have your exam, and yours.
Dialogue: 0,0:00:10.38,0:00:11.37,Default,,0000,0000,0000,,And you have to wait.
Dialogue: 0,0:00:11.37,0:00:12.61,Default,,0000,0000,0000,,I don't have it with me.
Dialogue: 0,0:00:12.61,0:00:15.82,Default,,0000,0000,0000,,I have it in my office.
Dialogue: 0,0:00:15.82,0:00:19.59,Default,,0000,0000,0000,,If you have questions\Nabout the score,
Dialogue: 0,0:00:19.59,0:00:23.03,Default,,0000,0000,0000,,why don't you go ahead and\Nemail me right after class.
Dialogue: 0,0:00:23.03,0:00:31.38,Default,,0000,0000,0000,,Chapter 13 is a very\Nphysical chapter.
Dialogue: 0,0:00:31.38,0:00:34.26,Default,,0000,0000,0000,,It has a lot to do with\Nmechanical engineering,
Dialogue: 0,0:00:34.26,0:00:36.97,Default,,0000,0000,0000,,with mechanics,\Nphysics, electricity.
Dialogue: 0,0:00:36.97,0:00:40.67,Default,,0000,0000,0000,,
Dialogue: 0,0:00:40.67,0:00:44.98,Default,,0000,0000,0000,,You're going to see things,\Nweird things like work.
Dialogue: 0,0:00:44.98,0:00:47.11,Default,,0000,0000,0000,,You've already seen work.
Dialogue: 0,0:00:47.11,0:00:49.78,Default,,0000,0000,0000,,Do you remember the definition?
Dialogue: 0,0:00:49.78,0:00:52.96,Default,,0000,0000,0000,,So we define the work\Nas a path integral
Dialogue: 0,0:00:52.96,0:00:55.12,Default,,0000,0000,0000,,along the regular curve.
Dialogue: 0,0:00:55.12,0:00:58.60,Default,,0000,0000,0000,,And by regular curve-- I'm\Nsorry if I'm repeating myself,
Dialogue: 0,0:00:58.60,0:01:02.10,Default,,0000,0000,0000,,but this is part of the\Ndeal-- R is the position
Dialogue: 0,0:01:02.10,0:01:10.00,Default,,0000,0000,0000,,vector in R3 that is class C1.
Dialogue: 0,0:01:10.00,0:01:15.24,Default,,0000,0000,0000,,That means differentiable and\Nderivatives are continuous.
Dialogue: 0,0:01:15.24,0:01:18.62,Default,,0000,0000,0000,,Plus you are not\Nallowed to stop.
Dialogue: 0,0:01:18.62,0:01:24.58,Default,,0000,0000,0000,,So no matter how drunk,\Nthe bug has to keep flying,
Dialogue: 0,0:01:24.58,0:01:27.80,Default,,0000,0000,0000,,and not even for a\Nfraction of a second is he
Dialogue: 0,0:01:27.80,0:01:31.09,Default,,0000,0000,0000,,or she allowed to\Nhave velocity 0.
Dialogue: 0,0:01:31.09,0:01:34.20,Default,,0000,0000,0000,,At no point I want\Nto have velocity 0.
Dialogue: 0,0:01:34.20,0:01:36.30,Default,,0000,0000,0000,,And that's the position vector.
Dialogue: 0,0:01:36.30,0:01:43.62,Default,,0000,0000,0000,,And then you have some force\Nfield acting on you-- no,
Dialogue: 0,0:01:43.62,0:01:47.53,Default,,0000,0000,0000,,acting on the particle\Nat every moment.
Dialogue: 0,0:01:47.53,0:01:54.01,Default,,0000,0000,0000,,So you have an F that is\Nacting at location xy.
Dialogue: 0,0:01:54.01,0:01:58.14,Default,,0000,0000,0000,,Maybe if you are in space,\Nlet's talk about the xyz,
Dialogue: 0,0:01:58.14,0:02:02.13,Default,,0000,0000,0000,,where x is a function of\Nt, y is a functional of t,
Dialogue: 0,0:02:02.13,0:02:06.52,Default,,0000,0000,0000,,z is a function of t,\Nwhich is the same as saying
Dialogue: 0,0:02:06.52,0:02:11.91,Default,,0000,0000,0000,,that R of t, which is the given\Nposition vector, is x of t
Dialogue: 0,0:02:11.91,0:02:12.70,Default,,0000,0000,0000,,y of t.
Dialogue: 0,0:02:12.70,0:02:15.58,Default,,0000,0000,0000,,Let me put angular bracket,\Nalthough I hate them,
Dialogue: 0,0:02:15.58,0:02:20.14,Default,,0000,0000,0000,,because you like angular\Nbrackets for vectors.
Dialogue: 0,0:02:20.14,0:02:22.91,Default,,0000,0000,0000,,F is also a nice function.
Dialogue: 0,0:02:22.91,0:02:24.49,Default,,0000,0000,0000,,How nice?
Dialogue: 0,0:02:24.49,0:02:26.58,Default,,0000,0000,0000,,We discussed a\Nlittle bit last time.
Dialogue: 0,0:02:26.58,0:02:28.86,Default,,0000,0000,0000,,It really doesn't\Nhave to be continuous.
Dialogue: 0,0:02:28.86,0:02:30.65,Default,,0000,0000,0000,,The book assumes it continues.
Dialogue: 0,0:02:30.65,0:02:33.42,Default,,0000,0000,0000,,It has to be\Nintegrable, so maybe it
Dialogue: 0,0:02:33.42,0:02:35.72,Default,,0000,0000,0000,,could be piecewise continuous.
Dialogue: 0,0:02:35.72,0:02:42.60,Default,,0000,0000,0000,,So I had nice enough, was\Nit continues piecewise.
Dialogue: 0,0:02:42.60,0:02:46.39,Default,,0000,0000,0000,,
Dialogue: 0,0:02:46.39,0:02:51.48,Default,,0000,0000,0000,,And we define the work as\Nbeing the path integral over c.
Dialogue: 0,0:02:51.48,0:02:54.73,Default,,0000,0000,0000,,I keep repeating, because\Nthat's going to be on the final
Dialogue: 0,0:02:54.73,0:02:55.79,Default,,0000,0000,0000,,as well.
Dialogue: 0,0:02:55.79,0:02:58.78,Default,,0000,0000,0000,,So all the notions\Nthat are important
Dialogue: 0,0:02:58.78,0:03:02.65,Default,,0000,0000,0000,,should be given enough\Nattention in this class.
Dialogue: 0,0:03:02.65,0:03:03.43,Default,,0000,0000,0000,,Hi.
Dialogue: 0,0:03:03.43,0:03:09.27,Default,,0000,0000,0000,,So do you guys remember\Nhow we denoted F?
Dialogue: 0,0:03:09.27,0:03:15.02,Default,,0000,0000,0000,,F was, in general, three\Ncomponents in our F1, F2, F3.
Dialogue: 0,0:03:15.02,0:03:18.83,Default,,0000,0000,0000,,They are functions of\Nthe position vector,
Dialogue: 0,0:03:18.83,0:03:22.01,Default,,0000,0000,0000,,or the position xyz.
Dialogue: 0,0:03:22.01,0:03:24.74,Default,,0000,0000,0000,,And the position is\Na function of time.
Dialogue: 0,0:03:24.74,0:03:28.38,Default,,0000,0000,0000,,So all in all, after\Nyou do all the work,
Dialogue: 0,0:03:28.38,0:03:35.23,Default,,0000,0000,0000,,keep in mind that when you\Nmultiply with a dot product,
Dialogue: 0,0:03:35.23,0:03:40.09,Default,,0000,0000,0000,,the integral will give you what?
Dialogue: 0,0:03:40.09,0:03:42.42,Default,,0000,0000,0000,,A time integral?
Dialogue: 0,0:03:42.42,0:03:47.21,Default,,0000,0000,0000,,From a time T0 to a time\NT1, you are here at time T0
Dialogue: 0,0:03:47.21,0:03:49.33,Default,,0000,0000,0000,,and you are here at time T1.
Dialogue: 0,0:03:49.33,0:03:51.87,Default,,0000,0000,0000,,
Dialogue: 0,0:03:51.87,0:03:55.65,Default,,0000,0000,0000,,Maybe your curve is\Npiecewise, differentiable,
Dialogue: 0,0:03:55.65,0:03:56.73,Default,,0000,0000,0000,,you don't know what it is.
Dialogue: 0,0:03:56.73,0:04:01.89,Default,,0000,0000,0000,,But let's assume just a\Nvery nice, smooth arc here.
Dialogue: 0,0:04:01.89,0:04:03.03,Default,,0000,0000,0000,,Of what?
Dialogue: 0,0:04:03.03,0:04:11.01,Default,,0000,0000,0000,,Of F1 times what is that?\Nx prime of t plus F2 times
Dialogue: 0,0:04:11.01,0:04:18.16,Default,,0000,0000,0000,,y prime of t, plus F3\Ntimes z prime of t dt.
Dialogue: 0,0:04:18.16,0:04:21.44,Default,,0000,0000,0000,,So keep in mind that\NMr. dR is your friend.
Dialogue: 0,0:04:21.44,0:04:23.42,Default,,0000,0000,0000,,And he was-- what was he?
Dialogue: 0,0:04:23.42,0:04:29.11,Default,,0000,0000,0000,,Was defined as the\Nvelocity vector multiplied
Dialogue: 0,0:04:29.11,0:04:32.38,Default,,0000,0000,0000,,by the infinitesimal element dt.
Dialogue: 0,0:04:32.38,0:04:35.84,Default,,0000,0000,0000,,Say again, the\Nvelocity vector prime
Dialogue: 0,0:04:35.84,0:04:41.61,Default,,0000,0000,0000,,was a vector in F3 quantified\Nby the infinitesimal element dt.
Dialogue: 0,0:04:41.61,0:04:46.14,Default,,0000,0000,0000,,So we reduce this Calc\N3 notion path integral
Dialogue: 0,0:04:46.14,0:04:53.20,Default,,0000,0000,0000,,to a Calc 1 notion, which was a\Nsimple integral from t0 to t1.
Dialogue: 0,0:04:53.20,0:04:55.16,Default,,0000,0000,0000,,And we've done a\Nlot of applications.
Dialogue: 0,0:04:55.16,0:04:56.63,Default,,0000,0000,0000,,What else have we done?
Dialogue: 0,0:04:56.63,0:05:00.43,Default,,0000,0000,0000,,We've done some\Nintegral of this type
Dialogue: 0,0:05:00.43,0:05:03.85,Default,,0000,0000,0000,,over another curve, script c.
Dialogue: 0,0:05:03.85,0:05:06.03,Default,,0000,0000,0000,,I'm repeating mostly for Alex.
Dialogue: 0,0:05:06.03,0:05:10.04,Default,,0000,0000,0000,,You're caught in the process.
Dialogue: 0,0:05:10.04,0:05:14.31,Default,,0000,0000,0000,,And there are two or three\Npeople who need an update.
Dialogue: 0,0:05:14.31,0:05:19.38,Default,,0000,0000,0000,,Maybe I have another\Nfunction of g and ds.
Dialogue: 0,0:05:19.38,0:05:25.30,Default,,0000,0000,0000,,And this is an integral that\Nin the end will depend on s.
Dialogue: 0,0:05:25.30,0:05:27.70,Default,,0000,0000,0000,,But s itself depends on t.
Dialogue: 0,0:05:27.70,0:05:31.41,Default,,0000,0000,0000,,So if I were to re-express\Nthis in terms of d,
Dialogue: 0,0:05:31.41,0:05:34.91,Default,,0000,0000,0000,,how would I re-express\Nthe whole thing?
Dialogue: 0,0:05:34.91,0:05:40.56,Default,,0000,0000,0000,,g of s, of t, whatever that\Nis, then Mr. ds was what?
Dialogue: 0,0:05:40.56,0:05:42.04,Default,,0000,0000,0000,,STUDENT: s prime of t.
Dialogue: 0,0:05:42.04,0:05:42.96,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,0:05:42.96,0:05:47.56,Default,,0000,0000,0000,,So this was the-- that s\Nprime of t was the speed.
Dialogue: 0,0:05:47.56,0:05:51.30,Default,,0000,0000,0000,,The speed of the arc of a curve.
Dialogue: 0,0:05:51.30,0:05:59.31,Default,,0000,0000,0000,,So you have an R of\Nt and R3, a vector.
Dialogue: 0,0:05:59.31,0:06:04.15,Default,,0000,0000,0000,,And the speed was,\Nby definition,
Dialogue: 0,0:06:04.15,0:06:07.24,Default,,0000,0000,0000,,arc length element was\Nby definition integral
Dialogue: 0,0:06:07.24,0:06:09.17,Default,,0000,0000,0000,,from 2t0 to t.
Dialogue: 0,0:06:09.17,0:06:14.74,Default,,0000,0000,0000,,Of the speed R prime\Nmagnitude d tau.
Dialogue: 0,0:06:14.74,0:06:17.71,Default,,0000,0000,0000,,I'll have you put tau\Nbecause I'm Greek,
Dialogue: 0,0:06:17.71,0:06:19.09,Default,,0000,0000,0000,,and it's all Greek to me.
Dialogue: 0,0:06:19.09,0:06:23.36,Default,,0000,0000,0000,,So the tau, some people call\Nthe tau the dummy variable.
Dialogue: 0,0:06:23.36,0:06:24.93,Default,,0000,0000,0000,,I don't like to call it dumb.
Dialogue: 0,0:06:24.93,0:06:27.68,Default,,0000,0000,0000,,It's a very smart variable.
Dialogue: 0,0:06:27.68,0:06:31.16,Default,,0000,0000,0000,,It goes from t0 to t, so what\Nyou have is a function of t.
Dialogue: 0,0:06:31.16,0:06:32.90,Default,,0000,0000,0000,,This guy is speed.
Dialogue: 0,0:06:32.90,0:06:38.88,Default,,0000,0000,0000,,So when you do that\Nhere, ds becomes speed,
Dialogue: 0,0:06:38.88,0:06:43.23,Default,,0000,0000,0000,,R prime of t times dt.
Dialogue: 0,0:06:43.23,0:06:45.29,Default,,0000,0000,0000,,This was your old friend ds.
Dialogue: 0,0:06:45.29,0:06:50.75,Default,,0000,0000,0000,,And let me put it on top\Nof this guy with speed.
Dialogue: 0,0:06:50.75,0:06:53.63,Default,,0000,0000,0000,,Because he was so\Nimportant to you,
Dialogue: 0,0:06:53.63,0:06:56.34,Default,,0000,0000,0000,,you cannot forget about him.
Dialogue: 0,0:06:56.34,0:07:03.85,Default,,0000,0000,0000,,So that was review of--\Nreviewing of 13.1 and 13.2
Dialogue: 0,0:07:03.85,0:07:10.24,Default,,0000,0000,0000,,There were some things in\N13.3 that I pointed out
Dialogue: 0,0:07:10.24,0:07:12.40,Default,,0000,0000,0000,,to you are important.
Dialogue: 0,0:07:12.40,0:07:16.99,Default,,0000,0000,0000,,13.3 was independence of path.
Dialogue: 0,0:07:16.99,0:07:18.92,Default,,0000,0000,0000,,Everybody write, magic-- no.
Dialogue: 0,0:07:18.92,0:07:20.80,Default,,0000,0000,0000,,Magic section.
Dialogue: 0,0:07:20.80,0:07:22.47,Default,,0000,0000,0000,,No, have to be serious.
Dialogue: 0,0:07:22.47,0:07:31.95,Default,,0000,0000,0000,,So that's independence of path\Nof certain type of integrals,
Dialogue: 0,0:07:31.95,0:07:35.08,Default,,0000,0000,0000,,of some integrals.
Dialogue: 0,0:07:35.08,0:07:37.73,Default,,0000,0000,0000,,And an integral like\Nthat, a path integral
Dialogue: 0,0:07:37.73,0:07:41.06,Default,,0000,0000,0000,,is independent of path.
Dialogue: 0,0:07:41.06,0:07:44.53,Default,,0000,0000,0000,,When would such an animal--\Nlook at this pink animal,
Dialogue: 0,0:07:44.53,0:07:47.03,Default,,0000,0000,0000,,inside-- when would\Nthis not depend
Dialogue: 0,0:07:47.03,0:07:51.23,Default,,0000,0000,0000,,on the path you are taking\Nbetween two given points?
Dialogue: 0,0:07:51.23,0:07:55.13,Default,,0000,0000,0000,,So I can move on another\Narc and another arc
Dialogue: 0,0:07:55.13,0:07:59.32,Default,,0000,0000,0000,,and another regular arc, and\Nall sorts of regular arcs.
Dialogue: 0,0:07:59.32,0:08:02.40,Default,,0000,0000,0000,,It doesn't matter\Nwhich path I'm taking--
Dialogue: 0,0:08:02.40,0:08:04.12,Default,,0000,0000,0000,,STUDENT: If that\Nforce is conservative.
Dialogue: 0,0:08:04.12,0:08:05.100,Default,,0000,0000,0000,,MAGDALENA TODA: If the\Nforce is conservative.
Dialogue: 0,0:08:05.100,0:08:07.16,Default,,0000,0000,0000,,Excellent, Alex.
Dialogue: 0,0:08:07.16,0:08:12.77,Default,,0000,0000,0000,,And what did it mean for a\Nforce to be conservative?
Dialogue: 0,0:08:12.77,0:08:15.88,Default,,0000,0000,0000,,How many of you\Nknow-- it's no shame.
Dialogue: 0,0:08:15.88,0:08:17.35,Default,,0000,0000,0000,,Just raise hands.
Dialogue: 0,0:08:17.35,0:08:19.75,Default,,0000,0000,0000,,If you forgot what it is,\Ndon't raise your hand.
Dialogue: 0,0:08:19.75,0:08:23.35,Default,,0000,0000,0000,,But if you remember what it\Nmeans for a force F force
Dialogue: 0,0:08:23.35,0:08:27.34,Default,,0000,0000,0000,,field-- may the force be\Nwith you-- be conservative,
Dialogue: 0,0:08:27.34,0:08:30.16,Default,,0000,0000,0000,,then what do you do?
Dialogue: 0,0:08:30.16,0:08:33.15,Default,,0000,0000,0000,,Say F is conservative\Nby definition.
Dialogue: 0,0:08:33.15,0:08:38.75,Default,,0000,0000,0000,,
Dialogue: 0,0:08:38.75,0:08:50.08,Default,,0000,0000,0000,,When, if and only, F\Nthere is a so-called--
Dialogue: 0,0:08:50.08,0:08:50.75,Default,,0000,0000,0000,,STUDENT: Scalar.
Dialogue: 0,0:08:50.75,0:08:51.92,Default,,0000,0000,0000,,MAGDALENA TODA: --potential.
Dialogue: 0,0:08:51.92,0:08:53.53,Default,,0000,0000,0000,,Scalar potential, thank you.
Dialogue: 0,0:08:53.53,0:08:54.11,Default,,0000,0000,0000,,I'll fix that.
Dialogue: 0,0:08:54.11,0:09:01.77,Default,,0000,0000,0000,,A scalar potential function f.
Dialogue: 0,0:09:01.77,0:09:07.13,Default,,0000,0000,0000,,
Dialogue: 0,0:09:07.13,0:09:09.85,Default,,0000,0000,0000,,Instead of there is, I\Ndidn't want to put this.
Dialogue: 0,0:09:09.85,0:09:11.65,Default,,0000,0000,0000,,Because a few\Npeople told me they
Dialogue: 0,0:09:11.65,0:09:14.57,Default,,0000,0000,0000,,got scared about\Nthe symbolistics.
Dialogue: 0,0:09:14.57,0:09:17.15,Default,,0000,0000,0000,,This means "there exists."
Dialogue: 0,0:09:17.15,0:09:22.38,Default,,0000,0000,0000,,OK, smooth potential\Nsuch that-- at least
Dialogue: 0,0:09:22.38,0:09:26.80,Default,,0000,0000,0000,,is differential [INAUDIBLE]\N1 such that the nabla of f--
Dialogue: 0,0:09:26.80,0:09:28.24,Default,,0000,0000,0000,,what the heck is that?
Dialogue: 0,0:09:28.24,0:09:32.18,Default,,0000,0000,0000,,The gradient of this little\Nf will be the given F.
Dialogue: 0,0:09:32.18,0:09:38.50,Default,,0000,0000,0000,,And we saw all sorts of wizards\Nhere, like, Harry Potter,
Dialogue: 0,0:09:38.50,0:09:42.93,Default,,0000,0000,0000,,[INAUDIBLE] well,\Nthere are many,
Dialogue: 0,0:09:42.93,0:09:47.67,Default,,0000,0000,0000,,Alex, Erin, many,\Nmany-- Matthew.
Dialogue: 0,0:09:47.67,0:09:49.54,Default,,0000,0000,0000,,So what did they do?
Dialogue: 0,0:09:49.54,0:09:50.96,Default,,0000,0000,0000,,They guessed the\Nscalar potential.
Dialogue: 0,0:09:50.96,0:09:53.37,Default,,0000,0000,0000,,I had to stop because\Nthere are 10 of them.
Dialogue: 0,0:09:53.37,0:09:55.94,Default,,0000,0000,0000,,It's a whole school\Nof Harry Potter.
Dialogue: 0,0:09:55.94,0:09:58.76,Default,,0000,0000,0000,,How do they find the little f?
Dialogue: 0,0:09:58.76,0:10:00.79,Default,,0000,0000,0000,,Through witchcraft.
Dialogue: 0,0:10:00.79,0:10:01.86,Default,,0000,0000,0000,,No.
Dialogue: 0,0:10:01.86,0:10:02.74,Default,,0000,0000,0000,,Normally you should--
Dialogue: 0,0:10:02.74,0:10:04.82,Default,,0000,0000,0000,,STUDENT: I've actually\Ndone it through witchcraft.
Dialogue: 0,0:10:04.82,0:10:05.51,Default,,0000,0000,0000,,Tell you that?
Dialogue: 0,0:10:05.51,0:10:06.50,Default,,0000,0000,0000,,MAGDALENA TODA: You did.
Dialogue: 0,0:10:06.50,0:10:08.77,Default,,0000,0000,0000,,I think you can do it\Nthrough witchcraft.
Dialogue: 0,0:10:08.77,0:10:15.08,Default,,0000,0000,0000,,But practically everybody\Nhas the ability to guess.
Dialogue: 0,0:10:15.08,0:10:17.51,Default,,0000,0000,0000,,Why do we have the ability\Nto guess and check?
Dialogue: 0,0:10:17.51,0:10:21.21,Default,,0000,0000,0000,,Because our brain does\Nthe integration for you.
Dialogue: 0,0:10:21.21,0:10:24.16,Default,,0000,0000,0000,,Whether you tell your\Nbrain to stop or not,
Dialogue: 0,0:10:24.16,0:10:27.25,Default,,0000,0000,0000,,when your brain, for example,\Nsees is kind of function--
Dialogue: 0,0:10:27.25,0:10:30.34,Default,,0000,0000,0000,,and now I'm gonna\Ntest your magic skills
Dialogue: 0,0:10:30.34,0:10:32.04,Default,,0000,0000,0000,,on a little harder one.
Dialogue: 0,0:10:32.04,0:10:36.04,Default,,0000,0000,0000,,I didn't want to do an\NR2 value vector function.
Dialogue: 0,0:10:36.04,0:10:38.07,Default,,0000,0000,0000,,Let me go to R3.
Dialogue: 0,0:10:38.07,0:10:42.07,Default,,0000,0000,0000,,But I know that you have\Nyour witchcraft handy.
Dialogue: 0,0:10:42.07,0:10:47.65,Default,,0000,0000,0000,,So let's say somebody\Ngave you a force field
Dialogue: 0,0:10:47.65,0:10:55.62,Default,,0000,0000,0000,,that is yz i plus xzj plus xyk.
Dialogue: 0,0:10:55.62,0:10:58.90,Default,,0000,0000,0000,,And you're going to jump and\Nsay this is a piece of cake.
Dialogue: 0,0:10:58.90,0:11:03.68,Default,,0000,0000,0000,,I can see the scalar potential\Nand just wave my magic wand,
Dialogue: 0,0:11:03.68,0:11:06.95,Default,,0000,0000,0000,,and I get it.
Dialogue: 0,0:11:06.95,0:11:08.10,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:11:08.10,0:11:09.39,Default,,0000,0000,0000,,MAGDALENA TODA: Oh my god, yes.
Dialogue: 0,0:11:09.39,0:11:11.00,Default,,0000,0000,0000,,Guys, you saw it fast.
Dialogue: 0,0:11:11.00,0:11:12.78,Default,,0000,0000,0000,,OK, I should be proud of you.
Dialogue: 0,0:11:12.78,0:11:14.30,Default,,0000,0000,0000,,And I am proud of you.
Dialogue: 0,0:11:14.30,0:11:18.61,Default,,0000,0000,0000,,I've had made classes\Nwhere the students couldn't
Dialogue: 0,0:11:18.61,0:11:21.93,Default,,0000,0000,0000,,see any of the scalar\Npotentials that I gave them,
Dialogue: 0,0:11:21.93,0:11:23.93,Default,,0000,0000,0000,,that I asked them to guess.
Dialogue: 0,0:11:23.93,0:11:25.38,Default,,0000,0000,0000,,How did you deal with it?
Dialogue: 0,0:11:25.38,0:11:27.50,Default,,0000,0000,0000,,You integrate this\Nwith respect to F?
Dialogue: 0,0:11:27.50,0:11:29.93,Default,,0000,0000,0000,,In the back of\Nyour mind you did.
Dialogue: 0,0:11:29.93,0:11:31.51,Default,,0000,0000,0000,,And then you guessed\None, and then you
Dialogue: 0,0:11:31.51,0:11:34.12,Default,,0000,0000,0000,,said, OK so should be xyz.
Dialogue: 0,0:11:34.12,0:11:36.79,Default,,0000,0000,0000,,Does it verify my\Nother two conditions?
Dialogue: 0,0:11:36.79,0:11:38.46,Default,,0000,0000,0000,,And you say, oh yeah, it does.
Dialogue: 0,0:11:38.46,0:11:42.55,Default,,0000,0000,0000,,Because of I prime with respect\Nto y, I have exactly xz.
Dialogue: 0,0:11:42.55,0:11:46.51,Default,,0000,0000,0000,,If I prime with respect to c I\Nhave exactly xy, so I got it.
Dialogue: 0,0:11:46.51,0:11:49.56,Default,,0000,0000,0000,,And even if somebody\Nsaid xyz plus 7,
Dialogue: 0,0:11:49.56,0:11:51.16,Default,,0000,0000,0000,,they would still be right.
Dialogue: 0,0:11:51.16,0:11:56.06,Default,,0000,0000,0000,,In the end you can have\Nany xyz plus a constant.
Dialogue: 0,0:11:56.06,0:11:58.26,Default,,0000,0000,0000,,In general it's not\Nso easy to guess.
Dialogue: 0,0:11:58.26,0:12:02.02,Default,,0000,0000,0000,,But there are lots of examples\Nof conservative forces where
Dialogue: 0,0:12:02.02,0:12:06.59,Default,,0000,0000,0000,,you simply cannot see the scalar\Npotential or cannot deduce it
Dialogue: 0,0:12:06.59,0:12:09.85,Default,,0000,0000,0000,,like in a few seconds.
Dialogue: 0,0:12:09.85,0:12:12.97,Default,,0000,0000,0000,,Expect something easy,\Nthough, like that,
Dialogue: 0,0:12:12.97,0:12:14.94,Default,,0000,0000,0000,,something that you can see.
Dialogue: 0,0:12:14.94,0:12:15.82,Default,,0000,0000,0000,,Let's see an example.
Dialogue: 0,0:12:15.82,0:12:19.26,Default,,0000,0000,0000,,Assume this is your force field\Nacting on a particle that's
Dialogue: 0,0:12:19.26,0:12:23.51,Default,,0000,0000,0000,,moving on a curving space.
Dialogue: 0,0:12:23.51,0:12:29.13,Default,,0000,0000,0000,,And it's stubborn and it\Ndecides to move on a helix,
Dialogue: 0,0:12:29.13,0:12:32.56,Default,,0000,0000,0000,,because it's a-- I don't\Nknow what kind of particle
Dialogue: 0,0:12:32.56,0:12:34.70,Default,,0000,0000,0000,,would move on a\Nhelix, but suppose
Dialogue: 0,0:12:34.70,0:12:39.49,Default,,0000,0000,0000,,a lot of particles, just a\Nlittle train or a drunken bug
Dialogue: 0,0:12:39.49,0:12:40.69,Default,,0000,0000,0000,,or something.
Dialogue: 0,0:12:40.69,0:12:45.07,Default,,0000,0000,0000,,And you were moving\Non another helix.
Dialogue: 0,0:12:45.07,0:12:52.17,Default,,0000,0000,0000,,Now suppose that helix will\Nbe R of t equals cosine t
Dialogue: 0,0:12:52.17,0:13:01.11,Default,,0000,0000,0000,,sine t and t where you\Nhave t as 0 to start with.
Dialogue: 0,0:13:01.11,0:13:02.35,Default,,0000,0000,0000,,What do I have at 0?
Dialogue: 0,0:13:02.35,0:13:05.64,Default,,0000,0000,0000,,The point 1, 0, 0.
Dialogue: 0,0:13:05.64,0:13:09.12,Default,,0000,0000,0000,,That's the point,\Nlet's call it A.
Dialogue: 0,0:13:09.12,0:13:12.87,Default,,0000,0000,0000,,And let's call this B. I\Ndon't know what I want to do.
Dialogue: 0,0:13:12.87,0:13:16.58,Default,,0000,0000,0000,,I'll just do a\Ncomplete rotation,
Dialogue: 0,0:13:16.58,0:13:18.51,Default,,0000,0000,0000,,just to make my life easier.
Dialogue: 0,0:13:18.51,0:13:23.12,Default,,0000,0000,0000,,And this is B. And that\Nwill be A at t equals 0
Dialogue: 0,0:13:23.12,0:13:25.16,Default,,0000,0000,0000,,and B equals 2 pi.
Dialogue: 0,0:13:25.16,0:13:32.84,Default,,0000,0000,0000,,
Dialogue: 0,0:13:32.84,0:13:35.35,Default,,0000,0000,0000,,So what will this be at B?
Dialogue: 0,0:13:35.35,0:13:37.12,Default,,0000,0000,0000,,STUDENT: 1, 0, 2 pi.
Dialogue: 0,0:13:37.12,0:13:40.43,Default,,0000,0000,0000,,MAGDALENA TODA: 1, 0, and 2 pi.
Dialogue: 0,0:13:40.43,0:13:44.32,Default,,0000,0000,0000,,So you perform a complete\Nrotation and come back.
Dialogue: 0,0:13:44.32,0:13:49.75,Default,,0000,0000,0000,,Now, if your force is\Nconservative, you are lucky.
Dialogue: 0,0:13:49.75,0:13:53.10,Default,,0000,0000,0000,,Because you know the theorem\Nthat says in that case
Dialogue: 0,0:13:53.10,0:13:57.82,Default,,0000,0000,0000,,the work integral will\Nbe independent of path.
Dialogue: 0,0:13:57.82,0:14:03.91,Default,,0000,0000,0000,,And due to the theorem in-- what\Nsection was that again-- 13.3,
Dialogue: 0,0:14:03.91,0:14:06.89,Default,,0000,0000,0000,,independence of path,\Nyou know that this
Dialogue: 0,0:14:06.89,0:14:11.72,Default,,0000,0000,0000,,is going to be-- let me rewrite\Nit one more time with gradient
Dialogue: 0,0:14:11.72,0:14:16.15,Default,,0000,0000,0000,,of f instead of big F.
Dialogue: 0,0:14:16.15,0:14:19.62,Default,,0000,0000,0000,,And this will become what,\Nf of the q-- not the q.
Dialogue: 0,0:14:19.62,0:14:24.06,Default,,0000,0000,0000,,In the book it's f of q minus\Nf of q. f of B minus f of A,
Dialogue: 0,0:14:24.06,0:14:24.56,Default,,0000,0000,0000,,right?
Dialogue: 0,0:14:24.56,0:14:28.05,Default,,0000,0000,0000,,
Dialogue: 0,0:14:28.05,0:14:29.03,Default,,0000,0000,0000,,What does this mean?
Dialogue: 0,0:14:29.03,0:14:33.41,Default,,0000,0000,0000,,You have to measure\Nthe-- to evaluate
Dialogue: 0,0:14:33.41,0:14:39.44,Default,,0000,0000,0000,,the coordinates of\Nthis function xyz
Dialogue: 0,0:14:39.44,0:14:50.08,Default,,0000,0000,0000,,where t equals 2 pi minus\Nxyz where t equals what?
Dialogue: 0,0:14:50.08,0:14:52.40,Default,,0000,0000,0000,,0.
Dialogue: 0,0:14:52.40,0:14:56.83,Default,,0000,0000,0000,,And now I have to be\Ncareful, because I
Dialogue: 0,0:14:56.83,0:14:58.20,Default,,0000,0000,0000,,have to evaluate them.
Dialogue: 0,0:14:58.20,0:15:06.63,Default,,0000,0000,0000,,So when t is 0 I have x\Nis 1, y is 0, and t is 0.
Dialogue: 0,0:15:06.63,0:15:08.10,Default,,0000,0000,0000,,In the end it doesn't matter.
Dialogue: 0,0:15:08.10,0:15:12.53,Default,,0000,0000,0000,,I can get 0-- I\Ncan get 0 for this
Dialogue: 0,0:15:12.53,0:15:14.70,Default,,0000,0000,0000,,and get 0 for that as well.
Dialogue: 0,0:15:14.70,0:15:20.60,Default,,0000,0000,0000,,So when this is 2 pi I get\Nx equals 1, y equals 0,
Dialogue: 0,0:15:20.60,0:15:23.21,Default,,0000,0000,0000,,and t equals 2 pi.
Dialogue: 0,0:15:23.21,0:15:26.65,Default,,0000,0000,0000,,So in the end, both products\Nare 0 and I got a 0.
Dialogue: 0,0:15:26.65,0:15:31.89,Default,,0000,0000,0000,,So although the [INAUDIBLE]\Nworks very hard-- I mean,
Dialogue: 0,0:15:31.89,0:15:36.45,Default,,0000,0000,0000,,works hard in our perception\Nto get from a point
Dialogue: 0,0:15:36.45,0:15:38.73,Default,,0000,0000,0000,,to another-- the work is 0.
Dialogue: 0,0:15:38.73,0:15:39.38,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:15:39.38,0:15:42.47,Default,,0000,0000,0000,,Because it's a vector\Nvalue thing inside.
Dialogue: 0,0:15:42.47,0:15:46.56,Default,,0000,0000,0000,,And there are some\Nannihilations going on.
Dialogue: 0,0:15:46.56,0:15:50.47,Default,,0000,0000,0000,,So that reminds me\Nof another example.
Dialogue: 0,0:15:50.47,0:15:52.83,Default,,0000,0000,0000,,So we are done\Nwith this example.
Dialogue: 0,0:15:52.83,0:15:55.65,Default,,0000,0000,0000,,Let's go back to our washer.
Dialogue: 0,0:15:55.65,0:15:57.56,Default,,0000,0000,0000,,I was just doing\Nlaundry last night
Dialogue: 0,0:15:57.56,0:16:01.36,Default,,0000,0000,0000,,and I was thinking of\Nthe washer example.
Dialogue: 0,0:16:01.36,0:16:04.96,Default,,0000,0000,0000,,And I thought of a small\Nvariation of the washer
Dialogue: 0,0:16:04.96,0:16:09.43,Default,,0000,0000,0000,,example, just assuming that\NI would give you a pop quiz.
Dialogue: 0,0:16:09.43,0:16:12.12,Default,,0000,0000,0000,,And I'm not giving you\Na pop quiz right now.
Dialogue: 0,0:16:12.12,0:16:14.94,Default,,0000,0000,0000,,But if I gave you\Na pop quiz now,
Dialogue: 0,0:16:14.94,0:16:20.85,Default,,0000,0000,0000,,I would ask you example\Ntwo, the washer.
Dialogue: 0,0:16:20.85,0:16:24.41,Default,,0000,0000,0000,,
Dialogue: 0,0:16:24.41,0:16:27.77,Default,,0000,0000,0000,,It is performing\Na circular motion,
Dialogue: 0,0:16:27.77,0:16:30.00,Default,,0000,0000,0000,,and I want to know\Nthe work performed
Dialogue: 0,0:16:30.00,0:16:36.32,Default,,0000,0000,0000,,by the centrifugal force\Nbetween various points.
Dialogue: 0,0:16:36.32,0:16:48.18,Default,,0000,0000,0000,,So have the circular motion,\Nthe centrifugal force.
Dialogue: 0,0:16:48.18,0:16:50.89,Default,,0000,0000,0000,,This is the\Ncentrifugal, I'm sorry.
Dialogue: 0,0:16:50.89,0:16:54.06,Default,,0000,0000,0000,,I'll take the centrifugal force.
Dialogue: 0,0:16:54.06,0:16:56.48,Default,,0000,0000,0000,,And that was last\Ntime we discussed
Dialogue: 0,0:16:56.48,0:17:04.96,Default,,0000,0000,0000,,that, that was extending\Nthe radius of the initial--
Dialogue: 0,0:17:04.96,0:17:07.22,Default,,0000,0000,0000,,the vector value position.
Dialogue: 0,0:17:07.22,0:17:11.56,Default,,0000,0000,0000,,So you have that in\Nevery point, xi plus yj.
Dialogue: 0,0:17:11.56,0:17:14.67,Default,,0000,0000,0000,,And you want F to\Nbe able xi plus yj.
Dialogue: 0,0:17:14.67,0:17:20.37,Default,,0000,0000,0000,,But it points outside\Nfrom the point
Dialogue: 0,0:17:20.37,0:17:22.96,Default,,0000,0000,0000,,on the circular trajectory.
Dialogue: 0,0:17:22.96,0:17:26.04,Default,,0000,0000,0000,,
Dialogue: 0,0:17:26.04,0:17:31.35,Default,,0000,0000,0000,,And I asked you, find\Nout what you performed
Dialogue: 0,0:17:31.35,0:17:38.69,Default,,0000,0000,0000,,by F in one full rotation.
Dialogue: 0,0:17:38.69,0:17:43.76,Default,,0000,0000,0000,,
Dialogue: 0,0:17:43.76,0:17:48.98,Default,,0000,0000,0000,,We gave the equation of motion,\Nbeing cosine t y sine t,
Dialogue: 0,0:17:48.98,0:17:51.56,Default,,0000,0000,0000,,if you remember from last time.
Dialogue: 0,0:17:51.56,0:17:57.39,Default,,0000,0000,0000,,And then W2, let's\Nsay, is performed by F
Dialogue: 0,0:17:57.39,0:18:01.91,Default,,0000,0000,0000,,from t equals 0 to t equals pi.
Dialogue: 0,0:18:01.91,0:18:03.50,Default,,0000,0000,0000,,I want that as well.
Dialogue: 0,0:18:03.50,0:18:12.57,Default,,0000,0000,0000,,And W2 performed by F from\Nt-- that makes t0 to t
Dialogue: 0,0:18:12.57,0:18:20.03,Default,,0000,0000,0000,,equals pi-- t equals 0\Nto t equals pi over 4.
Dialogue: 0,0:18:20.03,0:18:21.83,Default,,0000,0000,0000,,These are all very\Neasy questions,
Dialogue: 0,0:18:21.83,0:18:24.97,Default,,0000,0000,0000,,and you should be able to\Nanswer them in no time.
Dialogue: 0,0:18:24.97,0:18:28.43,Default,,0000,0000,0000,,Now, let me tell you something.
Dialogue: 0,0:18:28.43,0:18:29.71,Default,,0000,0000,0000,,We are in plane, not in space.
Dialogue: 0,0:18:29.71,0:18:30.79,Default,,0000,0000,0000,,But it doesn't matter.
Dialogue: 0,0:18:30.79,0:18:35.44,Default,,0000,0000,0000,,It's like the third quadrant\Nwould be 0, piece of cake.
Dialogue: 0,0:18:35.44,0:18:39.100,Default,,0000,0000,0000,,Your eye should be so\Nwell-trained that when
Dialogue: 0,0:18:39.100,0:18:41.58,Default,,0000,0000,0000,,you look at the force\Nfield like that,
Dialogue: 0,0:18:41.58,0:18:44.16,Default,,0000,0000,0000,,and people talk about what\Nyou should ask yourself,
Dialogue: 0,0:18:44.16,0:18:44.95,Default,,0000,0000,0000,,is it conservative?
Dialogue: 0,0:18:44.95,0:18:48.39,Default,,0000,0000,0000,,
Dialogue: 0,0:18:48.39,0:18:51.01,Default,,0000,0000,0000,,And it is conservative.
Dialogue: 0,0:18:51.01,0:18:53.83,Default,,0000,0000,0000,,And that means little f is what?
Dialogue: 0,0:18:53.83,0:18:56.90,Default,,0000,0000,0000,,
Dialogue: 0,0:18:56.90,0:18:59.70,Default,,0000,0000,0000,,Nitish said that yesterday.
Dialogue: 0,0:18:59.70,0:19:00.86,Default,,0000,0000,0000,,Why did you go there?
Dialogue: 0,0:19:00.86,0:19:02.63,Default,,0000,0000,0000,,You want to sleep today?
Dialogue: 0,0:19:02.63,0:19:05.51,Default,,0000,0000,0000,,I'm just teasing you.
Dialogue: 0,0:19:05.51,0:19:08.47,Default,,0000,0000,0000,,I got so comfortable with\Nyou sitting in the front row.
Dialogue: 0,0:19:08.47,0:19:10.07,Default,,0000,0000,0000,,STUDENT: I took his spot.
Dialogue: 0,0:19:10.07,0:19:12.07,Default,,0000,0000,0000,,STUDENT: She doesn't like\Nyou sitting over here.
Dialogue: 0,0:19:12.07,0:19:13.07,Default,,0000,0000,0000,,MAGDALENA TODA: It's OK.
Dialogue: 0,0:19:13.07,0:19:13.92,Default,,0000,0000,0000,,It's fine.
Dialogue: 0,0:19:13.92,0:19:16.71,Default,,0000,0000,0000,,I still give him credit\Nfor what he said last time.
Dialogue: 0,0:19:16.71,0:19:19.57,Default,,0000,0000,0000,,So do you guys remember,\Nhe gave us this answer?
Dialogue: 0,0:19:19.57,0:19:23.07,Default,,0000,0000,0000,,x squared plus y squared over\N2, and he found the scalar
Dialogue: 0,0:19:23.07,0:19:26.98,Default,,0000,0000,0000,,potential through witchcraft\Nin about a second and a half?
Dialogue: 0,0:19:26.98,0:19:28.19,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:19:28.19,0:19:31.43,Default,,0000,0000,0000,,We are gonna conclude something.
Dialogue: 0,0:19:31.43,0:19:36.42,Default,,0000,0000,0000,,Do you remember that I found the\Nanswer by find the explanation?
Dialogue: 0,0:19:36.42,0:19:39.16,Default,,0000,0000,0000,,I got W to be 0.
Dialogue: 0,0:19:39.16,0:19:45.34,Default,,0000,0000,0000,,But if I were to find another\Nexplanation why the work would
Dialogue: 0,0:19:45.34,0:19:50.18,Default,,0000,0000,0000,,be 0 in this case, it\Nwould have been 0 anyway
Dialogue: 0,0:19:50.18,0:19:53.22,Default,,0000,0000,0000,,for any force field.
Dialogue: 0,0:19:53.22,0:19:58.12,Default,,0000,0000,0000,,Even if I took the F\Nto be something else.
Dialogue: 0,0:19:58.12,0:20:03.32,Default,,0000,0000,0000,,Assume that F would be\NG. Really wild, crazy,
Dialogue: 0,0:20:03.32,0:20:06.97,Default,,0000,0000,0000,,but still differentiable\Nvector value function.
Dialogue: 0,0:20:06.97,0:20:08.70,Default,,0000,0000,0000,,G differential.
Dialogue: 0,0:20:08.70,0:20:15.05,Default,,0000,0000,0000,,Would the work that we\Nwant be the same for G?
Dialogue: 0,0:20:15.05,0:20:15.73,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:20:15.73,0:20:16.56,Default,,0000,0000,0000,,MAGDALENA TODA: Why?
Dialogue: 0,0:20:16.56,0:20:18.31,Default,,0000,0000,0000,,STUDENT: Because of\Ndisplacement scenario.
Dialogue: 0,0:20:18.31,0:20:20.68,Default,,0000,0000,0000,,MAGDALENA TODA: Since\Nit's conservative,
Dialogue: 0,0:20:20.68,0:20:23.15,Default,,0000,0000,0000,,you have a closed loop.
Dialogue: 0,0:20:23.15,0:20:25.80,Default,,0000,0000,0000,,So the closed loop\Nwill say, thick F
Dialogue: 0,0:20:25.80,0:20:30.34,Default,,0000,0000,0000,,at that terminal point minus\Nthick F at the initial point.
Dialogue: 0,0:20:30.34,0:20:33.62,Default,,0000,0000,0000,,But if a loop motion,\Nyour terminal point
Dialogue: 0,0:20:33.62,0:20:35.29,Default,,0000,0000,0000,,is the initial point.
Dialogue: 0,0:20:35.29,0:20:36.04,Default,,0000,0000,0000,,Duh.
Dialogue: 0,0:20:36.04,0:20:40.90,Default,,0000,0000,0000,,So you have the\Nsame point, the P
Dialogue: 0,0:20:40.90,0:20:44.49,Default,,0000,0000,0000,,equals qe if it's\Na closed curve.
Dialogue: 0,0:20:44.49,0:20:48.29,Default,,0000,0000,0000,,So for a closed curve--\Nwe also call that a loop.
Dialogue: 0,0:20:48.29,0:20:50.21,Default,,0000,0000,0000,,With a basketball, it\Nwould have been too easy
Dialogue: 0,0:20:50.21,0:20:54.92,Default,,0000,0000,0000,,and you would have gotten a\Ndollar for free like that.
Dialogue: 0,0:20:54.92,0:20:57.93,Default,,0000,0000,0000,,So any closed curve\Nis called a loop.
Dialogue: 0,0:20:57.93,0:21:01.61,Default,,0000,0000,0000,,If your force field is\Nconservative-- attention,
Dialogue: 0,0:21:01.61,0:21:05.39,Default,,0000,0000,0000,,you might have examples\Nlike that in the exams--
Dialogue: 0,0:21:05.39,0:21:08.99,Default,,0000,0000,0000,,then it doesn't matter\Nwho little f is,
Dialogue: 0,0:21:08.99,0:21:12.51,Default,,0000,0000,0000,,if p equals q you get 0 anyway.
Dialogue: 0,0:21:12.51,0:21:15.75,Default,,0000,0000,0000,,But the reason why I\Nsaid you would get 0
Dialogue: 0,0:21:15.75,0:21:20.95,Default,,0000,0000,0000,,on the example of last time\Nwas a slightly different one.
Dialogue: 0,0:21:20.95,0:21:24.37,Default,,0000,0000,0000,,What does the engineer\Nsay to himself?
Dialogue: 0,0:21:24.37,0:21:25.70,Default,,0000,0000,0000,,STUDENT: Force is perpendicular.
Dialogue: 0,0:21:25.70,0:21:26.36,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,0:21:26.36,0:21:27.00,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:21:27.00,0:21:28.99,Default,,0000,0000,0000,,Whenever the force\Nis perpendicular
Dialogue: 0,0:21:28.99,0:21:33.18,Default,,0000,0000,0000,,to the trajectory, I'm going\Nto get 0 for the force.
Dialogue: 0,0:21:33.18,0:21:36.71,Default,,0000,0000,0000,,Because at every\Nmoment the dot product
Dialogue: 0,0:21:36.71,0:21:41.16,Default,,0000,0000,0000,,between the force and the\Ndisplacement direction,
Dialogue: 0,0:21:41.16,0:21:45.19,Default,,0000,0000,0000,,which would be like dR, the\Ntangent to the displacement,
Dialogue: 0,0:21:45.19,0:21:46.75,Default,,0000,0000,0000,,would be [INAUDIBLE].
Dialogue: 0,0:21:46.75,0:21:50.19,Default,,0000,0000,0000,,And cosine of [INAUDIBLE] is 0.
Dialogue: 0,0:21:50.19,0:21:50.77,Default,,0000,0000,0000,,Duh.
Dialogue: 0,0:21:50.77,0:21:52.81,Default,,0000,0000,0000,,So that's another reason.
Dialogue: 0,0:21:52.81,0:21:59.82,Default,,0000,0000,0000,,Reason of last time\Nwas F perpendicular
Dialogue: 0,0:21:59.82,0:22:05.33,Default,,0000,0000,0000,,to the R prime\Ndirection, R prime
Dialogue: 0,0:22:05.33,0:22:11.03,Default,,0000,0000,0000,,being the velocity-- look,\Nwhen I'm moving in a circle,
Dialogue: 0,0:22:11.03,0:22:13.03,Default,,0000,0000,0000,,this is the force.
Dialogue: 0,0:22:13.03,0:22:14.52,Default,,0000,0000,0000,,And I'm moving.
Dialogue: 0,0:22:14.52,0:22:17.66,Default,,0000,0000,0000,,This is my velocity, is\Nthe tangent to the circle.
Dialogue: 0,0:22:17.66,0:22:22.46,Default,,0000,0000,0000,,And the velocity and the normal\Nare always perpendicular,
Dialogue: 0,0:22:22.46,0:22:23.11,Default,,0000,0000,0000,,at every point.
Dialogue: 0,0:22:23.11,0:22:24.00,Default,,0000,0000,0000,,That's why I have 0.
Dialogue: 0,0:22:24.00,0:22:26.60,Default,,0000,0000,0000,,
Dialogue: 0,0:22:26.60,0:22:32.10,Default,,0000,0000,0000,,So note that even if I\Ndidn't take a close look,
Dialogue: 0,0:22:32.10,0:22:36.02,Default,,0000,0000,0000,,why would the answer\Nbe from 0 to pi?
Dialogue: 0,0:22:36.02,0:22:38.67,Default,,0000,0000,0000,,Still?
Dialogue: 0,0:22:38.67,0:22:42.68,Default,,0000,0000,0000,,0 because of that.
Dialogue: 0,0:22:42.68,0:22:43.64,Default,,0000,0000,0000,,0.
Dialogue: 0,0:22:43.64,0:22:47.00,Default,,0000,0000,0000,,How about from 0 to pi over 4?
Dialogue: 0,0:22:47.00,0:22:49.89,Default,,0000,0000,0000,,Still 0.
Dialogue: 0,0:22:49.89,0:22:52.17,Default,,0000,0000,0000,,And of course if somebody\Nwould not believe them,
Dialogue: 0,0:22:52.17,0:22:54.86,Default,,0000,0000,0000,,if somebody would not\Nunderstand the theory,
Dialogue: 0,0:22:54.86,0:22:57.60,Default,,0000,0000,0000,,they would do the work and\Nthey would get to the answer
Dialogue: 0,0:22:57.60,0:23:01.05,Default,,0000,0000,0000,,and say, oh my\Ngod, yeah, I got 0.
Dialogue: 0,0:23:01.05,0:23:02.66,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:23:02.66,0:23:03.67,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:23:03.67,0:23:09.80,Default,,0000,0000,0000,,Now, what if somebody--\Nand I want to spray this.
Dialogue: 0,0:23:09.80,0:23:11.73,Default,,0000,0000,0000,,Can I go ahead and\Nerase the board
Dialogue: 0,0:23:11.73,0:23:15.16,Default,,0000,0000,0000,,and move onto example\Nthree or whatever?
Dialogue: 0,0:23:15.16,0:23:16.42,Default,,0000,0000,0000,,Yes?
Dialogue: 0,0:23:16.42,0:23:17.21,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:23:17.21,0:23:19.08,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:23:19.08,0:23:21.52,Default,,0000,0000,0000,,STUDENT: Could you say\Nnon-conservative force?
Dialogue: 0,0:23:21.52,0:23:23.57,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah,\Nthat's what I-- exactly.
Dialogue: 0,0:23:23.57,0:23:26.37,Default,,0000,0000,0000,,You are a mind reader.
Dialogue: 0,0:23:26.37,0:23:31.84,Default,,0000,0000,0000,,You are gonna guess my mind.
Dialogue: 0,0:23:31.84,0:23:44.26,Default,,0000,0000,0000,,
Dialogue: 0,0:23:44.26,0:23:46.80,Default,,0000,0000,0000,,And I'm going to\Npick a nasty one.
Dialogue: 0,0:23:46.80,0:23:49.84,Default,,0000,0000,0000,,And since I'm doing\Nreview anyway,
Dialogue: 0,0:23:49.84,0:23:50.97,Default,,0000,0000,0000,,you may have one like that.
Dialogue: 0,0:23:50.97,0:23:55.25,Default,,0000,0000,0000,,And you may have both one that\Ninvolves a conservative force
Dialogue: 0,0:23:55.25,0:24:00.11,Default,,0000,0000,0000,,field and one that does not\Ninvolve a conservative force
Dialogue: 0,0:24:00.11,0:24:00.69,Default,,0000,0000,0000,,field.
Dialogue: 0,0:24:00.69,0:24:07.45,Default,,0000,0000,0000,,And we can ask you, find us the\Nwork belong to different path.
Dialogue: 0,0:24:07.45,0:24:11.69,Default,,0000,0000,0000,,And I've done this\Ntype of example before.
Dialogue: 0,0:24:11.69,0:24:15.45,Default,,0000,0000,0000,,Let's take F of\Nx and y in plane.
Dialogue: 0,0:24:15.45,0:24:29.05,Default,,0000,0000,0000,,In our two I take xyi\Nplus x squared y of j.
Dialogue: 0,0:24:29.05,0:24:34.65,Default,,0000,0000,0000,,And the problem would\Ninvolve my favorite picture,
Dialogue: 0,0:24:34.65,0:24:39.00,Default,,0000,0000,0000,,y equals x squared and y\Nequals x, our two paths.
Dialogue: 0,0:24:39.00,0:24:40.52,Default,,0000,0000,0000,,One is the straight path.
Dialogue: 0,0:24:40.52,0:24:43.70,Default,,0000,0000,0000,,One is the [INAUDIBLE] path.
Dialogue: 0,0:24:43.70,0:24:46.28,Default,,0000,0000,0000,,They go from 0,\N0 to 1, 1 anyway.
Dialogue: 0,0:24:46.28,0:24:49.22,Default,,0000,0000,0000,,
Dialogue: 0,0:24:49.22,0:24:58.25,Default,,0000,0000,0000,,And I'm asking you to\Nfind W1 along path one
Dialogue: 0,0:24:58.25,0:25:01.25,Default,,0000,0000,0000,,and W2 along path two.
Dialogue: 0,0:25:01.25,0:25:06.28,Default,,0000,0000,0000,,And of course,\Nexample three, if this
Dialogue: 0,0:25:06.28,0:25:10.27,Default,,0000,0000,0000,,were conservative\Nyou would say, oh,
Dialogue: 0,0:25:10.27,0:25:11.90,Default,,0000,0000,0000,,it doesn't matter\Nwhat path I'm taking,
Dialogue: 0,0:25:11.90,0:25:14.93,Default,,0000,0000,0000,,I'm still getting\Nthe same answer.
Dialogue: 0,0:25:14.93,0:25:17.50,Default,,0000,0000,0000,,But is this conservative?
Dialogue: 0,0:25:17.50,0:25:18.42,Default,,0000,0000,0000,,STUDENT: No.
Dialogue: 0,0:25:18.42,0:25:20.18,Default,,0000,0000,0000,,Because you said it wasn't.
Dialogue: 0,0:25:20.18,0:25:21.26,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:25:21.26,0:25:22.87,Default,,0000,0000,0000,,So how do you know?
Dialogue: 0,0:25:22.87,0:25:26.22,Default,,0000,0000,0000,,That's one test\Nwhen you are in two.
Dialogue: 0,0:25:26.22,0:25:30.71,Default,,0000,0000,0000,,There is the magic test that\Nsays-- let's say this is M,
Dialogue: 0,0:25:30.71,0:25:36.70,Default,,0000,0000,0000,,and let's say this is N. You\Nwould have to check if M sub--
Dialogue: 0,0:25:36.70,0:25:37.27,Default,,0000,0000,0000,,STUDENT: y.
Dialogue: 0,0:25:37.27,0:25:38.02,Default,,0000,0000,0000,,MAGDALENA TODA: y.
Dialogue: 0,0:25:38.02,0:25:38.52,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:25:38.52,0:25:39.64,Default,,0000,0000,0000,,I'm proud of you.
Dialogue: 0,0:25:39.64,0:25:42.26,Default,,0000,0000,0000,,You're ready for\N3350, by the way.
Dialogue: 0,0:25:42.26,0:25:44.03,Default,,0000,0000,0000,,Is equal to N sub x.
Dialogue: 0,0:25:44.03,0:25:46.60,Default,,0000,0000,0000,,M sub y is x.
Dialogue: 0,0:25:46.60,0:25:49.18,Default,,0000,0000,0000,,N sub x is 2xy.
Dialogue: 0,0:25:49.18,0:25:51.53,Default,,0000,0000,0000,,They are not equal.
Dialogue: 0,0:25:51.53,0:25:55.24,Default,,0000,0000,0000,,So that's me crying that I have\Nto do the work twice and get--
Dialogue: 0,0:25:55.24,0:25:57.63,Default,,0000,0000,0000,,probably I'll get two\Ndifferent examples.
Dialogue: 0,0:25:57.63,0:26:00.97,Default,,0000,0000,0000,,
Dialogue: 0,0:26:00.97,0:26:03.51,Default,,0000,0000,0000,,If you read the book--\NI'm afraid to ask
Dialogue: 0,0:26:03.51,0:26:09.32,Default,,0000,0000,0000,,how many of you opened the\Nbook at section 13.2, 13.3.
Dialogue: 0,0:26:09.32,0:26:12.00,Default,,0000,0000,0000,,But did you read\Nit, any of them?
Dialogue: 0,0:26:12.00,0:26:13.00,Default,,0000,0000,0000,,STUDENT: Nitish read it.
Dialogue: 0,0:26:13.00,0:26:15.40,Default,,0000,0000,0000,,MAGDALENA TODA: Oh, good.
Dialogue: 0,0:26:15.40,0:26:20.14,Default,,0000,0000,0000,,There is another criteria for\Na force to be conservative.
Dialogue: 0,0:26:20.14,0:26:22.12,Default,,0000,0000,0000,,If you are, it's piece of cake.
Dialogue: 0,0:26:22.12,0:26:23.30,Default,,0000,0000,0000,,You do that, right?
Dialogue: 0,0:26:23.30,0:26:24.34,Default,,0000,0000,0000,,MAGDALENA TODA: Yes, sir?
Dialogue: 0,0:26:24.34,0:26:25.62,Default,,0000,0000,0000,,STUDENT: Curl has frequency 0.
Dialogue: 0,0:26:25.62,0:26:27.49,Default,,0000,0000,0000,,MAGDALENA TODA: The curl\Ncriteria, excellent.
Dialogue: 0,0:26:27.49,0:26:29.00,Default,,0000,0000,0000,,The curl has to be zero.
Dialogue: 0,0:26:29.00,0:26:38.98,Default,,0000,0000,0000,,So if F in R 3 is\Nconservative, then you'll
Dialogue: 0,0:26:38.98,0:26:40.06,Default,,0000,0000,0000,,get different order curve.
Dialogue: 0,0:26:40.06,0:26:42.05,Default,,0000,0000,0000,,Curl F is 0.
Dialogue: 0,0:26:42.05,0:26:44.64,Default,,0000,0000,0000,,Now let's check what\Nthe heck was curl.
Dialogue: 0,0:26:44.64,0:26:47.61,Default,,0000,0000,0000,,You see, mathematics\Nis not a bunch
Dialogue: 0,0:26:47.61,0:26:51.91,Default,,0000,0000,0000,,of these joint discussions\Nlike other sciences.
Dialogue: 0,0:26:51.91,0:26:55.34,Default,,0000,0000,0000,,In mathematics, if you don't\Nknow a section or you skipped
Dialogue: 0,0:26:55.34,0:26:58.45,Default,,0000,0000,0000,,it, you are sick, you\Nhave a date that day,
Dialogue: 0,0:26:58.45,0:27:02.70,Default,,0000,0000,0000,,you didn't study, then it's\Nall over because you cannot
Dialogue: 0,0:27:02.70,0:27:06.65,Default,,0000,0000,0000,,understand how to work out the\Nproblems and materials if you
Dialogue: 0,0:27:06.65,0:27:08.00,Default,,0000,0000,0000,,skip the section.
Dialogue: 0,0:27:08.00,0:27:12.02,Default,,0000,0000,0000,,Curl was the one\Nwhere we learned
Dialogue: 0,0:27:12.02,0:27:16.15,Default,,0000,0000,0000,,that we used the determinant.
Dialogue: 0,0:27:16.15,0:27:17.30,Default,,0000,0000,0000,,That's the easiest story.
Dialogue: 0,0:27:17.30,0:27:20.74,Default,,0000,0000,0000,,It came with a t-shirt,\Nbut that t-shirt really
Dialogue: 0,0:27:20.74,0:27:25.80,Default,,0000,0000,0000,,doesn't help because\Nit's easier to,
Dialogue: 0,0:27:25.80,0:27:28.27,Default,,0000,0000,0000,,instead of memorizing\Nthe formula,
Dialogue: 0,0:27:28.27,0:27:31.44,Default,,0000,0000,0000,,you set out the determinant.
Dialogue: 0,0:27:31.44,0:27:33.86,Default,,0000,0000,0000,,So you have the operator\Nderivative with respect
Dialogue: 0,0:27:33.86,0:27:40.76,Default,,0000,0000,0000,,to x, y z followed by what?
Dialogue: 0,0:27:40.76,0:27:43.10,Default,,0000,0000,0000,,F1, F2, F3.
Dialogue: 0,0:27:43.10,0:27:46.94,Default,,0000,0000,0000,,Now in your case, I'm\Nasking you if you did it
Dialogue: 0,0:27:46.94,0:27:51.68,Default,,0000,0000,0000,,for this F, what is\Nthe third component?
Dialogue: 0,0:27:51.68,0:27:52.51,Default,,0000,0000,0000,,STUDENT: The 0.
Dialogue: 0,0:27:52.51,0:27:54.80,Default,,0000,0000,0000,,MAGDALENA TODA: The\N0, so this guy is 0.
Dialogue: 0,0:27:54.80,0:27:59.74,Default,,0000,0000,0000,,This guy is X squared\NY, and this guy is xy.
Dialogue: 0,0:27:59.74,0:28:01.69,Default,,0000,0000,0000,,And it should be\Na piece of cake,
Dialogue: 0,0:28:01.69,0:28:03.99,Default,,0000,0000,0000,,but I want to do\Nit one more time.
Dialogue: 0,0:28:03.99,0:28:08.52,Default,,0000,0000,0000,,I times the minor derivative\Nof 0 with respect to y
Dialogue: 0,0:28:08.52,0:28:11.54,Default,,0000,0000,0000,,is 0 minus derivative\Nof x squared
Dialogue: 0,0:28:11.54,0:28:15.41,Default,,0000,0000,0000,,y respect to 0, all\Nright, plus j minus
Dialogue: 0,0:28:15.41,0:28:17.86,Default,,0000,0000,0000,,j because I'm alternating.
Dialogue: 0,0:28:17.86,0:28:19.96,Default,,0000,0000,0000,,You've known enough\Nin your algebra
Dialogue: 0,0:28:19.96,0:28:22.84,Default,,0000,0000,0000,,to know why I'm expanding\Nalong the first row.
Dialogue: 0,0:28:22.84,0:28:25.70,Default,,0000,0000,0000,,I have a minus, all\Nright, then the x
Dialogue: 0,0:28:25.70,0:28:33.60,Default,,0000,0000,0000,,of 0, 0 derivative of xy\Nrespect to the 0 plus k times
Dialogue: 0,0:28:33.60,0:28:37.55,Default,,0000,0000,0000,,the minor corresponding\Nto k derivative 2xy.
Dialogue: 0,0:28:37.55,0:28:45.05,Default,,0000,0000,0000,,
Dialogue: 0,0:28:45.05,0:28:46.32,Default,,0000,0000,0000,,Oh, and the derivative--
Dialogue: 0,0:28:46.32,0:28:49.06,Default,,0000,0000,0000,,
Dialogue: 0,0:28:49.06,0:28:52.50,Default,,0000,0000,0000,,STUDENT: Yeah, this\Nis the n equals 0.
Dialogue: 0,0:28:52.50,0:28:54.17,Default,,0000,0000,0000,,MAGDALENA TODA: Oh,\Nyeah, that's the one
Dialogue: 0,0:28:54.17,0:28:58.70,Default,,0000,0000,0000,,where it's not a because\Nthat's not conservative.
Dialogue: 0,0:28:58.70,0:28:59.83,Default,,0000,0000,0000,,So what do you get.
Dialogue: 0,0:28:59.83,0:29:04.95,Default,,0000,0000,0000,,You get 2xy minus x, right?
Dialogue: 0,0:29:04.95,0:29:07.10,Default,,0000,0000,0000,,But I don't know how to\Nwrite it better than that.
Dialogue: 0,0:29:07.10,0:29:08.10,Default,,0000,0000,0000,,Well, it doesn't matter.
Dialogue: 0,0:29:08.10,0:29:09.51,Default,,0000,0000,0000,,Leave it like that.
Dialogue: 0,0:29:09.51,0:29:18.08,Default,,0000,0000,0000,,So this would be 0 if it only\Nif x would be 0, but otherwise y
Dialogue: 0,0:29:18.08,0:29:18.86,Default,,0000,0000,0000,,was 1/2.
Dialogue: 0,0:29:18.86,0:29:22.85,Default,,0000,0000,0000,,But in general, it\Nis not a 0, good.
Dialogue: 0,0:29:22.85,0:29:30.47,Default,,0000,0000,0000,,So F is not\Nconservative, and then we
Dialogue: 0,0:29:30.47,0:29:32.40,Default,,0000,0000,0000,,can say goodbye\Nto the whole thing
Dialogue: 0,0:29:32.40,0:29:39.68,Default,,0000,0000,0000,,here and move on to\Ncomputing the works.
Dialogue: 0,0:29:39.68,0:29:42.02,Default,,0000,0000,0000,,What is the only\Nway we can do that?
Dialogue: 0,0:29:42.02,0:29:46.49,Default,,0000,0000,0000,,By parameterizing\Nthe first path,
Dialogue: 0,0:29:46.49,0:29:48.93,Default,,0000,0000,0000,,but I didn't say which\None is the first path.
Dialogue: 0,0:29:48.93,0:29:52.58,Default,,0000,0000,0000,,This is the first path, so\Nx of t equals t, and y of t
Dialogue: 0,0:29:52.58,0:29:55.26,Default,,0000,0000,0000,,equals t is your\Nparameterization.
Dialogue: 0,0:29:55.26,0:30:03.91,Default,,0000,0000,0000,,The simplest one, and then\NW1 will be integral of-- I'm
Dialogue: 0,0:30:03.91,0:30:10.14,Default,,0000,0000,0000,,too lazy to write down x of t,\Ny of t, but this is what it is.
Dialogue: 0,0:30:10.14,0:30:14.54,Default,,0000,0000,0000,,Times x prime of\Nt plus x squared
Dialogue: 0,0:30:14.54,0:30:21.55,Default,,0000,0000,0000,,y times y prime of t dt where--
Dialogue: 0,0:30:21.55,0:30:31.37,Default,,0000,0000,0000,,STUDENT: Isn't that just\Nxy dx y-- never mind.
Dialogue: 0,0:30:31.37,0:30:33.56,Default,,0000,0000,0000,,MAGDALENA TODA: This is F2.
Dialogue: 0,0:30:33.56,0:30:36.56,Default,,0000,0000,0000,,And this is x prime,\Nand this is y prime
Dialogue: 0,0:30:36.56,0:30:40.39,Default,,0000,0000,0000,,because this thing is\Njust-- I have no idea.
Dialogue: 0,0:30:40.39,0:30:41.85,Default,,0000,0000,0000,,STUDENT: Right,\Nbut what I'm asking
Dialogue: 0,0:30:41.85,0:30:46.72,Default,,0000,0000,0000,,is that not the same as just\NF1 dx because we're going
Dialogue: 0,0:30:46.72,0:30:49.63,Default,,0000,0000,0000,,to do a chain rule anyway.
Dialogue: 0,0:30:49.63,0:30:53.75,Default,,0000,0000,0000,,MAGDALENA TODA: If I put\Nthe x, I cannot put this.
Dialogue: 0,0:30:53.75,0:30:57.44,Default,,0000,0000,0000,,OK, this times that is dx.
Dialogue: 0,0:30:57.44,0:30:59.90,Default,,0000,0000,0000,,This guy times this guy is dx.
Dialogue: 0,0:30:59.90,0:31:01.40,Default,,0000,0000,0000,,STUDENT: But then\Nyou can't use your
Dialogue: 0,0:31:01.40,0:31:03.98,Default,,0000,0000,0000,,MAGDALENA TODA: Then I\Ncannot use the t's then.
Dialogue: 0,0:31:03.98,0:31:06.49,Default,,0000,0000,0000,,STUDENT: All right, there we go.
Dialogue: 0,0:31:06.49,0:31:11.16,Default,,0000,0000,0000,,MAGDALENA TODA: All right, so\NI have integral from 0 to 1 t,
Dialogue: 0,0:31:11.16,0:31:15.39,Default,,0000,0000,0000,,t times 1 t squared.
Dialogue: 0,0:31:15.39,0:31:18.26,Default,,0000,0000,0000,,If I make a mistake, that would\Nbe a silly algebra mistake
Dialogue: 0,0:31:18.26,0:31:18.76,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,0:31:18.76,0:31:21.60,Default,,0000,0000,0000,,
Dialogue: 0,0:31:21.60,0:31:23.98,Default,,0000,0000,0000,,All right, class.
Dialogue: 0,0:31:23.98,0:31:33.54,Default,,0000,0000,0000,,t cubed times 1dt,\Nhow much is this?
Dialogue: 0,0:31:33.54,0:31:38.00,Default,,0000,0000,0000,,t cubed over 3 plus t\Nto the fourth over 4.
Dialogue: 0,0:31:38.00,0:31:39.25,Default,,0000,0000,0000,,STUDENT: It's just 2-- oh, no.
Dialogue: 0,0:31:39.25,0:31:45.10,Default,,0000,0000,0000,,
Dialogue: 0,0:31:45.10,0:31:48.12,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:31:48.12,0:31:52.50,Default,,0000,0000,0000,,Do not expect that we kill you\Nwith computations on the exams,
Dialogue: 0,0:31:52.50,0:31:55.01,Default,,0000,0000,0000,,but that's not what we want.
Dialogue: 0,0:31:55.01,0:31:58.25,Default,,0000,0000,0000,,We want to test if you have\Nthe basic understanding of what
Dialogue: 0,0:31:58.25,0:32:03.27,Default,,0000,0000,0000,,this is all about, not to\Nkill you with, OK, that.
Dialogue: 0,0:32:03.27,0:32:05.35,Default,,0000,0000,0000,,I'm not going to say that\Nin front of the cameras,
Dialogue: 0,0:32:05.35,0:32:06.80,Default,,0000,0000,0000,,but everybody knows that.
Dialogue: 0,0:32:06.80,0:32:08.27,Default,,0000,0000,0000,,There are professors\Nwho would like
Dialogue: 0,0:32:08.27,0:32:09.69,Default,,0000,0000,0000,,to kill you with computations.
Dialogue: 0,0:32:09.69,0:32:12.18,Default,,0000,0000,0000,,Now, we're living in\Na different world.
Dialogue: 0,0:32:12.18,0:32:15.59,Default,,0000,0000,0000,,If I gave you a long\Npolynomial sausage here
Dialogue: 0,0:32:15.59,0:32:17.66,Default,,0000,0000,0000,,and I ask you to\Nwork with it, that
Dialogue: 0,0:32:17.66,0:32:21.09,Default,,0000,0000,0000,,doesn't mean that I'm smart\Nbecause MATLAB can do it.
Dialogue: 0,0:32:21.09,0:32:25.10,Default,,0000,0000,0000,,Mathematica, you get some\Nvery nice simplifications
Dialogue: 0,0:32:25.10,0:32:28.50,Default,,0000,0000,0000,,over there, so I'm\Njust trying to see
Dialogue: 0,0:32:28.50,0:32:35.73,Default,,0000,0000,0000,,if rather than being able\Nto compute with no error,
Dialogue: 0,0:32:35.73,0:32:40.67,Default,,0000,0000,0000,,you are having the basic\Nunderstanding of the concept.
Dialogue: 0,0:32:40.67,0:32:44.52,Default,,0000,0000,0000,,And the rest can been done\Nby the mathematical software,
Dialogue: 0,0:32:44.52,0:32:49.65,Default,,0000,0000,0000,,which, nowadays, most\Nmathematicians are using.
Dialogue: 0,0:32:49.65,0:32:52.78,Default,,0000,0000,0000,,If you asked me 15\Nyears ago, I think
Dialogue: 0,0:32:52.78,0:32:57.66,Default,,0000,0000,0000,,I knew colleagues at all the\Nranks in academia who would not
Dialogue: 0,0:32:57.66,0:33:01.59,Default,,0000,0000,0000,,touch Mathematica or\NMATLAB or Maple say
Dialogue: 0,0:33:01.59,0:33:04.81,Default,,0000,0000,0000,,that's like tool from\Nevil or something,
Dialogue: 0,0:33:04.81,0:33:08.07,Default,,0000,0000,0000,,but now everybody uses.
Dialogue: 0,0:33:08.07,0:33:10.76,Default,,0000,0000,0000,,Engineers use mostly\NMATLAB as I told you.
Dialogue: 0,0:33:10.76,0:33:15.91,Default,,0000,0000,0000,,Mathematicians use both\NMATLAB and Mathematica.
Dialogue: 0,0:33:15.91,0:33:19.45,Default,,0000,0000,0000,,Some of them use Maple,\Nespecially the ones who
Dialogue: 0,0:33:19.45,0:33:23.00,Default,,0000,0000,0000,,have demos for K-12\Nlevel teachers,
Dialogue: 0,0:33:23.00,0:33:26.14,Default,,0000,0000,0000,,but MATLAB is a wonderful\Ntool, very pretty powerful
Dialogue: 0,0:33:26.14,0:33:27.58,Default,,0000,0000,0000,,in many ways.
Dialogue: 0,0:33:27.58,0:33:30.83,Default,,0000,0000,0000,,If you are doing any kind\Nof linear algebra project--
Dialogue: 0,0:33:30.83,0:33:33.76,Default,,0000,0000,0000,,I noticed three or four of you\Nare taking linear algebra-- you
Dialogue: 0,0:33:33.76,0:33:39.07,Default,,0000,0000,0000,,can always rely on MATLAB being\Nthe best of all of the above.
Dialogue: 0,0:33:39.07,0:33:40.23,Default,,0000,0000,0000,,OK, W2.
Dialogue: 0,0:33:40.23,0:33:42.97,Default,,0000,0000,0000,,
Dialogue: 0,0:33:42.97,0:33:49.83,Default,,0000,0000,0000,,For W2, I have a parabola, and\Nit's, again, a piece of cake.
Dialogue: 0,0:33:49.83,0:33:54.54,Default,,0000,0000,0000,,X prime will be 1,\Ny prime will be 2t.
Dialogue: 0,0:33:54.54,0:33:56.48,Default,,0000,0000,0000,,When I write down\Nthe whole thing,
Dialogue: 0,0:33:56.48,0:33:58.67,Default,,0000,0000,0000,,I have to pay a little\Nbit of attention
Dialogue: 0,0:33:58.67,0:34:02.48,Default,,0000,0000,0000,,when I substitute\Nespecially when I'm
Dialogue: 0,0:34:02.48,0:34:04.60,Default,,0000,0000,0000,,taking an exam under pressure.
Dialogue: 0,0:34:04.60,0:34:08.86,Default,,0000,0000,0000,,
Dialogue: 0,0:34:08.86,0:34:13.91,Default,,0000,0000,0000,,x squared is t\Nsquared, y is t squared
Dialogue: 0,0:34:13.91,0:34:17.27,Default,,0000,0000,0000,,times y prime, which is 2t.
Dialogue: 0,0:34:17.27,0:34:19.57,Default,,0000,0000,0000,,So now this is x prime.
Dialogue: 0,0:34:19.57,0:34:21.14,Default,,0000,0000,0000,,This is y prime.
Dialogue: 0,0:34:21.14,0:34:24.70,Default,,0000,0000,0000,,Let me change colors.
Dialogue: 0,0:34:24.70,0:34:26.92,Default,,0000,0000,0000,,All politicians change colors.
Dialogue: 0,0:34:26.92,0:34:29.30,Default,,0000,0000,0000,,But I'm not a\Npolitician, but I'm
Dialogue: 0,0:34:29.30,0:34:34.03,Default,,0000,0000,0000,,thinking it's useful for you\Nto see who everybody was.
Dialogue: 0,0:34:34.03,0:34:38.69,Default,,0000,0000,0000,,This is the F1 in terms of t.
Dialogue: 0,0:34:38.69,0:34:46.21,Default,,0000,0000,0000,,That's the idea of what that\Nis, and this is F2 in terms of t
Dialogue: 0,0:34:46.21,0:34:48.12,Default,,0000,0000,0000,,as well.
Dialogue: 0,0:34:48.12,0:34:49.81,Default,,0000,0000,0000,,Oh, my God, another answer?
Dialogue: 0,0:34:49.81,0:34:53.54,Default,,0000,0000,0000,,Absolutely, I'm going to\Nget an another answer.
Dialogue: 0,0:34:53.54,0:34:57.28,Default,,0000,0000,0000,,Is it obviously to everybody\NI'm going to get another answer?
Dialogue: 0,0:34:57.28,0:34:58.08,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:34:58.08,0:35:01.50,Default,,0000,0000,0000,,MAGDALENA TODA: So I don't\Nhave to put the t's here,
Dialogue: 0,0:35:01.50,0:35:03.79,Default,,0000,0000,0000,,but I thought it was\Nsort of neat to see
Dialogue: 0,0:35:03.79,0:35:05.95,Default,,0000,0000,0000,,that t goes from 0 to 1.
Dialogue: 0,0:35:05.95,0:35:08.76,Default,,0000,0000,0000,,And what do I get?
Dialogue: 0,0:35:08.76,0:35:16.36,Default,,0000,0000,0000,,This whole lot of them is t\Ncubed plus 2 t to the fifth.
Dialogue: 0,0:35:16.36,0:35:19.04,Default,,0000,0000,0000,,
Dialogue: 0,0:35:19.04,0:35:26.00,Default,,0000,0000,0000,,So when I do the integration,\NI get t to the 4 over 4 plus--
Dialogue: 0,0:35:26.00,0:35:27.34,Default,,0000,0000,0000,,shut up, Magdalena, get people--
Dialogue: 0,0:35:27.34,0:35:29.74,Default,,0000,0000,0000,,
Dialogue: 0,0:35:29.74,0:35:30.62,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:35:30.62,0:35:32.04,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:35:32.04,0:35:36.01,Default,,0000,0000,0000,,Yeah, he's done\Nthe simplification.
Dialogue: 0,0:35:36.01,0:35:37.57,Default,,0000,0000,0000,,STUDENT: You get\Nthe same values.
Dialogue: 0,0:35:37.57,0:35:40.33,Default,,0000,0000,0000,,
Dialogue: 0,0:35:40.33,0:35:45.34,Default,,0000,0000,0000,,Plug in 1, you get 7/12 again.
Dialogue: 0,0:35:45.34,0:35:48.19,Default,,0000,0000,0000,,MAGDALENA TODA: So I'm\Nasking you-- OK, what was it?
Dialogue: 0,0:35:48.19,0:35:56.83,Default,,0000,0000,0000,,Solve 0, 1-- so I'm\Nasking why do you
Dialogue: 0,0:35:56.83,0:36:00.89,Default,,0000,0000,0000,,think we get the same value?
Dialogue: 0,0:36:00.89,0:36:03.31,Default,,0000,0000,0000,,Because the force\Nis not conservative,
Dialogue: 0,0:36:03.31,0:36:06.87,Default,,0000,0000,0000,,and I went on another path.
Dialogue: 0,0:36:06.87,0:36:10.08,Default,,0000,0000,0000,,I went on one path, and\NI went on another path.
Dialogue: 0,0:36:10.08,0:36:15.66,Default,,0000,0000,0000,,And look, obviously my\Nexpression was different.
Dialogue: 0,0:36:15.66,0:36:18.87,Default,,0000,0000,0000,,It's like one of those\Nmath games or UIL games.
Dialogue: 0,0:36:18.87,0:36:20.97,Default,,0000,0000,0000,,And look at the algebra.
Dialogue: 0,0:36:20.97,0:36:23.52,Default,,0000,0000,0000,,The polynomials are different.
Dialogue: 0,0:36:23.52,0:36:25.96,Default,,0000,0000,0000,,What was my luck here?
Dialogue: 0,0:36:25.96,0:36:27.14,Default,,0000,0000,0000,,I took 1.
Dialogue: 0,0:36:27.14,0:36:27.73,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:36:27.73,0:36:29.49,Default,,0000,0000,0000,,MAGDALENA TODA: I\Ncould have taken 2.
Dialogue: 0,0:36:29.49,0:36:35.99,Default,,0000,0000,0000,,So if instead of 1, I would\Nhave taken another number,
Dialogue: 0,0:36:35.99,0:36:38.33,Default,,0000,0000,0000,,then the higher the power,\Nthe bigger the number
Dialogue: 0,0:36:38.33,0:36:39.49,Default,,0000,0000,0000,,would have been.
Dialogue: 0,0:36:39.49,0:36:40.41,Default,,0000,0000,0000,,I could have taken 2--
Dialogue: 0,0:36:40.41,0:36:42.11,Default,,0000,0000,0000,,STUDENT: You could\Nhave taken negative 1,
Dialogue: 0,0:36:42.11,0:36:44.24,Default,,0000,0000,0000,,and you still wouldn't\Nhave got the same answer.
Dialogue: 0,0:36:44.24,0:36:48.94,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah, there\Nare many reasons why that is.
Dialogue: 0,0:36:48.94,0:36:53.90,Default,,0000,0000,0000,,But anyway, know that when you\Ntake 1, 1 to every power is 1.
Dialogue: 0,0:36:53.90,0:36:55.47,Default,,0000,0000,0000,,And yeah, you were lucky.
Dialogue: 0,0:36:55.47,0:36:58.43,Default,,0000,0000,0000,,But in general, keep in\Nmind that if the force is
Dialogue: 0,0:36:58.43,0:37:01.50,Default,,0000,0000,0000,,conservative, in\Ngeneral, in most examples
Dialogue: 0,0:37:01.50,0:37:04.51,Default,,0000,0000,0000,,you're not going to get the\Nsame answer for the work
Dialogue: 0,0:37:04.51,0:37:10.68,Default,,0000,0000,0000,,because it does depend on\Nthe path you want to take.
Dialogue: 0,0:37:10.68,0:37:18.21,Default,,0000,0000,0000,,I think I have reviewed quite\Neverything that I wanted.
Dialogue: 0,0:37:18.21,0:37:27.33,Default,,0000,0000,0000,,
Dialogue: 0,0:37:27.33,0:37:29.87,Default,,0000,0000,0000,,So I should be ready\Nto move forward.
Dialogue: 0,0:37:29.87,0:37:32.63,Default,,0000,0000,0000,,
Dialogue: 0,0:37:32.63,0:37:42.33,Default,,0000,0000,0000,,So I'm saying we are done\Nwith sections 13.1, 13.2,
Dialogue: 0,0:37:42.33,0:37:48.92,Default,,0000,0000,0000,,and 13.3, which was my\Nfavorite because it's not
Dialogue: 0,0:37:48.92,0:37:50.97,Default,,0000,0000,0000,,just the integral of\Nthe path that I like,
Dialogue: 0,0:37:50.97,0:37:54.60,Default,,0000,0000,0000,,but it's the so-called\Nfundamental theorem of calculus
Dialogue: 0,0:37:54.60,0:38:04.87,Default,,0000,0000,0000,,3, which says, fundamental\Ntheorem of the path integral
Dialogue: 0,0:38:04.87,0:38:12.03,Default,,0000,0000,0000,,saying that you have f of the\Nendpoint minus f of the origin,
Dialogue: 0,0:38:12.03,0:38:14.43,Default,,0000,0000,0000,,where little f is\Nthat scalar potential
Dialogue: 0,0:38:14.43,0:38:17.31,Default,,0000,0000,0000,,as the linear function\Nwas concerned.
Dialogue: 0,0:38:17.31,0:38:24.38,Default,,0000,0000,0000,,I'm going to call it the\Nfundamental theorem of path
Dialogue: 0,0:38:24.38,0:38:26.06,Default,,0000,0000,0000,,integral.
Dialogue: 0,0:38:26.06,0:38:29.23,Default,,0000,0000,0000,,Last time I told you the\Nfundamental theorem of calculus
Dialogue: 0,0:38:29.23,0:38:31.80,Default,,0000,0000,0000,,is Federal Trade Commission.
Dialogue: 0,0:38:31.80,0:38:34.62,Default,,0000,0000,0000,,We refer to that in Calc 1.
Dialogue: 0,0:38:34.62,0:38:39.08,Default,,0000,0000,0000,,But this one is the fundamental\Ntheorem of path integral.
Dialogue: 0,0:38:39.08,0:38:42.82,Default,,0000,0000,0000,,Remember it because at\Nleast one problem out of 15
Dialogue: 0,0:38:42.82,0:38:44.65,Default,,0000,0000,0000,,or something on the\Nfinal, and there are not
Dialogue: 0,0:38:44.65,0:38:45.56,Default,,0000,0000,0000,,going to be very many.
Dialogue: 0,0:38:45.56,0:38:48.82,Default,,0000,0000,0000,,It's going to ask you to\Nknow that result. This is
Dialogue: 0,0:38:48.82,0:38:51.95,Default,,0000,0000,0000,,an important theorem.
Dialogue: 0,0:38:51.95,0:38:55.97,Default,,0000,0000,0000,,And another important theorem\Nthat is starting right now
Dialogue: 0,0:38:55.97,0:38:57.81,Default,,0000,0000,0000,,is Green's theorem.
Dialogue: 0,0:38:57.81,0:39:02.69,Default,,0000,0000,0000,,Green's theorem is\Na magic result. I
Dialogue: 0,0:39:02.69,0:39:04.90,Default,,0000,0000,0000,,have a t-shirt with it.
Dialogue: 0,0:39:04.90,0:39:06.41,Default,,0000,0000,0000,,I didn't bring it today.
Dialogue: 0,0:39:06.41,0:39:08.29,Default,,0000,0000,0000,,Maybe I'm going to bring\Nit next time First,
Dialogue: 0,0:39:08.29,0:39:12.38,Default,,0000,0000,0000,,I want you to see\Nthe result, and then
Dialogue: 0,0:39:12.38,0:39:15.23,Default,,0000,0000,0000,,I'll bring the t-shirt\Nto the exam, so OK.
Dialogue: 0,0:39:15.23,0:39:18.08,Default,,0000,0000,0000,,
Dialogue: 0,0:39:18.08,0:39:24.87,Default,,0000,0000,0000,,Assume that you have a\Nsoup called Jordan curve.
Dialogue: 0,0:39:24.87,0:39:27.96,Default,,0000,0000,0000,,
Dialogue: 0,0:39:27.96,0:39:32.38,Default,,0000,0000,0000,,You see, mathematicians don't\Nfollow mathematical objects
Dialogue: 0,0:39:32.38,0:39:34.37,Default,,0000,0000,0000,,by their names.
Dialogue: 0,0:39:34.37,0:39:37.37,Default,,0000,0000,0000,,We are crazy people, but\Nwe don't have a big ego.
Dialogue: 0,0:39:37.37,0:39:41.89,Default,,0000,0000,0000,,We would not say a theorem\Nof myself or whatever.
Dialogue: 0,0:39:41.89,0:39:45.34,Default,,0000,0000,0000,,We never give our names to that.
Dialogue: 0,0:39:45.34,0:39:50.93,Default,,0000,0000,0000,,But all through calculus you\Nsaw all sorts of results.
Dialogue: 0,0:39:50.93,0:39:57.09,Default,,0000,0000,0000,,Like you see the Jordan\Ncurve is a terminology,
Dialogue: 0,0:39:57.09,0:40:00.20,Default,,0000,0000,0000,,but then you see\Neverywhere the Linus rule.
Dialogue: 0,0:40:00.20,0:40:02.36,Default,,0000,0000,0000,,Did Linus get to\Ncall it his own rule?
Dialogue: 0,0:40:02.36,0:40:06.32,Default,,0000,0000,0000,,No, but Euler's\Nnumber, these are
Dialogue: 0,0:40:06.32,0:40:08.52,Default,,0000,0000,0000,,things that were\Ndiscovered, and in honor
Dialogue: 0,0:40:08.52,0:40:11.64,Default,,0000,0000,0000,,of that particular\Nmathematician,
Dialogue: 0,0:40:11.64,0:40:13.29,Default,,0000,0000,0000,,we call them names.
Dialogue: 0,0:40:13.29,0:40:15.59,Default,,0000,0000,0000,,We call them the name\Nof the mathematician.
Dialogue: 0,0:40:15.59,0:40:18.43,Default,,0000,0000,0000,,
Dialogue: 0,0:40:18.43,0:40:22.56,Default,,0000,0000,0000,,Out of curiosity for\N0.5 extra credit points,
Dialogue: 0,0:40:22.56,0:40:25.14,Default,,0000,0000,0000,,find out who Jordan was.
Dialogue: 0,0:40:25.14,0:40:32.81,Default,,0000,0000,0000,,Jordan curve is a closed\Ncurve that, in general,
Dialogue: 0,0:40:32.81,0:40:34.60,Default,,0000,0000,0000,,could be piecewise continuous.
Dialogue: 0,0:40:34.60,0:40:40.61,Default,,0000,0000,0000,,
Dialogue: 0,0:40:40.61,0:40:43.23,Default,,0000,0000,0000,,So you have a closed\Nloop over here.
Dialogue: 0,0:40:43.23,0:40:50.17,Default,,0000,0000,0000,,So in general, I could\Nhave something like that
Dialogue: 0,0:40:50.17,0:40:54.10,Default,,0000,0000,0000,,that does not enclose.
Dialogue: 0,0:40:54.10,0:40:56.54,Default,,0000,0000,0000,,That encloses a\Ndomain without holes.
Dialogue: 0,0:40:56.54,0:41:04.35,Default,,0000,0000,0000,,
Dialogue: 0,0:41:04.35,0:41:07.54,Default,,0000,0000,0000,,Holes are functions\Nof the same thing.
Dialogue: 0,0:41:07.54,0:41:10.24,Default,,0000,0000,0000,,STUDENT: So doesn't it\Nneed to be continuous?
Dialogue: 0,0:41:10.24,0:41:12.44,Default,,0000,0000,0000,,MAGDALENA TODA:\NNo, I said it is.
Dialogue: 0,0:41:12.44,0:41:13.78,Default,,0000,0000,0000,,STUDENT: You said, piecewise.
Dialogue: 0,0:41:13.78,0:41:15.17,Default,,0000,0000,0000,,MAGDALENA TODA: Ah, piecewise.
Dialogue: 0,0:41:15.17,0:41:16.45,Default,,0000,0000,0000,,This is piecewise.
Dialogue: 0,0:41:16.45,0:41:17.76,Default,,0000,0000,0000,,STUDENT: Oh, so it's piecewise.
Dialogue: 0,0:41:17.76,0:41:18.26,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:41:18.26,0:41:20.43,Default,,0000,0000,0000,,MAGDALENA TODA: So you\Nhave a bunch of arcs.
Dialogue: 0,0:41:20.43,0:41:23.05,Default,,0000,0000,0000,,Finitely many, let's\Nsay, in your case.
Dialogue: 0,0:41:23.05,0:41:26.23,Default,,0000,0000,0000,,Finitely many arcs,\Nthey have corners,
Dialogue: 0,0:41:26.23,0:41:29.74,Default,,0000,0000,0000,,but you can see define the\Nintegral along such a path.
Dialogue: 0,0:41:29.74,0:41:33.68,Default,,0000,0000,0000,,
Dialogue: 0,0:41:33.68,0:41:37.53,Default,,0000,0000,0000,,Oh, and also for another\N0.5 extra credit,
Dialogue: 0,0:41:37.53,0:41:40.14,Default,,0000,0000,0000,,find out who Mr.\NGreen was because he
Dialogue: 0,0:41:40.14,0:41:43.49,Default,,0000,0000,0000,,has several theorems that\Nare through mathematics
Dialogue: 0,0:41:43.49,0:41:46.36,Default,,0000,0000,0000,,and free mechanics and\Nvariation calculus.
Dialogue: 0,0:41:46.36,0:41:50.86,Default,,0000,0000,0000,,There are several identities\Nthat are called Greens.
Dialogue: 0,0:41:50.86,0:41:52.42,Default,,0000,0000,0000,,There is this famous\NGreen's theorem,
Dialogue: 0,0:41:52.42,0:41:54.89,Default,,0000,0000,0000,,but there are Green's\Nfirst identity,
Dialogue: 0,0:41:54.89,0:41:57.84,Default,,0000,0000,0000,,Green's second identity,\Nand all sorts of things.
Dialogue: 0,0:41:57.84,0:42:01.82,Default,,0000,0000,0000,,And find out who Mr.\NGreen was, and as a total,
Dialogue: 0,0:42:01.82,0:42:03.83,Default,,0000,0000,0000,,you have 1 point extra credit.
Dialogue: 0,0:42:03.83,0:42:08.87,Default,,0000,0000,0000,,And you can turn in a regular\Nessay like a two-page thing.
Dialogue: 0,0:42:08.87,0:42:12.60,Default,,0000,0000,0000,,You want biography of these\Nmathematicians if you want,
Dialogue: 0,0:42:12.60,0:42:15.50,Default,,0000,0000,0000,,just a few paragraphs.
Dialogue: 0,0:42:15.50,0:42:19.21,Default,,0000,0000,0000,,So what does Green's theorem do?
Dialogue: 0,0:42:19.21,0:42:26.20,Default,,0000,0000,0000,,Green's theorem is\Na remarkable result
Dialogue: 0,0:42:26.20,0:42:31.08,Default,,0000,0000,0000,,which links the path integral\Nto the double integral.
Dialogue: 0,0:42:31.08,0:42:38.27,Default,,0000,0000,0000,,It's a remarkable\Nand famous result.
Dialogue: 0,0:42:38.27,0:42:48.33,Default,,0000,0000,0000,,And that links the path\Nintegral on the closed
Dialogue: 0,0:42:48.33,0:43:07.26,Default,,0000,0000,0000,,curve to a double integral\Nover the domain enclosed.
Dialogue: 0,0:43:07.26,0:43:09.76,Default,,0000,0000,0000,,I can see the domain\Ninside, but you
Dialogue: 0,0:43:09.76,0:43:15.16,Default,,0000,0000,0000,,have to understand it's\Nenclosed by the curve.
Dialogue: 0,0:43:15.16,0:43:20.74,Default,,0000,0000,0000,,
Dialogue: 0,0:43:20.74,0:43:24.44,Default,,0000,0000,0000,,All right, and assume\Nthat you have--
Dialogue: 0,0:43:24.44,0:43:35.97,Default,,0000,0000,0000,,M and N are C1 functions of\Nx and y, what does it mean?
Dialogue: 0,0:43:35.97,0:43:37.62,Default,,0000,0000,0000,,M is a function of xy.
Dialogue: 0,0:43:37.62,0:43:40.30,Default,,0000,0000,0000,,N is a function of xy in plane.
Dialogue: 0,0:43:40.30,0:43:43.36,Default,,0000,0000,0000,,Both of them are differentiable\Nwith continuous derivative.
Dialogue: 0,0:43:43.36,0:43:46.69,Default,,0000,0000,0000,,
Dialogue: 0,0:43:46.69,0:43:47.83,Default,,0000,0000,0000,,They are differentiable.
Dialogue: 0,0:43:47.83,0:43:49.52,Default,,0000,0000,0000,,You can take the\Npartial derivatives,
Dialogue: 0,0:43:49.52,0:43:51.84,Default,,0000,0000,0000,,and all the partial\Nderivatives are continuous.
Dialogue: 0,0:43:51.84,0:43:55.50,Default,,0000,0000,0000,,That's what we mean\Nby being C1 functions.
Dialogue: 0,0:43:55.50,0:43:58.61,Default,,0000,0000,0000,,And there the magic\Nhappens, so let me show you
Dialogue: 0,0:43:58.61,0:44:02.32,Default,,0000,0000,0000,,where the magic happens.
Dialogue: 0,0:44:02.32,0:44:06.36,Default,,0000,0000,0000,,This in the box,\Nthe path integral
Dialogue: 0,0:44:06.36,0:44:20.64,Default,,0000,0000,0000,,over c of M dx plus Ndy is\Nequal to the double integral
Dialogue: 0,0:44:20.64,0:44:22.39,Default,,0000,0000,0000,,over the domain enclosed.
Dialogue: 0,0:44:22.39,0:44:24.25,Default,,0000,0000,0000,,OK, this is the c.
Dialogue: 0,0:44:24.25,0:44:27.16,Default,,0000,0000,0000,,On the boundary you\Ngo counterclockwise
Dialogue: 0,0:44:27.16,0:44:29.42,Default,,0000,0000,0000,,like any respectable\Nmathematician
Dialogue: 0,0:44:29.42,0:44:33.88,Default,,0000,0000,0000,,would go in a trigonometric\Nsense, just counterclockwise.
Dialogue: 0,0:44:33.88,0:44:36.78,Default,,0000,0000,0000,,And this is the domain\Nbeing closed by c.
Dialogue: 0,0:44:36.78,0:44:40.06,Default,,0000,0000,0000,,
Dialogue: 0,0:44:40.06,0:44:44.26,Default,,0000,0000,0000,,And you put here the\Nintegral, which is magic.
Dialogue: 0,0:44:44.26,0:44:46.24,Default,,0000,0000,0000,,This is easy to\Nremember for you.
Dialogue: 0,0:44:46.24,0:44:48.17,Default,,0000,0000,0000,,This is not easy to\Nremember unless I
Dialogue: 0,0:44:48.17,0:44:49.98,Default,,0000,0000,0000,,take the t-shirt to\Nthe exam, and you
Dialogue: 0,0:44:49.98,0:44:52.07,Default,,0000,0000,0000,,cheat by looking at my t-shirt.
Dialogue: 0,0:44:52.07,0:44:54.59,Default,,0000,0000,0000,,No, by the time of the\Nexam, I promised you
Dialogue: 0,0:44:54.59,0:44:59.31,Default,,0000,0000,0000,,you are going to have at least\None week, seven days or more,
Dialogue: 0,0:44:59.31,0:45:03.31,Default,,0000,0000,0000,,10-day period in which\Nwe will study samples,
Dialogue: 0,0:45:03.31,0:45:05.69,Default,,0000,0000,0000,,various samples of old finals.
Dialogue: 0,0:45:05.69,0:45:08.32,Default,,0000,0000,0000,,I'm going to go ahead and\Nsend you some by email.
Dialogue: 0,0:45:08.32,0:45:11.44,Default,,0000,0000,0000,,Do you mind?
Dialogue: 0,0:45:11.44,0:45:13.68,Default,,0000,0000,0000,,In the next week\Nafter this week, we
Dialogue: 0,0:45:13.68,0:45:15.40,Default,,0000,0000,0000,,are going to start reviewing.
Dialogue: 0,0:45:15.40,0:45:20.24,Default,,0000,0000,0000,,And by dA I mean dxdy, the\Nusual area limit in Cartesian
Dialogue: 0,0:45:20.24,0:45:23.71,Default,,0000,0000,0000,,coordinates the way you\Nare used to it the most.
Dialogue: 0,0:45:23.71,0:45:27.32,Default,,0000,0000,0000,,
Dialogue: 0,0:45:27.32,0:45:29.72,Default,,0000,0000,0000,,And then, Alex is looking\Nat it and said, well,
Dialogue: 0,0:45:29.72,0:45:32.38,Default,,0000,0000,0000,,then I tell her that\Nthe most elegant way
Dialogue: 0,0:45:32.38,0:45:34.69,Default,,0000,0000,0000,,is to put it with dxdy.
Dialogue: 0,0:45:34.69,0:45:38.48,Default,,0000,0000,0000,,This is what we call a\None form in mathematics.
Dialogue: 0,0:45:38.48,0:45:39.75,Default,,0000,0000,0000,,What is a one form.
Dialogue: 0,0:45:39.75,0:45:43.94,Default,,0000,0000,0000,,It is a linear combination of\Nthis infinitesimal elements
Dialogue: 0,0:45:43.94,0:45:47.49,Default,,0000,0000,0000,,dxdy in plane with some\Nscalar functions of x
Dialogue: 0,0:45:47.49,0:45:49.49,Default,,0000,0000,0000,,and y in front of her.
Dialogue: 0,0:45:49.49,0:45:50.75,Default,,0000,0000,0000,,OK, so what do we do?
Dialogue: 0,0:45:50.75,0:45:52.50,Default,,0000,0000,0000,,We integrate the one form.
Dialogue: 0,0:45:52.50,0:45:57.46,Default,,0000,0000,0000,,The book doesn't talk about one\Nforms because the is actually
Dialogue: 0,0:45:57.46,0:46:00.73,Default,,0000,0000,0000,,written for the average\Nstudent, the average freshman
Dialogue: 0,0:46:00.73,0:46:04.92,Default,,0000,0000,0000,,or the average\Nsophomore, but I think
Dialogue: 0,0:46:04.92,0:46:08.25,Default,,0000,0000,0000,,we have an exposure to\Nthe notion of one form,
Dialogue: 0,0:46:08.25,0:46:10.77,Default,,0000,0000,0000,,so I can get a little bit\Nmore elegant and more rigorous
Dialogue: 0,0:46:10.77,0:46:12.29,Default,,0000,0000,0000,,in my speech.
Dialogue: 0,0:46:12.29,0:46:15.75,Default,,0000,0000,0000,,If you are a graduate\Nstudent, you most likely
Dialogue: 0,0:46:15.75,0:46:18.14,Default,,0000,0000,0000,,would know this is a one form.
Dialogue: 0,0:46:18.14,0:46:22.14,Default,,0000,0000,0000,,That's actually the\Ndefinition of a one form.
Dialogue: 0,0:46:22.14,0:46:23.48,Default,,0000,0000,0000,,And you'll say, what is this?
Dialogue: 0,0:46:23.48,0:46:27.28,Default,,0000,0000,0000,,This is actually two\Nform, but you are
Dialogue: 0,0:46:27.28,0:46:28.45,Default,,0000,0000,0000,,going to say, wait a minute.
Dialogue: 0,0:46:28.45,0:46:30.61,Default,,0000,0000,0000,,I have a scalar\Nfunction, whatever
Dialogue: 0,0:46:30.61,0:46:34.43,Default,,0000,0000,0000,,that is, from the integration\Nin front of the dxdy
Dialogue: 0,0:46:34.43,0:46:39.74,Default,,0000,0000,0000,,you want but you never said\Nthat dxdy is a two form.
Dialogue: 0,0:46:39.74,0:46:44.99,Default,,0000,0000,0000,,Actually, I did, and I\Ndidn't call it a two form.
Dialogue: 0,0:46:44.99,0:46:47.28,Default,,0000,0000,0000,,Do you remember that\NI introduced to you
Dialogue: 0,0:46:47.28,0:46:50.43,Default,,0000,0000,0000,,some magic wedge product?
Dialogue: 0,0:46:50.43,0:46:53.64,Default,,0000,0000,0000,,
Dialogue: 0,0:46:53.64,0:46:57.75,Default,,0000,0000,0000,,And we said, this is\Na tiny displacement.
Dialogue: 0,0:46:57.75,0:46:59.75,Default,,0000,0000,0000,,Dx infinitesimal is small.
Dialogue: 0,0:46:59.75,0:47:02.36,Default,,0000,0000,0000,,Imagine how much the\Nvideo we'll there
Dialogue: 0,0:47:02.36,0:47:04.96,Default,,0000,0000,0000,,is an infinitesimal\Ndisplacement dx
Dialogue: 0,0:47:04.96,0:47:07.57,Default,,0000,0000,0000,,and an infinitesimal\Ndisplacement dy,
Dialogue: 0,0:47:07.57,0:47:10.70,Default,,0000,0000,0000,,and you have some\Nsort of a sign area.
Dialogue: 0,0:47:10.70,0:47:15.13,Default,,0000,0000,0000,,So we said, we don't\Njust take dxdy,
Dialogue: 0,0:47:15.13,0:47:19.07,Default,,0000,0000,0000,,but we take a product\Nbetween dxdy with a wedge,
Dialogue: 0,0:47:19.07,0:47:21.70,Default,,0000,0000,0000,,meaning that if I\Nchange the order,
Dialogue: 0,0:47:21.70,0:47:24.14,Default,,0000,0000,0000,,I'm going to have minus dy here.
Dialogue: 0,0:47:24.14,0:47:29.14,Default,,0000,0000,0000,,This is typical exterior\Nderivative theory-- exterior
Dialogue: 0,0:47:29.14,0:47:31.46,Default,,0000,0000,0000,,derivative theory.
Dialogue: 0,0:47:31.46,0:47:34.94,Default,,0000,0000,0000,,And it's a theory that\Nstarts more or less
Dialogue: 0,0:47:34.94,0:47:36.45,Default,,0000,0000,0000,,at the graduate level.
Dialogue: 0,0:47:36.45,0:47:39.51,Default,,0000,0000,0000,,And many people get their\Nmaster's degree in math
Dialogue: 0,0:47:39.51,0:47:43.45,Default,,0000,0000,0000,,and never get to see it, and\NI pity them, but this life.
Dialogue: 0,0:47:43.45,0:47:47.47,Default,,0000,0000,0000,,On the other hand, when\Nyou have dx, which dx--
Dialogue: 0,0:47:47.47,0:47:50.55,Default,,0000,0000,0000,,the area between dx and dx is 0.
Dialogue: 0,0:47:50.55,0:47:53.40,Default,,0000,0000,0000,,So we're all very happy\NI get rid of those.
Dialogue: 0,0:47:53.40,0:47:55.85,Default,,0000,0000,0000,,When I have the sign\Nbetween the displacement,
Dialogue: 0,0:47:55.85,0:47:57.57,Default,,0000,0000,0000,,dy and itself is 0.
Dialogue: 0,0:47:57.57,0:47:59.99,Default,,0000,0000,0000,,So these are the\Nbasic properties
Dialogue: 0,0:47:59.99,0:48:05.42,Default,,0000,0000,0000,,that we started\Nabout the sign area.
Dialogue: 0,0:48:05.42,0:48:07.72,Default,,0000,0000,0000,,I want to show you what happens.
Dialogue: 0,0:48:07.72,0:48:16.36,Default,,0000,0000,0000,,I'm going to-- yeah,\NI'm going to erase here.
Dialogue: 0,0:48:16.36,0:48:22.69,Default,,0000,0000,0000,,
Dialogue: 0,0:48:22.69,0:48:25.86,Default,,0000,0000,0000,,I'm going to show\Nyou later I'm going
Dialogue: 0,0:48:25.86,0:48:32.70,Default,,0000,0000,0000,,to prove this theorem to you\Nlater using these tricks that I
Dialogue: 0,0:48:32.70,0:48:35.10,Default,,0000,0000,0000,,just showed you here.
Dialogue: 0,0:48:35.10,0:48:53.66,Default,,0000,0000,0000,,I will provide proof\Nto this formula, OK?
Dialogue: 0,0:48:53.66,0:48:57.50,Default,,0000,0000,0000,,And let's take a look at\Nthat, and we say, well,
Dialogue: 0,0:48:57.50,0:49:00.01,Default,,0000,0000,0000,,can I memorize that by\Nthe time of the final?
Dialogue: 0,0:49:00.01,0:49:01.66,Default,,0000,0000,0000,,Yes, you can.
Dialogue: 0,0:49:01.66,0:49:13.07,Default,,0000,0000,0000,,What is beautiful about\Nthis, it can actually
Dialogue: 0,0:49:13.07,0:49:18.51,Default,,0000,0000,0000,,help you solve problems that you\Ndidn't think would be possible.
Dialogue: 0,0:49:18.51,0:49:20.92,Default,,0000,0000,0000,,For example, example\N1, and I say,
Dialogue: 0,0:49:20.92,0:49:26.40,Default,,0000,0000,0000,,that would be one of\Nthe most basic ones.
Dialogue: 0,0:49:26.40,0:49:38.71,Default,,0000,0000,0000,,Find the geometric meaning of\Nthe integral over a c where
Dialogue: 0,0:49:38.71,0:49:39.89,Default,,0000,0000,0000,,c is a closed loop.
Dialogue: 0,0:49:39.89,0:49:41.92,Default,,0000,0000,0000,,OK, c is a loop.
Dialogue: 0,0:49:41.92,0:49:47.39,Default,,0000,0000,0000,,Piecewise define Jordan\Ncurve-- Jordan curve.
Dialogue: 0,0:49:47.39,0:49:49.64,Default,,0000,0000,0000,,And I integrate out\Nof something weird.
Dialogue: 0,0:49:49.64,0:49:51.14,Default,,0000,0000,0000,,And you say, oh, my God.
Dialogue: 0,0:49:51.14,0:49:51.95,Default,,0000,0000,0000,,Look at her.
Dialogue: 0,0:49:51.95,0:49:59.36,Default,,0000,0000,0000,,She picked some weird function\Nwhere the path from the dx
Dialogue: 0,0:49:59.36,0:50:05.97,Default,,0000,0000,0000,,is M, and the path in front of\Ndy is N, the M and N functions.
Dialogue: 0,0:50:05.97,0:50:07.78,Default,,0000,0000,0000,,Why would pick like that?
Dialogue: 0,0:50:07.78,0:50:11.26,Default,,0000,0000,0000,,You wouldn't know yet, but\Nif you apply Green's theorem,
Dialogue: 0,0:50:11.26,0:50:14.04,Default,,0000,0000,0000,,assuming you believe\Nit's true, you
Dialogue: 0,0:50:14.04,0:50:18.27,Default,,0000,0000,0000,,have double integral over the\Ndomain enclosed by this loop.
Dialogue: 0,0:50:18.27,0:50:24.82,Default,,0000,0000,0000,,The loop is enclosing\Nthis domain of what?
Dialogue: 0,0:50:24.82,0:50:32.08,Default,,0000,0000,0000,,Now, I'm trying to shut up,\Nand I'm want you to talk.
Dialogue: 0,0:50:32.08,0:50:35.54,Default,,0000,0000,0000,,What am I going to\Nwrite over here?
Dialogue: 0,0:50:35.54,0:50:36.94,Default,,0000,0000,0000,,STUDENT: 1 plus 1.
Dialogue: 0,0:50:36.94,0:50:40.60,Default,,0000,0000,0000,,MAGDALENA TODA: 1 plus\N1, how fun is that?
Dialogue: 0,0:50:40.60,0:50:46.13,Default,,0000,0000,0000,,Y minus 1, 1 plus 1 equals\N2 last time I checked,
Dialogue: 0,0:50:46.13,0:50:49.37,Default,,0000,0000,0000,,and this is dA.
Dialogue: 0,0:50:49.37,0:50:52.90,Default,,0000,0000,0000,,And what do you think\Nthis animal would be?
Dialogue: 0,0:50:52.90,0:50:56.09,Default,,0000,0000,0000,,The cast of 2 always can escape.
Dialogue: 0,0:50:56.09,0:51:00.57,Default,,0000,0000,0000,,So if we don't want\Nit, just kick it out.
Dialogue: 0,0:51:00.57,0:51:04.48,Default,,0000,0000,0000,,What is the remaining\Ndouble integral for d of DA?
Dialogue: 0,0:51:04.48,0:51:07.31,Default,,0000,0000,0000,,You have seen this guy all\Nthrough the Calculus 3 course.
Dialogue: 0,0:51:07.31,0:51:09.85,Default,,0000,0000,0000,,You're tired of it.
Dialogue: 0,0:51:09.85,0:51:13.78,Default,,0000,0000,0000,,You said, I cannot wait for\Nthis semester to be over
Dialogue: 0,0:51:13.78,0:51:19.06,Default,,0000,0000,0000,,because this is the double\Nintegral of 1dA over d.
Dialogue: 0,0:51:19.06,0:51:21.94,Default,,0000,0000,0000,,What in the world is that?
Dialogue: 0,0:51:21.94,0:51:24.36,Default,,0000,0000,0000,,That is the--
Dialogue: 0,0:51:24.36,0:51:25.33,Default,,0000,0000,0000,,STUDENT: --area.
Dialogue: 0,0:51:25.33,0:51:26.70,Default,,0000,0000,0000,,MAGDALENA TODA: Area, very good.
Dialogue: 0,0:51:26.70,0:51:31.15,Default,,0000,0000,0000,,This is the area of the\Ndomain d inside the curve.
Dialogue: 0,0:51:31.15,0:51:34.98,Default,,0000,0000,0000,,The shaded area is this.
Dialogue: 0,0:51:34.98,0:51:39.06,Default,,0000,0000,0000,,So you have discovered\Nsomething beautiful
Dialogue: 0,0:51:39.06,0:51:46.53,Default,,0000,0000,0000,,that the area of the domain\Nenclosed by a Jordan curve
Dialogue: 0,0:51:46.53,0:51:51.04,Default,,0000,0000,0000,,equals 1/2 because you pull\Nthe two out in front here,
Dialogue: 0,0:51:51.04,0:51:56.32,Default,,0000,0000,0000,,it's going to be 1/2 of the path\Nintegrals over the boundary.
Dialogue: 0,0:51:56.32,0:51:58.59,Default,,0000,0000,0000,,This is called\Nboundary of a domain.
Dialogue: 0,0:51:58.59,0:52:00.38,Default,,0000,0000,0000,,c is the boundary of the domain.
Dialogue: 0,0:52:00.38,0:52:04.99,Default,,0000,0000,0000,,
Dialogue: 0,0:52:04.99,0:52:06.94,Default,,0000,0000,0000,,Some mathematicians--\NI don't know
Dialogue: 0,0:52:06.94,0:52:10.95,Default,,0000,0000,0000,,how far you want to go with your\Neducation, but in a few years
Dialogue: 0,0:52:10.95,0:52:13.30,Default,,0000,0000,0000,,you might become\Ngraduate students.
Dialogue: 0,0:52:13.30,0:52:18.70,Default,,0000,0000,0000,,And even some engineers use this\Nnotation boundary of d, del d.
Dialogue: 0,0:52:18.70,0:52:22.23,Default,,0000,0000,0000,,That means the boundaries,\Nthe frontier of a domain.
Dialogue: 0,0:52:22.23,0:52:24.43,Default,,0000,0000,0000,,The fence of a ranch.
Dialogue: 0,0:52:24.43,0:52:27.05,Default,,0000,0000,0000,,That is the del d, but\Ndon't tell the rancher
Dialogue: 0,0:52:27.05,0:52:30.60,Default,,0000,0000,0000,,because he will take his gun\Nout and shoot you thinking
Dialogue: 0,0:52:30.60,0:52:33.86,Default,,0000,0000,0000,,that you are off the hook or\Nyou are after something weird.
Dialogue: 0,0:52:33.86,0:52:38.34,Default,,0000,0000,0000,,So that's the boundary\Nof the domain.
Dialogue: 0,0:52:38.34,0:52:42.44,Default,,0000,0000,0000,,And then you have\Nminus ydx plus xdy.
Dialogue: 0,0:52:42.44,0:52:46.21,Default,,0000,0000,0000,,
Dialogue: 0,0:52:46.21,0:52:48.21,Default,,0000,0000,0000,,MAGDALENA TODA: We discover\Nsomething beautiful.
Dialogue: 0,0:52:48.21,0:52:50.21,Default,,0000,0000,0000,,Something important.
Dialogue: 0,0:52:50.21,0:52:52.76,Default,,0000,0000,0000,,And now I'm asking,\Nwith this exercise--
Dialogue: 0,0:52:52.76,0:52:59.48,Default,,0000,0000,0000,,one which I could even--\NI could even call a lemma.
Dialogue: 0,0:52:59.48,0:53:03.96,Default,,0000,0000,0000,,Lemma is not quite a\Ntheorem, because it's based--
Dialogue: 0,0:53:03.96,0:53:05.13,Default,,0000,0000,0000,,could be based on a theorem.
Dialogue: 0,0:53:05.13,0:53:09.46,Default,,0000,0000,0000,,It's a little result that can\Nbe proved in just a few lines
Dialogue: 0,0:53:09.46,0:53:12.48,Default,,0000,0000,0000,,without being something\Nsophisticated based
Dialogue: 0,0:53:12.48,0:53:15.80,Default,,0000,0000,0000,,on something you\Nknew from before.
Dialogue: 0,0:53:15.80,0:53:19.95,Default,,0000,0000,0000,,So this is called a lemma.
Dialogue: 0,0:53:19.95,0:53:26.25,Default,,0000,0000,0000,,When you have a sophisticated\Narea to compute--
Dialogue: 0,0:53:26.25,0:53:30.02,Default,,0000,0000,0000,,or even can you prove-- if you\Nbelieve in Green's theorem,
Dialogue: 0,0:53:30.02,0:53:33.41,Default,,0000,0000,0000,,can you prove that the\Narea inside the circle
Dialogue: 0,0:53:33.41,0:53:34.78,Default,,0000,0000,0000,,is pi r squared?
Dialogue: 0,0:53:34.78,0:53:39.97,Default,,0000,0000,0000,,Can you prove that the\Narea inside of an ellipse
Dialogue: 0,0:53:39.97,0:53:41.49,Default,,0000,0000,0000,,is-- I don't know what.
Dialogue: 0,0:53:41.49,0:53:44.37,Default,,0000,0000,0000,,Do you know the area\Ninside of an ellipse?
Dialogue: 0,0:53:44.37,0:53:47.56,Default,,0000,0000,0000,,Nobody taught me in school.
Dialogue: 0,0:53:47.56,0:53:50.80,Default,,0000,0000,0000,,I don't know why\Nit's so beautiful.
Dialogue: 0,0:53:50.80,0:53:55.15,Default,,0000,0000,0000,,I learned what an ellipse\Nwas in eleventh grade
Dialogue: 0,0:53:55.15,0:53:59.19,Default,,0000,0000,0000,,in high school and again\Na review as a freshman
Dialogue: 0,0:53:59.19,0:54:01.31,Default,,0000,0000,0000,,analytic geometry.
Dialogue: 0,0:54:01.31,0:54:03.07,Default,,0000,0000,0000,,So we've seen conics again--
Dialogue: 0,0:54:03.07,0:54:04.94,Default,,0000,0000,0000,,STUDENT: I think we did\Nconics in 10th grade.
Dialogue: 0,0:54:04.94,0:54:06.68,Default,,0000,0000,0000,,We might have seen it.
Dialogue: 0,0:54:06.68,0:54:08.10,Default,,0000,0000,0000,,MAGDALENA TODA:\NBut nobody told me
Dialogue: 0,0:54:08.10,0:54:10.17,Default,,0000,0000,0000,,like-- I give you an ellipse.
Dialogue: 0,0:54:10.17,0:54:12.04,Default,,0000,0000,0000,,Compute the area inside.
Dialogue: 0,0:54:12.04,0:54:13.17,Default,,0000,0000,0000,,I had no idea.
Dialogue: 0,0:54:13.17,0:54:15.42,Default,,0000,0000,0000,,And I didn't know\Nthe formula until I
Dialogue: 0,0:54:15.42,0:54:17.82,Default,,0000,0000,0000,,became an assistant professor.
Dialogue: 0,0:54:17.82,0:54:19.49,Default,,0000,0000,0000,,I was already in my thirties.
Dialogue: 0,0:54:19.49,0:54:23.97,Default,,0000,0000,0000,,That's a shame to see that\Nthing for the first time OK.
Dialogue: 0,0:54:23.97,0:54:27.90,Default,,0000,0000,0000,,So let's see if we believe\Nthis lemma, and the Green's
Dialogue: 0,0:54:27.90,0:54:28.74,Default,,0000,0000,0000,,theorem of course.
Dialogue: 0,0:54:28.74,0:54:31.62,Default,,0000,0000,0000,,But let's apply the\Nlemma, primarily
Dialogue: 0,0:54:31.62,0:54:33.49,Default,,0000,0000,0000,,from the Green's theorem.
Dialogue: 0,0:54:33.49,0:54:36.60,Default,,0000,0000,0000,,Can we actually prove\Nthat the area of the disk
Dialogue: 0,0:54:36.60,0:54:40.89,Default,,0000,0000,0000,,is pi r squared and the\Narea of the ellipse--
Dialogue: 0,0:54:40.89,0:54:43.33,Default,,0000,0000,0000,,inside the ellipse\Nwill be god knows what.
Dialogue: 0,0:54:43.33,0:54:47.18,Default,,0000,0000,0000,,And we will discover\Nthat by ourselves.
Dialogue: 0,0:54:47.18,0:54:49.38,Default,,0000,0000,0000,,I think that's the\Nbeauty of mathematics.
Dialogue: 0,0:54:49.38,0:54:53.63,Default,,0000,0000,0000,,Because every now and then\Neven if you discover things
Dialogue: 0,0:54:53.63,0:54:56.28,Default,,0000,0000,0000,,that people have known\Nfor hundreds of years,
Dialogue: 0,0:54:56.28,0:54:58.08,Default,,0000,0000,0000,,it still gives you\Nthe satisfaction
Dialogue: 0,0:54:58.08,0:55:01.84,Default,,0000,0000,0000,,that you did something by\Nyourself-- all on yourself.
Dialogue: 0,0:55:01.84,0:55:06.35,Default,,0000,0000,0000,,Like, when you feel\Nbuild a helicopter or you
Dialogue: 0,0:55:06.35,0:55:07.73,Default,,0000,0000,0000,,build a table.
Dialogue: 0,0:55:07.73,0:55:09.90,Default,,0000,0000,0000,,There are many more\Nbeautiful tables
Dialogue: 0,0:55:09.90,0:55:12.05,Default,,0000,0000,0000,,that were built before\Nyou, but still it's
Dialogue: 0,0:55:12.05,0:55:14.98,Default,,0000,0000,0000,,a lot of satisfaction that\Nyou do all by yourself.
Dialogue: 0,0:55:14.98,0:55:16.85,Default,,0000,0000,0000,,It's the same with mathematics.
Dialogue: 0,0:55:16.85,0:55:23.39,Default,,0000,0000,0000,,So let's see what we can\Ndo now for the first time.
Dialogue: 0,0:55:23.39,0:55:24.59,Default,,0000,0000,0000,,Not for the first time.
Dialogue: 0,0:55:24.59,0:55:28.03,Default,,0000,0000,0000,,We do it in other ways.
Dialogue: 0,0:55:28.03,0:55:37.61,Default,,0000,0000,0000,,Can you prove using the lemma\Nor Green's theorem-- which
Dialogue: 0,0:55:37.61,0:55:43.12,Default,,0000,0000,0000,,is the same thing-- either\None-- that the area of the disk
Dialogue: 0,0:55:43.12,0:55:47.85,Default,,0000,0000,0000,,of radius r-- this is the r.
Dialogue: 0,0:55:47.85,0:55:52.33,Default,,0000,0000,0000,,so this the radius\Nr is pi r squared.
Dialogue: 0,0:55:52.33,0:55:56.29,Default,,0000,0000,0000,,
Dialogue: 0,0:55:56.29,0:55:57.53,Default,,0000,0000,0000,,I hope so.
Dialogue: 0,0:55:57.53,0:55:59.70,Default,,0000,0000,0000,,And the answer is, I hope so.
Dialogue: 0,0:55:59.70,0:56:00.66,Default,,0000,0000,0000,,And that's all.
Dialogue: 0,0:56:00.66,0:56:03.54,Default,,0000,0000,0000,,
Dialogue: 0,0:56:03.54,0:56:09.46,Default,,0000,0000,0000,,Area of the disk of radius r.
Dialogue: 0,0:56:09.46,0:56:10.24,Default,,0000,0000,0000,,Oh my god.
Dialogue: 0,0:56:10.24,0:56:12.62,Default,,0000,0000,0000,,So you go, well.
Dialogue: 0,0:56:12.62,0:56:19.42,Default,,0000,0000,0000,,If I knew the parameterization\Nof that boundary C,
Dialogue: 0,0:56:19.42,0:56:20.60,Default,,0000,0000,0000,,it would be a piece of cake.
Dialogue: 0,0:56:20.60,0:56:25.64,Default,,0000,0000,0000,,Because I would just-- I know\Nhow to do a path integral now.
Dialogue: 0,0:56:25.64,0:56:27.62,Default,,0000,0000,0000,,I've learned in the\Nprevious sections,
Dialogue: 0,0:56:27.62,0:56:30.46,Default,,0000,0000,0000,,so this should be easy.
Dialogue: 0,0:56:30.46,0:56:32.69,Default,,0000,0000,0000,,Can we do that?
Dialogue: 0,0:56:32.69,0:56:33.33,Default,,0000,0000,0000,,So let's see.
Dialogue: 0,0:56:33.33,0:56:36.36,Default,,0000,0000,0000,,
Dialogue: 0,0:56:36.36,0:56:38.29,Default,,0000,0000,0000,,Without computing\Nthe double integral,
Dialogue: 0,0:56:38.29,0:56:41.27,Default,,0000,0000,0000,,because I can always do\Nthat with polar coordinates.
Dialogue: 0,0:56:41.27,0:56:42.88,Default,,0000,0000,0000,,And we are going to do that.
Dialogue: 0,0:56:42.88,0:56:47.64,Default,,0000,0000,0000,,
Dialogue: 0,0:56:47.64,0:56:49.54,Default,,0000,0000,0000,,Let's do that as\Nwell, as practice.
Dialogue: 0,0:56:49.54,0:56:53.80,Default,,0000,0000,0000,,Because so you\Nreview for the exam.
Dialogue: 0,0:56:53.80,0:56:57.10,Default,,0000,0000,0000,,
Dialogue: 0,0:56:57.10,0:57:00.94,Default,,0000,0000,0000,,But another way to\Ndo it would be what?
Dialogue: 0,0:57:00.94,0:57:06.64,Default,,0000,0000,0000,,1/2 integral over the circle.
Dialogue: 0,0:57:06.64,0:57:13.90,Default,,0000,0000,0000,,And how do I parametrize a\Ncircle of fixed radius r?
Dialogue: 0,0:57:13.90,0:57:14.83,Default,,0000,0000,0000,,Who tells me?
Dialogue: 0,0:57:14.83,0:57:18.02,Default,,0000,0000,0000,,x of t will be--\Nthat was Chapter 10.
Dialogue: 0,0:57:18.02,0:57:20.87,Default,,0000,0000,0000,,Everything is a\Ncircle in mathematics.
Dialogue: 0,0:57:20.87,0:57:21.80,Default,,0000,0000,0000,,STUDENT: r cosine t.
Dialogue: 0,0:57:21.80,0:57:22.92,Default,,0000,0000,0000,,MAGDALENA TODA: r cosine t.
Dialogue: 0,0:57:22.92,0:57:24.21,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:57:24.21,0:57:25.79,Default,,0000,0000,0000,,y of t is?
Dialogue: 0,0:57:25.79,0:57:26.64,Default,,0000,0000,0000,,STUDENT: r sine t.
Dialogue: 0,0:57:26.64,0:57:29.42,Default,,0000,0000,0000,,MAGDALENA TODA: r sine t.
Dialogue: 0,0:57:29.42,0:57:32.36,Default,,0000,0000,0000,,So, finally I'm going to\Ngo ahead and use this one.
Dialogue: 0,0:57:32.36,0:57:37.06,Default,,0000,0000,0000,,And I'm going to say, well,\Nminus y to be plugged in.
Dialogue: 0,0:57:37.06,0:57:39.80,Default,,0000,0000,0000,,
Dialogue: 0,0:57:39.80,0:57:43.00,Default,,0000,0000,0000,,This is minus y.
Dialogue: 0,0:57:43.00,0:57:44.58,Default,,0000,0000,0000,,Multiply by dx.
Dialogue: 0,0:57:44.58,0:57:47.55,Default,,0000,0000,0000,,Well, you say, wait a\Nminute. dx with respect.
Dialogue: 0,0:57:47.55,0:57:48.75,Default,,0000,0000,0000,,What is dx?
Dialogue: 0,0:57:48.75,0:57:52.41,Default,,0000,0000,0000,,dx is just x prime dt.
Dialogue: 0,0:57:52.41,0:57:54.69,Default,,0000,0000,0000,,Dy is just y prime dt.
Dialogue: 0,0:57:54.69,0:57:56.37,Default,,0000,0000,0000,,And t goes out.
Dialogue: 0,0:57:56.37,0:57:57.52,Default,,0000,0000,0000,,It's banished.
Dialogue: 0,0:57:57.52,0:57:59.62,Default,,0000,0000,0000,,No, he's the most important guy.
Dialogue: 0,0:57:59.62,0:58:02.97,Default,,0000,0000,0000,,So t goes from something\Nto something else.
Dialogue: 0,0:58:02.97,0:58:05.26,Default,,0000,0000,0000,,We will see that later.
Dialogue: 0,0:58:05.26,0:58:06.97,Default,,0000,0000,0000,,What is x prime dt?
Dialogue: 0,0:58:06.97,0:58:12.85,Default,,0000,0000,0000,,X prime is minus r sine\Ntheta-- sine t, Magdalena.
Dialogue: 0,0:58:12.85,0:58:15.74,Default,,0000,0000,0000,,Minus r sine t.
Dialogue: 0,0:58:15.74,0:58:18.26,Default,,0000,0000,0000,,That was x prime.
Dialogue: 0,0:58:18.26,0:58:19.43,Default,,0000,0000,0000,,Change the color.
Dialogue: 0,0:58:19.43,0:58:23.24,Default,,0000,0000,0000,,Give people some\Nvariation in their life.
Dialogue: 0,0:58:23.24,0:58:32.06,Default,,0000,0000,0000,,Plus r cosine t,\Nbecause this x--
Dialogue: 0,0:58:32.06,0:58:32.93,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:58:32.93,0:58:39.31,Default,,0000,0000,0000,,
Dialogue: 0,0:58:39.31,0:58:46.44,Default,,0000,0000,0000,,MAGDALENA TODA: --times\Nthe y, which is r cosine t.
Dialogue: 0,0:58:46.44,0:58:49.15,Default,,0000,0000,0000,,So it suddenly became beautiful.
Dialogue: 0,0:58:49.15,0:58:52.48,Default,,0000,0000,0000,,It looks-- first it looks ugly,\Nbut now it became beautiful.
Dialogue: 0,0:58:52.48,0:58:52.98,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:58:52.98,0:58:54.36,Default,,0000,0000,0000,,How come it became beautiful?
Dialogue: 0,0:58:54.36,0:58:56.74,Default,,0000,0000,0000,,STUDENT: Because you got sine\Nsquared plus cosine square.
Dialogue: 0,0:58:56.74,0:58:58.28,Default,,0000,0000,0000,,MAGDALENA TODA:\NBecause I got a plus.
Dialogue: 0,0:58:58.28,0:59:01.85,Default,,0000,0000,0000,,If you pay attention, plus sine\Nsquared plus cosine squared.
Dialogue: 0,0:59:01.85,0:59:04.97,Default,,0000,0000,0000,,So I have, what is sine\Nsquared plus cosine squared?
Dialogue: 0,0:59:04.97,0:59:07.80,Default,,0000,0000,0000,,I heard that our\Nstudents in trig--
Dialogue: 0,0:59:07.80,0:59:11.86,Default,,0000,0000,0000,,Poly told me-- who still don't\Nknow that this is the most
Dialogue: 0,0:59:11.86,0:59:13.65,Default,,0000,0000,0000,,important thing you\Nlearn in trigonometry--
Dialogue: 0,0:59:13.65,0:59:15.15,Default,,0000,0000,0000,,is Pythagorean theorem.
Dialogue: 0,0:59:15.15,0:59:15.86,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:59:15.86,0:59:24.24,Default,,0000,0000,0000,,So you have 1/2 integral
Dialogue: 0,0:59:24.24,0:59:27.08,Default,,0000,0000,0000,,STUDENT: r squared--
Dialogue: 0,0:59:27.08,0:59:28.45,Default,,0000,0000,0000,,MAGDALENA TODA:\Nr-- no, I'm lazy.
Dialogue: 0,0:59:28.45,0:59:30.92,Default,,0000,0000,0000,,I'm going slow-- r.
Dialogue: 0,0:59:30.92,0:59:32.73,Default,,0000,0000,0000,,dt.
Dialogue: 0,0:59:32.73,0:59:34.92,Default,,0000,0000,0000,,T from what to what?
Dialogue: 0,0:59:34.92,0:59:36.78,Default,,0000,0000,0000,,From 0 times 0.
Dialogue: 0,0:59:36.78,0:59:39.88,Default,,0000,0000,0000,,I'm starting whatever\NI want, actually.
Dialogue: 0,0:59:39.88,0:59:44.21,Default,,0000,0000,0000,,I go counterclockwise\NI'm into pi.
Dialogue: 0,0:59:44.21,0:59:45.99,Default,,0000,0000,0000,,STUDENT: Why is\Nthat not r squared?
Dialogue: 0,0:59:45.99,0:59:47.81,Default,,0000,0000,0000,,It should be r squared.
Dialogue: 0,0:59:47.81,0:59:49.18,Default,,0000,0000,0000,,MAGDALENA TODA: I'm sorry, guys.
Dialogue: 0,0:59:49.18,0:59:50.09,Default,,0000,0000,0000,,I'm sorry.
Dialogue: 0,0:59:50.09,0:59:53.55,Default,,0000,0000,0000,,I don't know what\NI am-- r squared.
Dialogue: 0,0:59:53.55,0:59:57.06,Default,,0000,0000,0000,,1/2 r squared times 2 pi.
Dialogue: 0,0:59:57.06,1:00:00.74,Default,,0000,0000,0000,,
Dialogue: 0,1:00:00.74,1:00:03.41,Default,,0000,0000,0000,,So we have pi r squared.
Dialogue: 0,1:00:03.41,1:00:06.87,Default,,0000,0000,0000,,And if you did not\Ntell me it's r squared,
Dialogue: 0,1:00:06.87,1:00:09.71,Default,,0000,0000,0000,,we wouldn't have\Ngotten the answer.
Dialogue: 0,1:00:09.71,1:00:10.21,Default,,0000,0000,0000,,That's good.
Dialogue: 0,1:00:10.21,1:00:14.13,Default,,0000,0000,0000,,
Dialogue: 0,1:00:14.13,1:00:16.26,Default,,0000,0000,0000,,What's the other way to do it?
Dialogue: 0,1:00:16.26,1:00:18.53,Default,,0000,0000,0000,,If a problem on\Nthe final would ask
Dialogue: 0,1:00:18.53,1:00:22.17,Default,,0000,0000,0000,,you prove in two different\Nways that the rubber
Dialogue: 0,1:00:22.17,1:00:25.13,Default,,0000,0000,0000,,disk is pi r squared using\NCalc 3, or whatever--
Dialogue: 0,1:00:25.13,1:00:26.13,Default,,0000,0000,0000,,STUDENT: Would require--
Dialogue: 0,1:00:26.13,1:00:28.35,Default,,0000,0000,0000,,MAGDALENA TODA: The\Ndouble integral, right?
Dialogue: 0,1:00:28.35,1:00:28.85,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:00:28.85,1:00:31.14,Default,,0000,0000,0000,,STUDENT: Could have done\NCartesian coordinates as well.
Dialogue: 0,1:00:31.14,1:00:33.10,Default,,0000,0000,0000,,If that counts as a second way.
Dialogue: 0,1:00:33.10,1:00:33.98,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:00:33.98,1:00:34.83,Default,,0000,0000,0000,,You can-- OK.
Dialogue: 0,1:00:34.83,1:00:36.36,Default,,0000,0000,0000,,What could this be?
Dialogue: 0,1:00:36.36,1:00:37.14,Default,,0000,0000,0000,,Oh my god.
Dialogue: 0,1:00:37.14,1:00:42.28,Default,,0000,0000,0000,,This would be minus 1\Nto 1 minus square root
Dialogue: 0,1:00:42.28,1:00:45.97,Default,,0000,0000,0000,,1 minus x squared to square\Nroot 1 minus x squared.
Dialogue: 0,1:00:45.97,1:00:46.85,Default,,0000,0000,0000,,Am i right guys?
Dialogue: 0,1:00:46.85,1:00:47.39,Default,,0000,0000,0000,,STUDENT: Yep.
Dialogue: 0,1:00:47.39,1:00:48.70,Default,,0000,0000,0000,,MAGDALENA TODA: 1 dy dx.
Dialogue: 0,1:00:48.70,1:00:51.16,Default,,0000,0000,0000,,Of course it's a pain.
Dialogue: 0,1:00:51.16,1:00:53.71,Default,,0000,0000,0000,,STUDENT: You could double that\Nand set the bottoms both equal
Dialogue: 0,1:00:53.71,1:00:55.07,Default,,0000,0000,0000,,to 0.
Dialogue: 0,1:00:55.07,1:00:55.99,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,1:00:55.99,1:01:01.42,Default,,0000,0000,0000,,So we can do by symmetry--
Dialogue: 0,1:01:01.42,1:01:02.72,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,1:01:02.72,1:01:05.06,Default,,0000,0000,0000,,MAGDALENA TODA: I'm--\Nshall I erase or leave it.
Dialogue: 0,1:01:05.06,1:01:07.79,Default,,0000,0000,0000,,Are you understand\Nwhat Alex is saying?
Dialogue: 0,1:01:07.79,1:01:12.19,Default,,0000,0000,0000,,This is 2i is the integral\Nthat you will get.
Dialogue: 0,1:01:12.19,1:01:13.65,Default,,0000,0000,0000,,STUDENT: Just write\Nit next to it--
Dialogue: 0,1:01:13.65,1:01:14.99,Default,,0000,0000,0000,,MAGDALENA TODA: I tell\Nyou four times, you
Dialogue: 0,1:01:14.99,1:01:16.49,Default,,0000,0000,0000,,see, Alex, because you have--
Dialogue: 0,1:01:16.49,1:01:16.82,Default,,0000,0000,0000,,STUDENT: Oh, yeah.
Dialogue: 0,1:01:16.82,1:01:19.07,Default,,0000,0000,0000,,MAGDALENA TODA: --symmetry\Nwith respect to the x-axis,
Dialogue: 0,1:01:19.07,1:01:21.42,Default,,0000,0000,0000,,and symmetry with\Nrespect to y-axis.
Dialogue: 0,1:01:21.42,1:01:26.72,Default,,0000,0000,0000,,And you can take 0\Nto 1 and 0 to that.
Dialogue: 0,1:01:26.72,1:01:29.21,Default,,0000,0000,0000,,And you have x from 0 to 1.
Dialogue: 0,1:01:29.21,1:01:33.74,Default,,0000,0000,0000,,You have y from 0 to stop.
Dialogue: 0,1:01:33.74,1:01:35.44,Default,,0000,0000,0000,,Square root of 1 minus x square.
Dialogue: 0,1:01:35.44,1:01:37.46,Default,,0000,0000,0000,,Like the strips.
Dialogue: 0,1:01:37.46,1:01:41.60,Default,,0000,0000,0000,,And you have 4\Ntimes that A1, which
Dialogue: 0,1:01:41.60,1:01:44.92,Default,,0000,0000,0000,,would be the area of\Nthe first quadratic.
Dialogue: 0,1:01:44.92,1:01:46.47,Default,,0000,0000,0000,,You can do that, too.
Dialogue: 0,1:01:46.47,1:01:46.97,Default,,0000,0000,0000,,It's easier.
Dialogue: 0,1:01:46.97,1:01:50.10,Default,,0000,0000,0000,,But the best way to do that is\Nnot in Cartesian coordinates.
Dialogue: 0,1:01:50.10,1:01:52.77,Default,,0000,0000,0000,,The best way is to do\Nit in polar coordinates.
Dialogue: 0,1:01:52.77,1:01:56.57,Default,,0000,0000,0000,,Always remember\Nyour Jacobian is r.
Dialogue: 0,1:01:56.57,1:02:00.89,Default,,0000,0000,0000,,So if you have\NJacobian r-- erase.
Dialogue: 0,1:02:00.89,1:02:03.42,Default,,0000,0000,0000,,Let's put r here again.
Dialogue: 0,1:02:03.42,1:02:08.27,Default,,0000,0000,0000,,And then dr d theta.
Dialogue: 0,1:02:08.27,1:02:10.28,Default,,0000,0000,0000,,But now you say, wait\Na minute, Magdalena.
Dialogue: 0,1:02:10.28,1:02:11.78,Default,,0000,0000,0000,,You said r is fixed.
Dialogue: 0,1:02:11.78,1:02:12.44,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:02:12.44,1:02:13.98,Default,,0000,0000,0000,,And that's why I\Nneed to learn Greek,
Dialogue: 0,1:02:13.98,1:02:15.76,Default,,0000,0000,0000,,because it's all Greek to me.
Dialogue: 0,1:02:15.76,1:02:18.86,Default,,0000,0000,0000,,Instead of r I put\Nrho as a variable.
Dialogue: 0,1:02:18.86,1:02:23.56,Default,,0000,0000,0000,,And I say, rho is\Nbetween 0 and r.
Dialogue: 0,1:02:23.56,1:02:25.30,Default,,0000,0000,0000,,r is fixed.
Dialogue: 0,1:02:25.30,1:02:27.20,Default,,0000,0000,0000,,That's my [INAUDIBLE].
Dialogue: 0,1:02:27.20,1:02:32.13,Default,,0000,0000,0000,,Big r is not usually written\Nas a variable from 0 to some.
Dialogue: 0,1:02:32.13,1:02:33.48,Default,,0000,0000,0000,,I cannot use that.
Dialogue: 0,1:02:33.48,1:02:37.26,Default,,0000,0000,0000,,So I have to us a Greek letter,\Nwhether I like it or not.
Dialogue: 0,1:02:37.26,1:02:39.58,Default,,0000,0000,0000,,And theta is from 0 to 2 pi.
Dialogue: 0,1:02:39.58,1:02:41.75,Default,,0000,0000,0000,,And I still get the same thing.
Dialogue: 0,1:02:41.75,1:02:47.05,Default,,0000,0000,0000,,I get r-- rho squared\Nover 2 between 0 and r.
Dialogue: 0,1:02:47.05,1:02:48.61,Default,,0000,0000,0000,,And I have 2 pi.
Dialogue: 0,1:02:48.61,1:02:53.35,Default,,0000,0000,0000,,And in the end that means\Npi r squared, and I'm back.
Dialogue: 0,1:02:53.35,1:02:56.26,Default,,0000,0000,0000,,And you say, wait,\Nthis is Example 4.
Dialogue: 0,1:02:56.26,1:02:57.46,Default,,0000,0000,0000,,Whatever example.
Dialogue: 0,1:02:57.46,1:02:59.18,Default,,0000,0000,0000,,Is it Example 4, 5?
Dialogue: 0,1:02:59.18,1:03:01.09,Default,,0000,0000,0000,,You say, this is\Na piece of cake.
Dialogue: 0,1:03:01.09,1:03:05.61,Default,,0000,0000,0000,,I have two methods showing\Nme that area of the disk
Dialogue: 0,1:03:05.61,1:03:06.96,Default,,0000,0000,0000,,is so pi r squared.
Dialogue: 0,1:03:06.96,1:03:08.20,Default,,0000,0000,0000,,It's so trivial.
Dialogue: 0,1:03:08.20,1:03:12.14,Default,,0000,0000,0000,,Yeah, then let's move\Non and do the ellipse.
Dialogue: 0,1:03:12.14,1:03:14.74,Default,,0000,0000,0000,,Or we could have been\Nsmart and done the ellipse
Dialogue: 0,1:03:14.74,1:03:16.71,Default,,0000,0000,0000,,from the beginning.
Dialogue: 0,1:03:16.71,1:03:18.60,Default,,0000,0000,0000,,And then the circular\Ndisk would have
Dialogue: 0,1:03:18.60,1:03:23.01,Default,,0000,0000,0000,,been just a trivial, particular\Nexample of the ellipse.
Dialogue: 0,1:03:23.01,1:03:25.01,Default,,0000,0000,0000,,But let's do the ellipse\Nwith this magic formula
Dialogue: 0,1:03:25.01,1:03:26.74,Default,,0000,0000,0000,,that I just taught you.
Dialogue: 0,1:03:26.74,1:03:29.83,Default,,0000,0000,0000,,
Dialogue: 0,1:03:29.83,1:03:34.25,Default,,0000,0000,0000,,In the finals-- I'm going to\Nsend you a bunch of finals.
Dialogue: 0,1:03:34.25,1:03:36.63,Default,,0000,0000,0000,,You're going to be\Namused, because you're
Dialogue: 0,1:03:36.63,1:03:38.46,Default,,0000,0000,0000,,going to look at\Nthem and you say,
Dialogue: 0,1:03:38.46,1:03:41.65,Default,,0000,0000,0000,,regardless of the year and\Nsemester when the final was
Dialogue: 0,1:03:41.65,1:03:44.22,Default,,0000,0000,0000,,given for Calc 3,\Nthere was always
Dialogue: 0,1:03:44.22,1:03:49.04,Default,,0000,0000,0000,,one of the problems at the\Nend using direct application
Dialogue: 0,1:03:49.04,1:03:50.96,Default,,0000,0000,0000,,of Green's theorem.
Dialogue: 0,1:03:50.96,1:03:53.29,Default,,0000,0000,0000,,So Green's theorem\Nis an obsession,
Dialogue: 0,1:03:53.29,1:03:55.21,Default,,0000,0000,0000,,and not only at Tech.
Dialogue: 0,1:03:55.21,1:03:58.78,Default,,0000,0000,0000,,I was looking UT Austin,\NA&M, other schools--
Dialogue: 0,1:03:58.78,1:04:05.99,Default,,0000,0000,0000,,California Berkley-- all the\NCalc 3 courses on the final
Dialogue: 0,1:04:05.99,1:04:10.62,Default,,0000,0000,0000,,have at least one application--\Ndirect application
Dialogue: 0,1:04:10.62,1:04:12.36,Default,,0000,0000,0000,,applying principal.
Dialogue: 0,1:04:12.36,1:04:12.86,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:04:12.86,1:04:17.25,Default,,0000,0000,0000,,
Dialogue: 0,1:04:17.25,1:04:18.96,Default,,0000,0000,0000,,So what did I say?
Dialogue: 0,1:04:18.96,1:04:21.72,Default,,0000,0000,0000,,I said that we have\Nto draw an ellipse.
Dialogue: 0,1:04:21.72,1:04:25.43,Default,,0000,0000,0000,,How do we draw an ellipse\Nwithout making it up?
Dialogue: 0,1:04:25.43,1:04:26.84,Default,,0000,0000,0000,,That's the question.
Dialogue: 0,1:04:26.84,1:04:28.74,Default,,0000,0000,0000,,STUDENT: Draw a circle.
Dialogue: 0,1:04:28.74,1:04:30.16,Default,,0000,0000,0000,,MAGDALENA TODA: Draw a circle.
Dialogue: 0,1:04:30.16,1:04:32.06,Default,,0000,0000,0000,,Good answer.
Dialogue: 0,1:04:32.06,1:04:33.58,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:04:33.58,1:04:35.06,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:04:35.06,1:04:40.01,Default,,0000,0000,0000,,And guys this\Nstarted really bad.
Dialogue: 0,1:04:40.01,1:04:43.43,Default,,0000,0000,0000,,So I'm doing what I can.
Dialogue: 0,1:04:43.43,1:04:46.39,Default,,0000,0000,0000,,
Dialogue: 0,1:04:46.39,1:04:49.35,Default,,0000,0000,0000,,I should have tried\Nmore coffee today,
Dialogue: 0,1:04:49.35,1:04:52.46,Default,,0000,0000,0000,,because I'm getting\Ninsecure and very shaky.
Dialogue: 0,1:04:52.46,1:04:52.96,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:04:52.96,1:04:58.10,Default,,0000,0000,0000,,So I have the ellipse\Nin standard form
Dialogue: 0,1:04:58.10,1:05:01.62,Default,,0000,0000,0000,,of center O, x squared over\Nx squared plus y squared
Dialogue: 0,1:05:01.62,1:05:05.30,Default,,0000,0000,0000,,over B squared equals 1.
Dialogue: 0,1:05:05.30,1:05:07.89,Default,,0000,0000,0000,,And now you are going to\Nme who is A and who is B?
Dialogue: 0,1:05:07.89,1:05:08.92,Default,,0000,0000,0000,,What are they called?
Dialogue: 0,1:05:08.92,1:05:10.31,Default,,0000,0000,0000,,Semi--
Dialogue: 0,1:05:10.31,1:05:11.10,Default,,0000,0000,0000,,STUDENT: Semiotics.
Dialogue: 0,1:05:11.10,1:05:12.19,Default,,0000,0000,0000,,MAGDALENA TODA: Semiotics.
Dialogue: 0,1:05:12.19,1:05:15.48,Default,,0000,0000,0000,,A and B. Good.
Dialogue: 0,1:05:15.48,1:05:18.90,Default,,0000,0000,0000,,Find the area.
Dialogue: 0,1:05:18.90,1:05:21.81,Default,,0000,0000,0000,,I don't like-- OK.
Dialogue: 0,1:05:21.81,1:05:27.08,Default,,0000,0000,0000,,Let's put B inside, and let's\Nput C outside the boundary.
Dialogue: 0,1:05:27.08,1:05:42.69,Default,,0000,0000,0000,,So area of the ellipse domain\ND will be-- by the lemma-- 1/2
Dialogue: 0,1:05:42.69,1:05:46.01,Default,,0000,0000,0000,,integral over C.
Dialogue: 0,1:05:46.01,1:05:47.25,Default,,0000,0000,0000,,This is C. Is not f.
Dialogue: 0,1:05:47.25,1:05:48.20,Default,,0000,0000,0000,,Don't confuse it.
Dialogue: 0,1:05:48.20,1:05:50.62,Default,,0000,0000,0000,,It is my beautiful\Nscript C. I've
Dialogue: 0,1:05:50.62,1:05:52.42,Default,,0000,0000,0000,,tried to use it many times.
Dialogue: 0,1:05:52.42,1:05:55.24,Default,,0000,0000,0000,,Going to be minus y dx plus xdy.
Dialogue: 0,1:05:55.24,1:05:58.38,Default,,0000,0000,0000,,
Dialogue: 0,1:05:58.38,1:05:59.49,Default,,0000,0000,0000,,Again, why was that?
Dialogue: 0,1:05:59.49,1:06:04.53,Default,,0000,0000,0000,,Because we said this\Nis M and this is N,
Dialogue: 0,1:06:04.53,1:06:09.11,Default,,0000,0000,0000,,and Green's theorem will give\Nyou double integral of N sub x
Dialogue: 0,1:06:09.11,1:06:10.57,Default,,0000,0000,0000,,minus M sub y.
Dialogue: 0,1:06:10.57,1:06:13.91,Default,,0000,0000,0000,,So you have 1 minus\Nminus 1, which is 2.
Dialogue: 0,1:06:13.91,1:06:15.91,Default,,0000,0000,0000,,And 2 knocked that out.
Dialogue: 0,1:06:15.91,1:06:16.41,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:06:16.41,1:06:19.09,Default,,0000,0000,0000,,That's how we prove it.
Dialogue: 0,1:06:19.09,1:06:19.72,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:06:19.72,1:06:24.21,Default,,0000,0000,0000,,Problem is that I do not the\Nparametrization of the ellipse.
Dialogue: 0,1:06:24.21,1:06:28.22,Default,,0000,0000,0000,,And if somebody doesn't help me,\NI'm going to be in big trouble.
Dialogue: 0,1:06:28.22,1:06:32.62,Default,,0000,0000,0000,,
Dialogue: 0,1:06:32.62,1:06:34.42,Default,,0000,0000,0000,,And I'll start\Ncursing and I'm not
Dialogue: 0,1:06:34.42,1:06:37.07,Default,,0000,0000,0000,,allowed to curse in\Nfront of the classroom.
Dialogue: 0,1:06:37.07,1:06:40.76,Default,,0000,0000,0000,,But you can help me on\Nthat, because this reminds
Dialogue: 0,1:06:40.76,1:06:46.04,Default,,0000,0000,0000,,you of a famous Greek identity.
Dialogue: 0,1:06:46.04,1:06:48.52,Default,,0000,0000,0000,,The fundamental trig identity.
Dialogue: 0,1:06:48.52,1:06:51.61,Default,,0000,0000,0000,,If this would be cosine\Nsquared of theta,
Dialogue: 0,1:06:51.61,1:06:55.26,Default,,0000,0000,0000,,and this would be sine squared\Nof theta, as two animals,
Dialogue: 0,1:06:55.26,1:06:56.71,Default,,0000,0000,0000,,their sum would be 1.
Dialogue: 0,1:06:56.71,1:07:00.63,Default,,0000,0000,0000,,And whenever you have sums\Nof sum squared thingies,
Dialogue: 0,1:07:00.63,1:07:03.83,Default,,0000,0000,0000,,then you have to think trig.
Dialogue: 0,1:07:03.83,1:07:06.71,Default,,0000,0000,0000,,So, what would be\Ngood as a parameter?
Dialogue: 0,1:07:06.71,1:07:07.30,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:07:07.30,1:07:10.24,Default,,0000,0000,0000,,What would be good\Nas a parametrization
Dialogue: 0,1:07:10.24,1:07:12.21,Default,,0000,0000,0000,,to make this come true?
Dialogue: 0,1:07:12.21,1:07:15.01,Default,,0000,0000,0000,,STUDENT: You have the cosine\Nof theta would equal x over x.
Dialogue: 0,1:07:15.01,1:07:15.97,Default,,0000,0000,0000,,MAGDALENA TODA: Uh-huh.
Dialogue: 0,1:07:15.97,1:07:18.13,Default,,0000,0000,0000,,So then x would be A times--
Dialogue: 0,1:07:18.13,1:07:19.41,Default,,0000,0000,0000,,STUDENT: The cosine of theta.
Dialogue: 0,1:07:19.41,1:07:21.59,Default,,0000,0000,0000,,MAGDALENA TODA:\NDo you like theta?
Dialogue: 0,1:07:21.59,1:07:23.69,Default,,0000,0000,0000,,You don't, because\Nyou're not Greek.
Dialogue: 0,1:07:23.69,1:07:25.05,Default,,0000,0000,0000,,That's the problem.
Dialogue: 0,1:07:25.05,1:07:26.84,Default,,0000,0000,0000,,If you were Greek,\Nyou would like it.
Dialogue: 0,1:07:26.84,1:07:29.26,Default,,0000,0000,0000,,We had a colleague who\Nis not here anymore.
Dialogue: 0,1:07:29.26,1:07:30.79,Default,,0000,0000,0000,,Greek from Cypress.
Dialogue: 0,1:07:30.79,1:07:38.25,Default,,0000,0000,0000,,And he could claim that the\Nmost important-- most important
Dialogue: 0,1:07:38.25,1:07:40.23,Default,,0000,0000,0000,,alphabet is the\NGreek one, and that's
Dialogue: 0,1:07:40.23,1:07:44.21,Default,,0000,0000,0000,,why the mathematicians\Nadopted it.
Dialogue: 0,1:07:44.21,1:07:45.15,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:07:45.15,1:07:47.15,Default,,0000,0000,0000,,B sine t.
Dialogue: 0,1:07:47.15,1:07:48.27,Default,,0000,0000,0000,,How do you check?
Dialogue: 0,1:07:48.27,1:07:49.29,Default,,0000,0000,0000,,You always think, OK.
Dialogue: 0,1:07:49.29,1:07:51.46,Default,,0000,0000,0000,,This over that is cosine.
Dialogue: 0,1:07:51.46,1:07:53.49,Default,,0000,0000,0000,,This over this is sine.
Dialogue: 0,1:07:53.49,1:07:54.34,Default,,0000,0000,0000,,I square them.
Dialogue: 0,1:07:54.34,1:07:56.21,Default,,0000,0000,0000,,I get exactly that\Nand I get a 1.
Dialogue: 0,1:07:56.21,1:07:56.71,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:07:56.71,1:07:57.67,Default,,0000,0000,0000,,I'm in good shape.
Dialogue: 0,1:07:57.67,1:08:01.38,Default,,0000,0000,0000,,I know that this\Nimplicit equation--
Dialogue: 0,1:08:01.38,1:08:04.71,Default,,0000,0000,0000,,this is an implicit\Nequation-- happens if and only
Dialogue: 0,1:08:04.71,1:08:11.08,Default,,0000,0000,0000,,if I have this system of\Nthe parametrization with t
Dialogue: 0,1:08:11.08,1:08:17.00,Default,,0000,0000,0000,,between-- anything I want,\Nincluding the basic 0 to 2
Dialogue: 0,1:08:17.00,1:08:18.42,Default,,0000,0000,0000,,pi interval.
Dialogue: 0,1:08:18.42,1:08:22.38,Default,,0000,0000,0000,,And then if I were to move\Nall around for time real t
Dialogue: 0,1:08:22.38,1:08:26.27,Default,,0000,0000,0000,,I would wind around that the\Ncircle infinitely many times.
Dialogue: 0,1:08:26.27,1:08:29.27,Default,,0000,0000,0000,,Between time equals\Nminus infinity--
Dialogue: 0,1:08:29.27,1:08:33.06,Default,,0000,0000,0000,,that nobody remembers-- and\Ntime equals plus infinity--
Dialogue: 0,1:08:33.06,1:08:36.18,Default,,0000,0000,0000,,that nobody will\Never get to know.
Dialogue: 0,1:08:36.18,1:08:38.55,Default,,0000,0000,0000,,So those are the values of it.
Dialogue: 0,1:08:38.55,1:08:41.37,Default,,0000,0000,0000,,All the real values, actually.
Dialogue: 0,1:08:41.37,1:08:44.96,Default,,0000,0000,0000,,I only needed from 0 to 2\Npi to wind one time around.
Dialogue: 0,1:08:44.96,1:08:46.66,Default,,0000,0000,0000,,And this is the idea.
Dialogue: 0,1:08:46.66,1:08:48.51,Default,,0000,0000,0000,,I wind one time around.
Dialogue: 0,1:08:48.51,1:08:51.15,Default,,0000,0000,0000,,Now people-- you're going\Nto see mathematicians
Dialogue: 0,1:08:51.15,1:08:52.74,Default,,0000,0000,0000,,are not the greatest people.
Dialogue: 0,1:08:52.74,1:09:01.25,Default,,0000,0000,0000,,I've seen engineers and\Nphysicists use a lot this sign.
Dialogue: 0,1:09:01.25,1:09:02.42,Default,,0000,0000,0000,,Do you know what this means?
Dialogue: 0,1:09:02.42,1:09:04.42,Default,,0000,0000,0000,,STUDENT: It means\None full revolution.
Dialogue: 0,1:09:04.42,1:09:06.55,Default,,0000,0000,0000,,MAGDALENA TODA: It\Nmeans a full revolution.
Dialogue: 0,1:09:06.55,1:09:10.41,Default,,0000,0000,0000,,You're going to have\Na loop-- loops, that's
Dialogue: 0,1:09:10.41,1:09:11.24,Default,,0000,0000,0000,,whatever you want.
Dialogue: 0,1:09:11.24,1:09:13.38,Default,,0000,0000,0000,,Here and goes counterclockwise.
Dialogue: 0,1:09:13.38,1:09:15.72,Default,,0000,0000,0000,,And they put this\Nlittle sign showing
Dialogue: 0,1:09:15.72,1:09:21.79,Default,,0000,0000,0000,,I'm going counterclockwise on\Na closed curved, or a loop.
Dialogue: 0,1:09:21.79,1:09:22.51,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:09:22.51,1:09:24.44,Default,,0000,0000,0000,,Don't think they are crazy.
Dialogue: 0,1:09:24.44,1:09:27.48,Default,,0000,0000,0000,,This was used in lots\Nof scientific papers
Dialogue: 0,1:09:27.48,1:09:30.81,Default,,0000,0000,0000,,in math, physics, and\Nengineering, and so on.
Dialogue: 0,1:09:30.81,1:09:31.31,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:09:31.31,1:09:34.85,Default,,0000,0000,0000,,
Dialogue: 0,1:09:34.85,1:09:36.66,Default,,0000,0000,0000,,Let's do it then.
Dialogue: 0,1:09:36.66,1:09:38.28,Default,,0000,0000,0000,,Can we do it by ourselves?
Dialogue: 0,1:09:38.28,1:09:39.21,Default,,0000,0000,0000,,I think so.
Dialogue: 0,1:09:39.21,1:09:39.81,Default,,0000,0000,0000,,That's see.
Dialogue: 0,1:09:39.81,1:09:42.37,Default,,0000,0000,0000,,1/2 is 1.
Dialogue: 0,1:09:42.37,1:09:45.31,Default,,0000,0000,0000,,And I don't like\Nthe pink marker.
Dialogue: 0,1:09:45.31,1:09:47.27,Default,,0000,0000,0000,,Integral log.
Dialogue: 0,1:09:47.27,1:09:51.93,Default,,0000,0000,0000,,Time from 0 to 2 pi\Nshould be measured.
Dialogue: 0,1:09:51.93,1:09:55.74,Default,,0000,0000,0000,,y minus B sine t.
Dialogue: 0,1:09:55.74,1:10:01.73,Default,,0000,0000,0000,,
Dialogue: 0,1:10:01.73,1:10:04.34,Default,,0000,0000,0000,,dx-- what tells me that?
Dialogue: 0,1:10:04.34,1:10:06.81,Default,,0000,0000,0000,,STUDENT: B minus--
Dialogue: 0,1:10:06.81,1:10:07.22,Default,,0000,0000,0000,,
Dialogue: 0,1:10:07.22,1:10:08.31,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,1:10:08.31,1:10:09.80,Default,,0000,0000,0000,,Minus A sine t.
Dialogue: 0,1:10:09.80,1:10:10.80,Default,,0000,0000,0000,,How hard is that?
Dialogue: 0,1:10:10.80,1:10:16.27,Default,,0000,0000,0000,,It's a piece of cake Plus x--
Dialogue: 0,1:10:16.27,1:10:18.18,Default,,0000,0000,0000,,STUDENT: A cosine.
Dialogue: 0,1:10:18.18,1:10:19.26,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,1:10:19.26,1:10:21.76,Default,,0000,0000,0000,,A cosine t.
Dialogue: 0,1:10:21.76,1:10:23.09,Default,,0000,0000,0000,,TImes--
Dialogue: 0,1:10:23.09,1:10:25.63,Default,,0000,0000,0000,,STUDENT: B cosine t.
Dialogue: 0,1:10:25.63,1:10:28.79,Default,,0000,0000,0000,,MAGDALENA TODA: --B cosine t.
Dialogue: 0,1:10:28.79,1:10:30.13,Default,,0000,0000,0000,,And dt.
Dialogue: 0,1:10:30.13,1:10:32.75,Default,,0000,0000,0000,,And this thing-- look at it.
Dialogue: 0,1:10:32.75,1:10:33.46,Default,,0000,0000,0000,,It's huge.
Dialogue: 0,1:10:33.46,1:10:35.70,Default,,0000,0000,0000,,It looks huge, but it's\Nso beautiful, because--
Dialogue: 0,1:10:35.70,1:10:36.57,Default,,0000,0000,0000,,STUDENT: AB.
Dialogue: 0,1:10:36.57,1:10:37.36,Default,,0000,0000,0000,,MAGDALENA TODA: AB.
Dialogue: 0,1:10:37.36,1:10:38.80,Default,,0000,0000,0000,,Why is it AB?
Dialogue: 0,1:10:38.80,1:10:44.03,Default,,0000,0000,0000,,It's AB because sine squared\Nplus cosine squared inside
Dialogue: 0,1:10:44.03,1:10:46.18,Default,,0000,0000,0000,,becomes 1.
Dialogue: 0,1:10:46.18,1:10:49.99,Default,,0000,0000,0000,,And I have plus AB,\Nplus AB, AB out.
Dialogue: 0,1:10:49.99,1:10:52.02,Default,,0000,0000,0000,,Kick out the AB.
Dialogue: 0,1:10:52.02,1:10:57.09,Default,,0000,0000,0000,,Kick out the A and\Nthe B and you get
Dialogue: 0,1:10:57.09,1:11:01.75,Default,,0000,0000,0000,,something beautiful-- sine\Nsquared t plus cosine squared
Dialogue: 0,1:11:01.75,1:11:03.42,Default,,0000,0000,0000,,t is your old friend.
Dialogue: 0,1:11:03.42,1:11:04.84,Default,,0000,0000,0000,,And he says, I'm 1.
Dialogue: 0,1:11:04.84,1:11:08.05,Default,,0000,0000,0000,,Look how beautiful\Nlife is for you.
Dialogue: 0,1:11:08.05,1:11:09.50,Default,,0000,0000,0000,,Finally, we proved it.
Dialogue: 0,1:11:09.50,1:11:10.95,Default,,0000,0000,0000,,What did we prove?
Dialogue: 0,1:11:10.95,1:11:11.91,Default,,0000,0000,0000,,We are almost there.
Dialogue: 0,1:11:11.91,1:11:12.88,Default,,0000,0000,0000,,We got a 1/2.
Dialogue: 0,1:11:12.88,1:11:15.78,Default,,0000,0000,0000,,
Dialogue: 0,1:11:15.78,1:11:18.60,Default,,0000,0000,0000,,A constant value kick out, AB.
Dialogue: 0,1:11:18.60,1:11:21.50,Default,,0000,0000,0000,,
Dialogue: 0,1:11:21.50,1:11:22.47,Default,,0000,0000,0000,,STUDENT: Times 2 pi.
Dialogue: 0,1:11:22.47,1:11:23.60,Default,,0000,0000,0000,,MAGDALENA TODA: Times 2 pi.
Dialogue: 0,1:11:23.60,1:11:26.84,Default,,0000,0000,0000,,
Dialogue: 0,1:11:26.84,1:11:28.01,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:11:28.01,1:11:30.11,Default,,0000,0000,0000,,2 goes away.
Dialogue: 0,1:11:30.11,1:11:33.30,Default,,0000,0000,0000,,And we got a magic thing that\Nnobody taught us in school,
Dialogue: 0,1:11:33.30,1:11:34.73,Default,,0000,0000,0000,,because they were mean.
Dialogue: 0,1:11:34.73,1:11:37.36,Default,,0000,0000,0000,,They really didn't want\Nus to learn too much.
Dialogue: 0,1:11:37.36,1:11:38.76,Default,,0000,0000,0000,,That's the thingy.
Dialogue: 0,1:11:38.76,1:11:40.54,Default,,0000,0000,0000,,AB pi.
Dialogue: 0,1:11:40.54,1:11:45.22,Default,,0000,0000,0000,,AB pi is what we were\Nhoping for, because, look.
Dialogue: 0,1:11:45.22,1:11:47.82,Default,,0000,0000,0000,,I mean it's almost\Ntoo good to be true.
Dialogue: 0,1:11:47.82,1:11:53.61,Default,,0000,0000,0000,,Well, it's a disk of radius\Nr, A and B are equal.
Dialogue: 0,1:11:53.61,1:11:55.87,Default,,0000,0000,0000,,And they are the\Nradius of the disk.
Dialogue: 0,1:11:55.87,1:11:58.73,Default,,0000,0000,0000,,And that's why we\Nhave pi r squared
Dialogue: 0,1:11:58.73,1:12:00.97,Default,,0000,0000,0000,,as a particular\Nexample of the disk
Dialogue: 0,1:12:00.97,1:12:04.92,Default,,0000,0000,0000,,of the area of this ellipse.
Dialogue: 0,1:12:04.92,1:12:07.64,Default,,0000,0000,0000,,When I saw it the first\Ntime, I was like, well,
Dialogue: 0,1:12:07.64,1:12:12.50,Default,,0000,0000,0000,,I'm glad that I lived to be\N30 or something to learn this.
Dialogue: 0,1:12:12.50,1:12:17.51,Default,,0000,0000,0000,,Because nobody had shown it\Nto me in K-12 or in college.
Dialogue: 0,1:12:17.51,1:12:22.53,Default,,0000,0000,0000,,And I was a completing-- I was\Na PhD and I didn't know it.
Dialogue: 0,1:12:22.53,1:12:25.38,Default,,0000,0000,0000,,And then I said, oh,\Nthat's why-- pi AB.
Dialogue: 0,1:12:25.38,1:12:26.80,Default,,0000,0000,0000,,Yes, OK.
Dialogue: 0,1:12:26.80,1:12:27.77,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:12:27.77,1:12:31.87,Default,,0000,0000,0000,,So it's so easy to\Nunderstand once you-- well.
Dialogue: 0,1:12:31.87,1:12:33.49,Default,,0000,0000,0000,,Once you learn the section.
Dialogue: 0,1:12:33.49,1:12:34.91,Default,,0000,0000,0000,,If you don't learn\Nthe section you
Dialogue: 0,1:12:34.91,1:12:38.97,Default,,0000,0000,0000,,will not be able to understand.
Dialogue: 0,1:12:38.97,1:12:39.47,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:12:39.47,1:12:39.97,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:12:39.97,1:12:42.30,Default,,0000,0000,0000,,I'm going to go\Nahead and erase this.
Dialogue: 0,1:12:42.30,1:12:44.95,Default,,0000,0000,0000,,And I'll show you\Nan example that
Dialogue: 0,1:12:44.95,1:12:49.50,Default,,0000,0000,0000,,was popping up like an obsession\Nwith the numbers changed
Dialogue: 0,1:12:49.50,1:12:53.19,Default,,0000,0000,0000,,in most of the final exams\Nthat happen in the last three
Dialogue: 0,1:12:53.19,1:12:58.90,Default,,0000,0000,0000,,years, regardless of\Nwho wrote the exam.
Dialogue: 0,1:12:58.90,1:13:04.59,Default,,0000,0000,0000,,Because this problem really\Nmatches the learning outcomes,
Dialogue: 0,1:13:04.59,1:13:08.53,Default,,0000,0000,0000,,oh, just about any university--\Nany good university
Dialogue: 0,1:13:08.53,1:13:10.69,Default,,0000,0000,0000,,around the world.
Dialogue: 0,1:13:10.69,1:13:12.24,Default,,0000,0000,0000,,So you'll say, wow.
Dialogue: 0,1:13:12.24,1:13:13.11,Default,,0000,0000,0000,,It's so easy.
Dialogue: 0,1:13:13.11,1:13:16.75,Default,,0000,0000,0000,,I could not believe it\Nthat-- how easy it is.
Dialogue: 0,1:13:16.75,1:13:25.66,Default,,0000,0000,0000,,But once you see it, you\Nwill-- you'll say, wow.
Dialogue: 0,1:13:25.66,1:13:26.66,Default,,0000,0000,0000,,It's easy.
Dialogue: 0,1:13:26.66,1:13:34.63,Default,,0000,0000,0000,,
Dialogue: 0,1:13:34.63,1:13:35.13,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:13:35.13,1:13:41.10,Default,,0000,0000,0000,,
Dialogue: 0,1:13:41.10,1:13:44.12,Default,,0000,0000,0000,,[CHATTER]
Dialogue: 0,1:13:44.12,1:13:46.03,Default,,0000,0000,0000,,Let's try this one.
Dialogue: 0,1:13:46.03,1:13:48.52,Default,,0000,0000,0000,,You have a circle.
Dialogue: 0,1:13:48.52,1:13:57.77,Default,,0000,0000,0000,,and the circle will be\Na circle radius r given
Dialogue: 0,1:13:57.77,1:14:03.59,Default,,0000,0000,0000,,and origin 0 of 4, 9, 0, and 0.
Dialogue: 0,1:14:03.59,1:14:08.52,Default,,0000,0000,0000,,
Dialogue: 0,1:14:08.52,1:14:17.29,Default,,0000,0000,0000,,And I'm going to\Nwrite-- I'm going
Dialogue: 0,1:14:17.29,1:14:19.97,Default,,0000,0000,0000,,to give you-- first I'm going\Nto give you a very simple one.
Dialogue: 0,1:14:19.97,1:14:31.17,Default,,0000,0000,0000,,
Dialogue: 0,1:14:31.17,1:14:37.99,Default,,0000,0000,0000,,Compute in the\Nsimplest possible way.
Dialogue: 0,1:14:37.99,1:14:41.57,Default,,0000,0000,0000,,If you don't want to\Nparametrize the circle--
Dialogue: 0,1:14:41.57,1:14:43.40,Default,,0000,0000,0000,,you can always\Nparametrize the circle.
Dialogue: 0,1:14:43.40,1:14:44.30,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:14:44.30,1:14:45.42,Default,,0000,0000,0000,,But you don't want to.
Dialogue: 0,1:14:45.42,1:14:49.27,Default,,0000,0000,0000,,You want to do it the\Nfastest possible way
Dialogue: 0,1:14:49.27,1:14:51.44,Default,,0000,0000,0000,,without parameterizing\Nthe circle.
Dialogue: 0,1:14:51.44,1:14:53.88,Default,,0000,0000,0000,,Without writing down\Nwhat I'm writing down.
Dialogue: 0,1:14:53.88,1:14:55.11,Default,,0000,0000,0000,,You are in a hurry.
Dialogue: 0,1:14:55.11,1:14:58.98,Default,,0000,0000,0000,,You have 20-- 15 minutes\Nleft of your final.
Dialogue: 0,1:14:58.98,1:15:00.70,Default,,0000,0000,0000,,And you're looking\Nat me and say, I
Dialogue: 0,1:15:00.70,1:15:02.22,Default,,0000,0000,0000,,hope I get an A in this final.
Dialogue: 0,1:15:02.22,1:15:05.89,Default,,0000,0000,0000,,So what do you have to\Nremember when you look at that?
Dialogue: 0,1:15:05.89,1:15:09.66,Default,,0000,0000,0000,,
Dialogue: 0,1:15:09.66,1:15:14.77,Default,,0000,0000,0000,,M and M. M and M.\NNo, M and N. OK.
Dialogue: 0,1:15:14.77,1:15:18.69,Default,,0000,0000,0000,,And you have to remember\Nthat you are over a circle
Dialogue: 0,1:15:18.69,1:15:20.15,Default,,0000,0000,0000,,so you have a closed loop.
Dialogue: 0,1:15:20.15,1:15:21.78,Default,,0000,0000,0000,,And that's a Jordan curve.
Dialogue: 0,1:15:21.78,1:15:24.13,Default,,0000,0000,0000,,That's enclosing a disk.
Dialogue: 0,1:15:24.13,1:15:28.07,Default,,0000,0000,0000,,So you have a relationship\Nbetween the path
Dialogue: 0,1:15:28.07,1:15:34.65,Default,,0000,0000,0000,,integral along the C and the\Narea along the D-- over D.
Dialogue: 0,1:15:34.65,1:15:36.35,Default,,0000,0000,0000,,Which is of what?
Dialogue: 0,1:15:36.35,1:15:38.72,Default,,0000,0000,0000,,Is N sub x minus M sub y.
Dialogue: 0,1:15:38.72,1:15:41.30,Default,,0000,0000,0000,,So let me write it\Nin this form, which
Dialogue: 0,1:15:41.30,1:15:46.12,Default,,0000,0000,0000,,is the same thing my students\Nmostly prefer to write it as.
Dialogue: 0,1:15:46.12,1:15:48.71,Default,,0000,0000,0000,,N sub x minus M sub y.
Dialogue: 0,1:15:48.71,1:15:51.79,Default,,0000,0000,0000,,The t-shirt I have\Nhas it written
Dialogue: 0,1:15:51.79,1:15:56.96,Default,,0000,0000,0000,,like that, because it was\Nbought from nerdytshirt.com
Dialogue: 0,1:15:56.96,1:16:01.22,Default,,0000,0000,0000,,And it was especially\Ncreated to impress nerds.
Dialogue: 0,1:16:01.22,1:16:04.30,Default,,0000,0000,0000,,And of course if you\Nlook at the del notation
Dialogue: 0,1:16:04.30,1:16:07.38,Default,,0000,0000,0000,,that gives you that kind\Nof snobbish attitude
Dialogue: 0,1:16:07.38,1:16:11.67,Default,,0000,0000,0000,,that you aren't a scientist.
Dialogue: 0,1:16:11.67,1:16:12.22,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:16:12.22,1:16:16.33,Default,,0000,0000,0000,,So what is this\Ngoing to be then?
Dialogue: 0,1:16:16.33,1:16:19.39,Default,,0000,0000,0000,,Double integral over d.
Dialogue: 0,1:16:19.39,1:16:22.39,Default,,0000,0000,0000,,And sub x is up\Nhere so it gave 5.
Dialogue: 0,1:16:22.39,1:16:24.75,Default,,0000,0000,0000,,And sub y is a piece of cake.
Dialogue: 0,1:16:24.75,1:16:36.53,Default,,0000,0000,0000,,3 dx dy equals 2 out times\Nthe area of the disk, which
Dialogue: 0,1:16:36.53,1:16:38.47,Default,,0000,0000,0000,,is something you know.
Dialogue: 0,1:16:38.47,1:16:40.89,Default,,0000,0000,0000,,And I'm not going to ask you\Nto prove that all over again.
Dialogue: 0,1:16:40.89,1:16:42.76,Default,,0000,0000,0000,,So you have to say 2.
Dialogue: 0,1:16:42.76,1:16:46.59,Default,,0000,0000,0000,,I know the area of the\Ndisk-- pi r squared.
Dialogue: 0,1:16:46.59,1:16:48.00,Default,,0000,0000,0000,,And that's the answer.
Dialogue: 0,1:16:48.00,1:16:49.42,Default,,0000,0000,0000,,And you leave the room.
Dialogue: 0,1:16:49.42,1:16:50.36,Default,,0000,0000,0000,,And that's it.
Dialogue: 0,1:16:50.36,1:16:52.26,Default,,0000,0000,0000,,It's almost too\Neasy to believe it,
Dialogue: 0,1:16:52.26,1:16:58.28,Default,,0000,0000,0000,,but it was always there in\Nthe simplest possible way.
Dialogue: 0,1:16:58.28,1:17:02.61,Default,,0000,0000,0000,,And now I'm wondering, if I\Nwere to give you something hard,
Dialogue: 0,1:17:02.61,1:17:08.04,Default,,0000,0000,0000,,because-- you know my theory\Nthat when you practice
Dialogue: 0,1:17:08.04,1:17:11.81,Default,,0000,0000,0000,,at something in\Nthe classroom you
Dialogue: 0,1:17:11.81,1:17:16.65,Default,,0000,0000,0000,,have to be working on harder\Nthings in the classroom
Dialogue: 0,1:17:16.65,1:17:19.31,Default,,0000,0000,0000,,to do better in the exam.
Dialogue: 0,1:17:19.31,1:17:22.99,Default,,0000,0000,0000,,So let me cook up\Nsomething ugly for you.
Dialogue: 0,1:17:22.99,1:17:25.83,Default,,0000,0000,0000,,The same kind of disk.
Dialogue: 0,1:17:25.83,1:17:28.25,Default,,0000,0000,0000,,And I'm changing the functions.
Dialogue: 0,1:17:28.25,1:17:33.06,Default,,0000,0000,0000,,And I'll make it\Nmore complicated.
Dialogue: 0,1:17:33.06,1:17:36.14,Default,,0000,0000,0000,,
Dialogue: 0,1:17:36.14,1:17:40.48,Default,,0000,0000,0000,,Let's see how you\Nperform on this one.
Dialogue: 0,1:17:40.48,1:17:47.24,Default,,0000,0000,0000,,
Dialogue: 0,1:17:47.24,1:17:49.71,Default,,0000,0000,0000,,We avoided that one,\Nprobably, on finals
Dialogue: 0,1:17:49.71,1:17:52.48,Default,,0000,0000,0000,,because I think the\Nmajority of students
Dialogue: 0,1:17:52.48,1:17:58.06,Default,,0000,0000,0000,,wouldn't have understood what\Ntheorem they needed to apply.
Dialogue: 0,1:17:58.06,1:17:59.69,Default,,0000,0000,0000,,It looks a little bit scary.
Dialogue: 0,1:17:59.69,1:18:01.91,Default,,0000,0000,0000,,But let's say that I've\Ngiven you the hint,
Dialogue: 0,1:18:01.91,1:18:05.00,Default,,0000,0000,0000,,apply Greens theorem\Non the same path
Dialogue: 0,1:18:05.00,1:18:10.38,Default,,0000,0000,0000,,integral, which is a circle\Nof origin 0 and radius r.
Dialogue: 0,1:18:10.38,1:18:14.41,Default,,0000,0000,0000,,I now draw counterclockwise.
Dialogue: 0,1:18:14.41,1:18:18.49,Default,,0000,0000,0000,,You apply Green's theorem and\Nyou say, I know how to do this,
Dialogue: 0,1:18:18.49,1:18:21.13,Default,,0000,0000,0000,,because now I know the theorem.
Dialogue: 0,1:18:21.13,1:18:27.27,Default,,0000,0000,0000,,This is M. This is N. And I--\Nmy t-shirt did not say M and N.
Dialogue: 0,1:18:27.27,1:18:30.87,Default,,0000,0000,0000,,It said P and Q. Do you\Nwant to put P and Q?
Dialogue: 0,1:18:30.87,1:18:31.75,Default,,0000,0000,0000,,I put P and Q.
Dialogue: 0,1:18:31.75,1:18:34.74,Default,,0000,0000,0000,,So I can-- I can have this\Nlike it is on my t-shirt.
Dialogue: 0,1:18:34.74,1:18:39.23,Default,,0000,0000,0000,,So this is going\Nto be P sub x-- no.
Dialogue: 0,1:18:39.23,1:18:39.73,Default,,0000,0000,0000,,Q sub x.
Dialogue: 0,1:18:39.73,1:18:40.73,Default,,0000,0000,0000,,Sorry.
Dialogue: 0,1:18:40.73,1:18:44.56,Default,,0000,0000,0000,,M and N. So the second\None with respect to x.
Dialogue: 0,1:18:44.56,1:18:48.56,Default,,0000,0000,0000,,The one that sticks to the y\Nis prime root respect to x.
Dialogue: 0,1:18:48.56,1:18:54.34,Default,,0000,0000,0000,,The one that sticks to dx is\Nprime root with respect to y.
Dialogue: 0,1:18:54.34,1:18:56.88,Default,,0000,0000,0000,,And I think one\Ntime-- the one time
Dialogue: 0,1:18:56.88,1:19:01.14,Default,,0000,0000,0000,,when that my friend and\Ncolleague wrote that,
Dialogue: 0,1:19:01.14,1:19:02.68,Default,,0000,0000,0000,,he did it differently.
Dialogue: 0,1:19:02.68,1:19:05.60,Default,,0000,0000,0000,,He wrote something\Nlike, just-- I'll
Dialogue: 0,1:19:05.60,1:19:09.75,Default,,0000,0000,0000,,put-- I don't remember what.
Dialogue: 0,1:19:09.75,1:19:10.80,Default,,0000,0000,0000,,He put this one.
Dialogue: 0,1:19:10.80,1:19:13.51,Default,,0000,0000,0000,,
Dialogue: 0,1:19:13.51,1:19:17.47,Default,,0000,0000,0000,,Then the student\Nwas used to dx/dy
Dialogue: 0,1:19:17.47,1:19:19.25,Default,,0000,0000,0000,,and got completely confused.
Dialogue: 0,1:19:19.25,1:19:25.62,Default,,0000,0000,0000,,So pay attention to\Nwhat you are saying.
Dialogue: 0,1:19:25.62,1:19:29.40,Default,,0000,0000,0000,,Most of us write it\Nin x and y first.
Dialogue: 0,1:19:29.40,1:19:32.96,Default,,0000,0000,0000,,And we can see that the\Nderivative with respect
Dialogue: 0,1:19:32.96,1:19:38.87,Default,,0000,0000,0000,,to x of q, because that is\Nthe one next to be the y.
Dialogue: 0,1:19:38.87,1:19:42.45,Default,,0000,0000,0000,,When he gave it to me\Nlike that, he messed up
Dialogue: 0,1:19:42.45,1:19:44.76,Default,,0000,0000,0000,,everybody's notations.
Dialogue: 0,1:19:44.76,1:19:46.00,Default,,0000,0000,0000,,No.
Dialogue: 0,1:19:46.00,1:19:47.19,Default,,0000,0000,0000,,Good students steal data.
Dialogue: 0,1:19:47.19,1:19:49.77,Default,,0000,0000,0000,,So you guys have to\Nput it in standard form
Dialogue: 0,1:19:49.77,1:19:52.82,Default,,0000,0000,0000,,and pay attention to\Nwhat you are doing.
Dialogue: 0,1:19:52.82,1:19:53.77,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:19:53.77,1:19:57.14,Default,,0000,0000,0000,,So that one form can\Nbe swapped by people
Dialogue: 0,1:19:57.14,1:19:58.62,Default,,0000,0000,0000,,who try to play games.
Dialogue: 0,1:19:58.62,1:20:02.57,Default,,0000,0000,0000,,
Dialogue: 0,1:20:02.57,1:20:08.31,Default,,0000,0000,0000,,Now in this one-- So you\Nhave q sub x minus b sub y.
Dialogue: 0,1:20:08.31,1:20:15.99,Default,,0000,0000,0000,,You have 3x squared minus\Nminus, or just plus, 3y squared.
Dialogue: 0,1:20:15.99,1:20:16.49,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:20:16.49,1:20:17.46,Default,,0000,0000,0000,,Wonderful.
Dialogue: 0,1:20:17.46,1:20:20.37,Default,,0000,0000,0000,,Am I happy, do you\Nthink I'm happy?
Dialogue: 0,1:20:20.37,1:20:22.74,Default,,0000,0000,0000,,Why would I be so happy?
Dialogue: 0,1:20:22.74,1:20:25.79,Default,,0000,0000,0000,,Why is this a happy thing?
Dialogue: 0,1:20:25.79,1:20:27.74,Default,,0000,0000,0000,,I could have had\Nsomething more wild.
Dialogue: 0,1:20:27.74,1:20:28.27,Default,,0000,0000,0000,,I don't.
Dialogue: 0,1:20:28.27,1:20:30.21,Default,,0000,0000,0000,,I'm happy I don't.
Dialogue: 0,1:20:30.21,1:20:31.76,Default,,0000,0000,0000,,Why am I so happy?
Dialogue: 0,1:20:31.76,1:20:34.36,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,1:20:34.36,1:20:39.85,Default,,0000,0000,0000,,3 out over the disk.
Dialogue: 0,1:20:39.85,1:20:41.71,Default,,0000,0000,0000,,Is this ringing a bell?
Dialogue: 0,1:20:41.71,1:20:48.85,Default,,0000,0000,0000,,
Dialogue: 0,1:20:48.85,1:20:49.88,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,1:20:49.88,1:20:53.15,Default,,0000,0000,0000,,It's r squared if I\Ndo this in former.
Dialogue: 0,1:20:53.15,1:20:58.64,Default,,0000,0000,0000,,So if I do this in former,\Nits going to be rdr, d theta.
Dialogue: 0,1:20:58.64,1:21:01.34,Default,,0000,0000,0000,,So life is not as\Nhard as you believe.
Dialogue: 0,1:21:01.34,1:21:04.04,Default,,0000,0000,0000,,It can look like\Na harder problem,
Dialogue: 0,1:21:04.04,1:21:06.25,Default,,0000,0000,0000,,but in reality, it's not really.
Dialogue: 0,1:21:06.25,1:21:11.97,Default,,0000,0000,0000,,So I have 3 times-- now, I\Nhave r squared, I have r cubed.
Dialogue: 0,1:21:11.97,1:21:16.78,Default,,0000,0000,0000,,r cubed dr d theta, r between.
Dialogue: 0,1:21:16.78,1:21:20.56,Default,,0000,0000,0000,,
Dialogue: 0,1:21:20.56,1:21:30.71,Default,,0000,0000,0000,,r was between 0 and big\NR. Theta will always
Dialogue: 0,1:21:30.71,1:21:34.15,Default,,0000,0000,0000,,be between 0 and 2 pi.
Dialogue: 0,1:21:34.15,1:21:42.32,Default,,0000,0000,0000,,So, I want you, without\Nme to compute the answer
Dialogue: 0,1:21:42.32,1:21:44.45,Default,,0000,0000,0000,,and tell me what you got.
Dialogue: 0,1:21:44.45,1:21:46.84,Default,,0000,0000,0000,,STUDENT: Just say it?
Dialogue: 0,1:21:46.84,1:21:48.27,Default,,0000,0000,0000,,MAGDALENA TODA: Yep.
Dialogue: 0,1:21:48.27,1:21:52.58,Default,,0000,0000,0000,,STUDENT: 3/2, pi\Nr to the fourth.
Dialogue: 0,1:21:52.58,1:21:54.35,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nhow did you do that?
Dialogue: 0,1:21:54.35,1:21:56.61,Default,,0000,0000,0000,,You said, r to the\N4 over 4, coming
Dialogue: 0,1:21:56.61,1:22:00.47,Default,,0000,0000,0000,,from integration times the 2 pi,\Ncoming from integration times
Dialogue: 0,1:22:00.47,1:22:01.97,Default,,0000,0000,0000,,3.
Dialogue: 0,1:22:01.97,1:22:05.31,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:22:05.31,1:22:08.69,Default,,0000,0000,0000,,Is everybody with me on this?
Dialogue: 0,1:22:08.69,1:22:12.13,Default,,0000,0000,0000,,OK so, we will simplify\Nthe answer, we'll do that.
Dialogue: 0,1:22:12.13,1:22:16.70,Default,,0000,0000,0000,,What regard is the\Nradius of the disk?
Dialogue: 0,1:22:16.70,1:22:18.32,Default,,0000,0000,0000,,STUDENT: How did he\Nsolve that integral
Dialogue: 0,1:22:18.32,1:22:20.01,Default,,0000,0000,0000,,without switching the poles?
Dialogue: 0,1:22:20.01,1:22:26.46,Default,,0000,0000,0000,,
Dialogue: 0,1:22:26.46,1:22:28.94,Default,,0000,0000,0000,,MAGDALENA TODA: It would\Nhave been a killer.
Dialogue: 0,1:22:28.94,1:22:30.92,Default,,0000,0000,0000,,Let me write it out.
Dialogue: 0,1:22:30.92,1:22:32.41,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,1:22:32.41,1:22:34.91,Default,,0000,0000,0000,,Because you want to\Nwrite it out, of course.
Dialogue: 0,1:22:34.91,1:22:40.55,Default,,0000,0000,0000,,OK, 3 integral, integral\Nx squared plus y
Dialogue: 0,1:22:40.55,1:22:45.45,Default,,0000,0000,0000,,squared, dy/dx, just to make\Nmy life a little bit funnier,
Dialogue: 0,1:22:45.45,1:22:50.28,Default,,0000,0000,0000,,and then y between minus\Nsquare root-- you're
Dialogue: 0,1:22:50.28,1:22:52.06,Default,,0000,0000,0000,,looking for trouble, huh?
Dialogue: 0,1:22:52.06,1:22:59.58,Default,,0000,0000,0000,,Y squared minus x squared to\Nr squared minus s squared.
Dialogue: 0,1:22:59.58,1:23:01.25,Default,,0000,0000,0000,,And again, you could\Ndo what you just
Dialogue: 0,1:23:01.25,1:23:04.54,Default,,0000,0000,0000,,said, split into four integrals\Nover four different domains,
Dialogue: 0,1:23:04.54,1:23:07.44,Default,,0000,0000,0000,,or two up and down.
Dialogue: 0,1:23:07.44,1:23:11.95,Default,,0000,0000,0000,,And minus r and are\Nyou guys with me?
Dialogue: 0,1:23:11.95,1:23:14.89,Default,,0000,0000,0000,,And then, when you go\Nand integrate that,
Dialogue: 0,1:23:14.89,1:23:22.96,Default,,0000,0000,0000,,you integrate with respect\Nto y-- [INAUDIBLE].
Dialogue: 0,1:23:22.96,1:23:25.30,Default,,0000,0000,0000,,Well he's right,\Nso you can get x
Dialogue: 0,1:23:25.30,1:23:27.84,Default,,0000,0000,0000,,squared y plus y cubed over 3.
Dialogue: 0,1:23:27.84,1:23:30.77,Default,,0000,0000,0000,,
Dialogue: 0,1:23:30.77,1:23:34.24,Default,,0000,0000,0000,,Between those points, minus 12.
Dialogue: 0,1:23:34.24,1:23:36.12,Default,,0000,0000,0000,,And from that moment,\Nthat would just
Dialogue: 0,1:23:36.12,1:23:39.56,Default,,0000,0000,0000,,leave it and go for a walk.
Dialogue: 0,1:23:39.56,1:23:43.26,Default,,0000,0000,0000,,I will not have the\Npatience to do this.
Dialogue: 0,1:23:43.26,1:23:44.80,Default,,0000,0000,0000,,Just a second, Matthew.
Dialogue: 0,1:23:44.80,1:23:46.68,Default,,0000,0000,0000,,For this kind of\Nstuff, of course
Dialogue: 0,1:23:46.68,1:23:50.41,Default,,0000,0000,0000,,I could put this in Maple.
Dialogue: 0,1:23:50.41,1:23:54.15,Default,,0000,0000,0000,,You know Maple has these\Nlittle interactive fields,
Dialogue: 0,1:23:54.15,1:23:55.94,Default,,0000,0000,0000,,like little squares?
Dialogue: 0,1:23:55.94,1:23:58.94,Default,,0000,0000,0000,,And you go inside there\Nand add your endpoints.
Dialogue: 0,1:23:58.94,1:24:02.74,Default,,0000,0000,0000,,And even if it looks very ugly,\NMaple will spit you the answer.
Dialogue: 0,1:24:02.74,1:24:05.67,Default,,0000,0000,0000,,If you know your\Nsyntax and do it right,
Dialogue: 0,1:24:05.67,1:24:07.62,Default,,0000,0000,0000,,even if you don't\Nswitch to polar
Dialogue: 0,1:24:07.62,1:24:10.06,Default,,0000,0000,0000,,coordinates or put\Nit in Cartesian.
Dialogue: 0,1:24:10.06,1:24:12.98,Default,,0000,0000,0000,,Give it the right data, and\Nit's going to spit the answer.
Dialogue: 0,1:24:12.98,1:24:13.94,Default,,0000,0000,0000,,Yes, Matthew?
Dialogue: 0,1:24:13.94,1:24:16.11,Default,,0000,0000,0000,,STUDENT: I was\Nout of the room, I
Dialogue: 0,1:24:16.11,1:24:19.24,Default,,0000,0000,0000,,was wondering why\Nit's now y cubed.
Dialogue: 0,1:24:19.24,1:24:21.45,Default,,0000,0000,0000,,MAGDALENA TODA: Because if\Nyou integrate with respect
Dialogue: 0,1:24:21.45,1:24:24.14,Default,,0000,0000,0000,,to y first--
Dialogue: 0,1:24:24.14,1:24:27.46,Default,,0000,0000,0000,,STUDENT: Because when I\Nwalked out, it was negative y.
Dialogue: 0,1:24:27.46,1:24:29.08,Default,,0000,0000,0000,,MAGDALENA TODA: If\NI didn't put minus.
Dialogue: 0,1:24:29.08,1:24:30.25,Default,,0000,0000,0000,,STUDENT: It's a new problem.
Dialogue: 0,1:24:30.25,1:24:32.12,Default,,0000,0000,0000,,That's what he's confused about.
Dialogue: 0,1:24:32.12,1:24:34.70,Default,,0000,0000,0000,,He walked out of the room\Nduring the previous problem
Dialogue: 0,1:24:34.70,1:24:36.42,Default,,0000,0000,0000,,and came back after this one.
Dialogue: 0,1:24:36.42,1:24:37.71,Default,,0000,0000,0000,,And now he's confused.
Dialogue: 0,1:24:37.71,1:24:40.00,Default,,0000,0000,0000,,MAGDALENA TODA: You don't\Ncare about what I just asked?
Dialogue: 0,1:24:40.00,1:24:40.38,Default,,0000,0000,0000,,STUDENT: Oh.
Dialogue: 0,1:24:40.38,1:24:40.88,Default,,0000,0000,0000,,No.
Dialogue: 0,1:24:40.88,1:24:44.12,Default,,0000,0000,0000,,
Dialogue: 0,1:24:44.12,1:24:45.94,Default,,0000,0000,0000,,I like the polar coordinates.
Dialogue: 0,1:24:45.94,1:24:47.65,Default,,0000,0000,0000,,MAGDALENA TODA: Let\Nme ask you a question
Dialogue: 0,1:24:47.65,1:24:50.10,Default,,0000,0000,0000,,before I talk any further.
Dialogue: 0,1:24:50.10,1:24:53.25,Default,,0000,0000,0000,,I was about to put a plus here.
Dialogue: 0,1:24:53.25,1:24:56.46,Default,,0000,0000,0000,,What would have been the problem\Nif I had put a plus here?
Dialogue: 0,1:24:56.46,1:24:59.81,Default,,0000,0000,0000,,
Dialogue: 0,1:24:59.81,1:25:02.56,Default,,0000,0000,0000,,If I worked this out,\NI would have gotten
Dialogue: 0,1:25:02.56,1:25:05.93,Default,,0000,0000,0000,,x squared minus y squared.
Dialogue: 0,1:25:05.93,1:25:08.30,Default,,0000,0000,0000,,Would that have been\Nthe end of the world?
Dialogue: 0,1:25:08.30,1:25:10.04,Default,,0000,0000,0000,,No.
Dialogue: 0,1:25:10.04,1:25:16.16,Default,,0000,0000,0000,,But it would have complicated\Nmy life a little bit more.
Dialogue: 0,1:25:16.16,1:25:20.99,Default,,0000,0000,0000,,Let's do that one as well.
Dialogue: 0,1:25:20.99,1:25:22.42,Default,,0000,0000,0000,,STUDENT: I was\Njust curious of how
Dialogue: 0,1:25:22.42,1:25:25.25,Default,,0000,0000,0000,,you do any of these problems\Nwhen you can't switch to polar.
Dialogue: 0,1:25:25.25,1:25:27.83,Default,,0000,0000,0000,,MAGDALENA TODA: Right, let's see\Nwhat-- because Actually, even
Dialogue: 0,1:25:27.83,1:25:32.44,Default,,0000,0000,0000,,in this case, life is not so\Nhard, not as hard as you think.
Dialogue: 0,1:25:32.44,1:25:34.75,Default,,0000,0000,0000,,The persistence in that matters.
Dialogue: 0,1:25:34.75,1:25:37.62,Default,,0000,0000,0000,,You never give up on a\Nproblem that freaks you out.
Dialogue: 0,1:25:37.62,1:25:41.24,Default,,0000,0000,0000,,That's the definition\Nof a mathematician.
Dialogue: 0,1:25:41.24,1:25:48.38,Default,,0000,0000,0000,,3x squared minus 3y\Nsquared over dx/dy.
Dialogue: 0,1:25:48.38,1:25:50.27,Default,,0000,0000,0000,,Do it slowly because\NI'm not in a hurry.
Dialogue: 0,1:25:50.27,1:25:55.67,Default,,0000,0000,0000,,We are almost done with 13.4.
Dialogue: 0,1:25:55.67,1:25:57.41,Default,,0000,0000,0000,,This is OK, right?
Dialogue: 0,1:25:57.41,1:25:59.36,Default,,0000,0000,0000,,Just the minus sign again?
Dialogue: 0,1:25:59.36,1:26:01.30,Default,,0000,0000,0000,,STUDENT: Well not\Nthe minus sign.
Dialogue: 0,1:26:01.30,1:26:03.74,Default,,0000,0000,0000,,I was just wondering because\Nin the previous problem
Dialogue: 0,1:26:03.74,1:26:07.15,Default,,0000,0000,0000,,you were doing the ellipse, you\Nstarted out with the equation
Dialogue: 0,1:26:07.15,1:26:08.61,Default,,0000,0000,0000,,with the negative y--
Dialogue: 0,1:26:08.61,1:26:10.56,Default,,0000,0000,0000,,MAGDALENA TODA:\NFor this one that's
Dialogue: 0,1:26:10.56,1:26:14.84,Default,,0000,0000,0000,,just the limit that says that\Nthis is the go double integral
Dialogue: 0,1:26:14.84,1:26:18.92,Default,,0000,0000,0000,,of the area of the domain.
Dialogue: 0,1:26:18.92,1:26:23.12,Default,,0000,0000,0000,,It's just a consequence--\Nor correlate if you want.
Dialogue: 0,1:26:23.12,1:26:27.72,Default,,0000,0000,0000,,It's a consequence\Nof Green's theorem.
Dialogue: 0,1:26:27.72,1:26:31.10,Default,,0000,0000,0000,,When you forget that consequence\Nof Green's theorem and we say
Dialogue: 0,1:26:31.10,1:26:32.00,Default,,0000,0000,0000,,goodbye to that.
Dialogue: 0,1:26:32.00,1:26:35.81,Default,,0000,0000,0000,,But while you were out,\Nthis is Green's theorem.
Dialogue: 0,1:26:35.81,1:26:38.64,Default,,0000,0000,0000,,The real Green's theorem,\Nthe one that was a teacher.
Dialogue: 0,1:26:38.64,1:26:41.31,Default,,0000,0000,0000,,There are several\NGreens I can give you.
Dialogue: 0,1:26:41.31,1:26:43.28,Default,,0000,0000,0000,,The famous Green\Ntheorem is the one
Dialogue: 0,1:26:43.28,1:26:46.85,Default,,0000,0000,0000,,I said when you have--\Nthis is what we apply here.
Dialogue: 0,1:26:46.85,1:26:50.97,Default,,0000,0000,0000,,The integral of M dx plus M dy.
Dialogue: 0,1:26:50.97,1:27:00.82,Default,,0000,0000,0000,,You have a double integral of\NM sub x minus M sub y over c.
Dialogue: 0,1:27:00.82,1:27:03.45,Default,,0000,0000,0000,,
Dialogue: 0,1:27:03.45,1:27:07.69,Default,,0000,0000,0000,,So I'm assuming we would have\Nhad this case of maybe me not
Dialogue: 0,1:27:07.69,1:27:10.99,Default,,0000,0000,0000,,paying attention, or\Nbeing mean and not wanting
Dialogue: 0,1:27:10.99,1:27:14.48,Default,,0000,0000,0000,,to give you a simple problem.
Dialogue: 0,1:27:14.48,1:27:17.66,Default,,0000,0000,0000,,And what do you\Ndo in such a case?
Dialogue: 0,1:27:17.66,1:27:19.66,Default,,0000,0000,0000,,It's not obvious to\Neverybody, but you will see.
Dialogue: 0,1:27:19.66,1:27:22.05,Default,,0000,0000,0000,,It's so pretty at some\Npoint, if you know
Dialogue: 0,1:27:22.05,1:27:24.03,Default,,0000,0000,0000,,how to get out of the mess.
Dialogue: 0,1:27:24.03,1:27:27.01,Default,,0000,0000,0000,,
Dialogue: 0,1:27:27.01,1:27:30.82,Default,,0000,0000,0000,,I was already thinking, but\NI'm using polar coordinates.
Dialogue: 0,1:27:30.82,1:27:35.62,Default,,0000,0000,0000,,So that's arc of sine, so I\Nhave to go back to the basics.
Dialogue: 0,1:27:35.62,1:27:40.00,Default,,0000,0000,0000,,If I go back to the\Nbasics, ideas come to me.
Dialogue: 0,1:27:40.00,1:27:41.76,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:27:41.76,1:27:45.93,Default,,0000,0000,0000,,So, OK.
Dialogue: 0,1:27:45.93,1:27:51.56,Default,,0000,0000,0000,,r-- let's put dr d theta,\Njust to get rid of it,
Dialogue: 0,1:27:51.56,1:27:53.78,Default,,0000,0000,0000,,because it's on my nerves.
Dialogue: 0,1:27:53.78,1:28:00.37,Default,,0000,0000,0000,,This is 0 to 2 pi,\Nthis is 0 to r.
Dialogue: 0,1:28:00.37,1:28:03.34,Default,,0000,0000,0000,,And now, you say,\NOK, in our mind,
Dialogue: 0,1:28:03.34,1:28:06.96,Default,,0000,0000,0000,,because we are lazy\Npeople, plug in those
Dialogue: 0,1:28:06.96,1:28:10.97,Default,,0000,0000,0000,,and pull out what you can.
Dialogue: 0,1:28:10.97,1:28:14.97,Default,,0000,0000,0000,,One 3 out equals for what?
Dialogue: 0,1:28:14.97,1:28:18.40,Default,,0000,0000,0000,,Inside, you have r squared.
Dialogue: 0,1:28:18.40,1:28:20.72,Default,,0000,0000,0000,,Do you agree?
Dialogue: 0,1:28:20.72,1:28:29.51,Default,,0000,0000,0000,,And times your favorite\Nexpression, which is cosine
Dialogue: 0,1:28:29.51,1:28:32.55,Default,,0000,0000,0000,,squared theta, minus\Ni squared theta.
Dialogue: 0,1:28:32.55,1:28:34.52,Default,,0000,0000,0000,,And you're going to ask me why.
Dialogue: 0,1:28:34.52,1:28:36.07,Default,,0000,0000,0000,,You shouldn't ask me why.
Dialogue: 0,1:28:36.07,1:28:40.30,Default,,0000,0000,0000,,You just square these\Nand subtract them,
Dialogue: 0,1:28:40.30,1:28:44.05,Default,,0000,0000,0000,,and see what in the world\Nyou're going to get.
Dialogue: 0,1:28:44.05,1:28:48.29,Default,,0000,0000,0000,,Because you get r squared\Ntimes cosine squared,
Dialogue: 0,1:28:48.29,1:28:49.51,Default,,0000,0000,0000,,minus i squared.
Dialogue: 0,1:28:49.51,1:28:51.43,Default,,0000,0000,0000,,I'm too lazy to write\Ndown the argument.
Dialogue: 0,1:28:51.43,1:28:52.85,Default,,0000,0000,0000,,But you know we\Nhave trigonometry.
Dialogue: 0,1:28:52.85,1:28:55.67,Default,,0000,0000,0000,,
Dialogue: 0,1:28:55.67,1:28:58.17,Default,,0000,0000,0000,,Yes, you see why it's\Nimportant for you
Dialogue: 0,1:28:58.17,1:29:02.01,Default,,0000,0000,0000,,to learn trigonometry\Nwhen you are little.
Dialogue: 0,1:29:02.01,1:29:05.73,Default,,0000,0000,0000,,You may be 50 or\N60, in high school,
Dialogue: 0,1:29:05.73,1:29:09.19,Default,,0000,0000,0000,,or you may be freshman year.
Dialogue: 0,1:29:09.19,1:29:11.98,Default,,0000,0000,0000,,I don't care when, but you\Nhave to learn that this is
Dialogue: 0,1:29:11.98,1:29:14.31,Default,,0000,0000,0000,,the cosine of the double angle.
Dialogue: 0,1:29:14.31,1:29:16.24,Default,,0000,0000,0000,,How many of you remember that?
Dialogue: 0,1:29:16.24,1:29:18.54,Default,,0000,0000,0000,,Maybe you learned that?
Dialogue: 0,1:29:18.54,1:29:19.46,Default,,0000,0000,0000,,Remember that?
Dialogue: 0,1:29:19.46,1:29:21.40,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:29:21.40,1:29:25.80,Default,,0000,0000,0000,,I don't blame you at all\Nwhen you don't remember,
Dialogue: 0,1:29:25.80,1:29:30.92,Default,,0000,0000,0000,,because since I've been\Nthe main checker of finals
Dialogue: 0,1:29:30.92,1:29:39.49,Default,,0000,0000,0000,,for the past five years--\Nit's a lot of finals.
Dialogue: 0,1:29:39.49,1:29:40.69,Default,,0000,0000,0000,,Yeah, the i is there.
Dialogue: 0,1:29:40.69,1:29:42.84,Default,,0000,0000,0000,,That's exactly what\NI wanted to tell you,
Dialogue: 0,1:29:42.84,1:29:46.45,Default,,0000,0000,0000,,that's why I left some room.
Dialogue: 0,1:29:46.45,1:29:52.57,Default,,0000,0000,0000,,This data would be t.
Dialogue: 0,1:29:52.57,1:29:56.37,Default,,0000,0000,0000,,The double angle formula did\Nnot appear on many finals.
Dialogue: 0,1:29:56.37,1:29:58.11,Default,,0000,0000,0000,,And I was thinking\Nit's a period.
Dialogue: 0,1:29:58.11,1:29:59.96,Default,,0000,0000,0000,,When I ask the\Ninstructors, generally they
Dialogue: 0,1:29:59.96,1:30:06.55,Default,,0000,0000,0000,,say students have trouble\Nremembering or understanding
Dialogue: 0,1:30:06.55,1:30:10.13,Default,,0000,0000,0000,,this later on, by\Navoiding the issue,
Dialogue: 0,1:30:10.13,1:30:14.01,Default,,0000,0000,0000,,you sort of bound to it for\Nthe first time in Cal 2,
Dialogue: 0,1:30:14.01,1:30:16.87,Default,,0000,0000,0000,,because there are any\Ngeometric formulas.
Dialogue: 0,1:30:16.87,1:30:21.63,Default,,0000,0000,0000,,And then, you bump again\Ninside it in Cal 3.
Dialogue: 0,1:30:21.63,1:30:23.31,Default,,0000,0000,0000,,And it never leaves you.
Dialogue: 0,1:30:23.31,1:30:27.56,Default,,0000,0000,0000,,So this, just knowing this\Nwill help you so much.
Dialogue: 0,1:30:27.56,1:30:30.85,Default,,0000,0000,0000,,Let me put the r nicely here.
Dialogue: 0,1:30:30.85,1:30:34.12,Default,,0000,0000,0000,,And now finally, we know\Nhow to solve it, because I'm
Dialogue: 0,1:30:34.12,1:30:35.29,Default,,0000,0000,0000,,going to go ahead and erase.
Dialogue: 0,1:30:35.29,1:30:44.66,Default,,0000,0000,0000,,
Dialogue: 0,1:30:44.66,1:30:49.25,Default,,0000,0000,0000,,So why it is good for us is\Nthat-- as Matthew observed
Dialogue: 0,1:30:49.25,1:30:52.81,Default,,0000,0000,0000,,a few moments ago,\Nwhenever you have
Dialogue: 0,1:30:52.81,1:30:56.73,Default,,0000,0000,0000,,a product of a function, you\Nnot only in a function in theta
Dialogue: 0,1:30:56.73,1:31:01.17,Default,,0000,0000,0000,,only, your life becomes easier\Nbecause you can separate them
Dialogue: 0,1:31:01.17,1:31:03.05,Default,,0000,0000,0000,,between the rhos.
Dialogue: 0,1:31:03.05,1:31:04.19,Default,,0000,0000,0000,,In two different products.
Dialogue: 0,1:31:04.19,1:31:05.95,Default,,0000,0000,0000,,So that's would be this theorem.
Dialogue: 0,1:31:05.95,1:31:11.27,Default,,0000,0000,0000,,And you have 3 times-- the\Npart that depends only on r,
Dialogue: 0,1:31:11.27,1:31:13.99,Default,,0000,0000,0000,,and the part that depends\Nonly on theta, let's
Dialogue: 0,1:31:13.99,1:31:14.89,Default,,0000,0000,0000,,put them separate.
Dialogue: 0,1:31:14.89,1:31:22.17,Default,,0000,0000,0000,,We need theta, and\Ndr. And what do you
Dialogue: 0,1:31:22.17,1:31:24.52,Default,,0000,0000,0000,,integrate when you integrate?
Dialogue: 0,1:31:24.52,1:31:25.47,Default,,0000,0000,0000,,r cubed.
Dialogue: 0,1:31:25.47,1:31:29.10,Default,,0000,0000,0000,,Attention, do not do rr.
Dialogue: 0,1:31:29.10,1:31:30.85,Default,,0000,0000,0000,,From 0 to r.
Dialogue: 0,1:31:30.85,1:31:32.36,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:31:32.36,1:31:33.65,Default,,0000,0000,0000,,STUDENT: And cosine theta?
Dialogue: 0,1:31:33.65,1:31:39.55,Default,,0000,0000,0000,,MAGDALENA TODA: And then you\Nhave a 0 to 2 pi, cosine 2.
Dialogue: 0,1:31:39.55,1:31:42.13,Default,,0000,0000,0000,,now, let me give\Nyou-- Let me tell you
Dialogue: 0,1:31:42.13,1:31:44.88,Default,,0000,0000,0000,,what it is, because when\NI was young, I was naive
Dialogue: 0,1:31:44.88,1:31:48.10,Default,,0000,0000,0000,,and I always started with that.
Dialogue: 0,1:31:48.10,1:31:52.49,Default,,0000,0000,0000,,You should always start with the\Npart, the trig part in theta.
Dialogue: 0,1:31:52.49,1:31:54.03,Default,,0000,0000,0000,,Because that becomes 0.
Dialogue: 0,1:31:54.03,1:31:56.55,Default,,0000,0000,0000,,So no matter how\Nugly this is, I've
Dialogue: 0,1:31:56.55,1:31:59.79,Default,,0000,0000,0000,,had professors who are\Nplaying games with us,
Dialogue: 0,1:31:59.79,1:32:03.80,Default,,0000,0000,0000,,and they were giving us\Nsome extremely ugly thing
Dialogue: 0,1:32:03.80,1:32:06.47,Default,,0000,0000,0000,,that would take you forever\Nfor you to integrate.
Dialogue: 0,1:32:06.47,1:32:09.29,Default,,0000,0000,0000,,Or sometimes, it would have\Nbeen impossible to integrate.
Dialogue: 0,1:32:09.29,1:32:11.57,Default,,0000,0000,0000,,But then, the whole\Nthing would have been 0
Dialogue: 0,1:32:11.57,1:32:14.30,Default,,0000,0000,0000,,because when you\Nintegrate cosine 2 theta,
Dialogue: 0,1:32:14.30,1:32:16.61,Default,,0000,0000,0000,,it goes to sine theta.
Dialogue: 0,1:32:16.61,1:32:20.60,Default,,0000,0000,0000,,Sine 2 theta at 2 pi and 0\Nare the same things, 0 minus 0
Dialogue: 0,1:32:20.60,1:32:21.12,Default,,0000,0000,0000,,equals z.
Dialogue: 0,1:32:21.12,1:32:23.80,Default,,0000,0000,0000,,So the answer is z.
Dialogue: 0,1:32:23.80,1:32:27.04,Default,,0000,0000,0000,,I cannot tell you how many\Nprofessors I've had who will
Dialogue: 0,1:32:27.04,1:32:28.57,Default,,0000,0000,0000,,play this game with us.
Dialogue: 0,1:32:28.57,1:32:30.36,Default,,0000,0000,0000,,They give us something\Nthat discouraged us.
Dialogue: 0,1:32:30.36,1:32:34.01,Default,,0000,0000,0000,,No, it's not a piece of cake,\Ncompared to what I have.
Dialogue: 0,1:32:34.01,1:32:37.45,Default,,0000,0000,0000,,Some integral value\Nwill go over two lines,
Dialogue: 0,1:32:37.45,1:32:40.49,Default,,0000,0000,0000,,with a huge polynomial\Nor something.
Dialogue: 0,1:32:40.49,1:32:44.00,Default,,0000,0000,0000,,But in the end, the integral\Nwas 0 for such a result. Yes?
Dialogue: 0,1:32:44.00,1:32:45.25,Default,,0000,0000,0000,,STUDENT: So I have a question.
Dialogue: 0,1:32:45.25,1:32:49.70,Default,,0000,0000,0000,,Could we take that force and\Nprove that it was conservative?
Dialogue: 0,1:32:49.70,1:32:54.94,Default,,0000,0000,0000,,MAGDALENA TODA: So now\Nthat I'm questioning this,
Dialogue: 0,1:32:54.94,1:32:59.66,Default,,0000,0000,0000,,I'm not questioning\Nyou, but I-- is
Dialogue: 0,1:32:59.66,1:33:05.63,Default,,0000,0000,0000,,the force, that is with you--\Nwhat is the original force
Dialogue: 0,1:33:05.63,1:33:08.36,Default,,0000,0000,0000,,that Alex is talking about?
Dialogue: 0,1:33:08.36,1:33:17.00,Default,,0000,0000,0000,,If I take y cubed i plus x cubed\Nj-- and you have to be careful.
Dialogue: 0,1:33:17.00,1:33:18.92,Default,,0000,0000,0000,,Is this conservative?
Dialogue: 0,1:33:18.92,1:33:22.67,Default,,0000,0000,0000,,
Dialogue: 0,1:33:22.67,1:33:25.13,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,1:33:25.13,1:33:27.74,Default,,0000,0000,0000,,MAGDALENA TODA: Really?
Dialogue: 0,1:33:27.74,1:33:30.94,Default,,0000,0000,0000,,Why would we pick\Na conservative?
Dialogue: 0,1:33:30.94,1:33:33.81,Default,,0000,0000,0000,,STUDENT: Y squared plus\Nx squared over 2 is--
Dialogue: 0,1:33:33.81,1:33:35.60,Default,,0000,0000,0000,,MAGDALENA TODA: Why is\Nit not conservative?
Dialogue: 0,1:33:35.60,1:33:38.86,Default,,0000,0000,0000,,
Dialogue: 0,1:33:38.86,1:33:40.56,Default,,0000,0000,0000,,IT doesn't pass the hole test.
Dialogue: 0,1:33:40.56,1:33:43.59,Default,,0000,0000,0000,,
Dialogue: 0,1:33:43.59,1:33:48.28,Default,,0000,0000,0000,,So p sub y is not\Nequal to q sub x.
Dialogue: 0,1:33:48.28,1:33:52.09,Default,,0000,0000,0000,,If you primed this with respect\Nto y, you get that dy squared.
Dialogue: 0,1:33:52.09,1:33:54.64,Default,,0000,0000,0000,,Prime this with this respect\Nto x, you get 3x squared.
Dialogue: 0,1:33:54.64,1:33:57.43,Default,,0000,0000,0000,,So it's not concerned with him.
Dialogue: 0,1:33:57.43,1:34:00.64,Default,,0000,0000,0000,,And still, I'm\Ngetting-- it's a loop,
Dialogue: 0,1:34:00.64,1:34:04.52,Default,,0000,0000,0000,,and I'm getting a 0, sort\Nof like I would expect it
Dialogue: 0,1:34:04.52,1:34:07.53,Default,,0000,0000,0000,,I had any dependence of that.
Dialogue: 0,1:34:07.53,1:34:09.25,Default,,0000,0000,0000,,What is the secret here?
Dialogue: 0,1:34:09.25,1:34:14.36,Default,,0000,0000,0000,,STUDENT: That is conservative,\Ngiven a condition.
Dialogue: 0,1:34:14.36,1:34:16.73,Default,,0000,0000,0000,,MAGDALENA TODA: Yes,\Ngiven a condition
Dialogue: 0,1:34:16.73,1:34:21.17,Default,,0000,0000,0000,,that your x and y are moving\Non the serpent's circle.
Dialogue: 0,1:34:21.17,1:34:25.90,Default,,0000,0000,0000,,And that happens, because this\Nis a symmetric expression,
Dialogue: 0,1:34:25.90,1:34:28.42,Default,,0000,0000,0000,,and x and y are\Nmoving on a circle,
Dialogue: 0,1:34:28.42,1:34:31.09,Default,,0000,0000,0000,,and one is the cosine theta\Nand one is sine theta.
Dialogue: 0,1:34:31.09,1:34:35.16,Default,,0000,0000,0000,,So in the end, it\Nsimplifies out.
Dialogue: 0,1:34:35.16,1:34:40.28,Default,,0000,0000,0000,,But in general, if I would\Nhave this kind of problem--
Dialogue: 0,1:34:40.28,1:34:43.95,Default,,0000,0000,0000,,if somebody asked me is this\Nconservative, the answer is no.
Dialogue: 0,1:34:43.95,1:34:46.12,Default,,0000,0000,0000,,Let me give you a\Nfew more examples.
Dialogue: 0,1:34:46.12,1:34:57.62,Default,,0000,0000,0000,,
Dialogue: 0,1:34:57.62,1:35:14.76,Default,,0000,0000,0000,,One example that maybe will look\Nhard to most people is here.
Dialogue: 0,1:35:14.76,1:35:36.54,Default,,0000,0000,0000,,
Dialogue: 0,1:35:36.54,1:35:49.75,Default,,0000,0000,0000,,The vector value function\Ngiven by f of x, y incline,
Dialogue: 0,1:35:49.75,1:35:51.95,Default,,0000,0000,0000,,are two values.
Dialogue: 0,1:35:51.95,1:35:54.97,Default,,0000,0000,0000,,No, I mean define two\Nvalues of [INAUDIBLE].
Dialogue: 0,1:35:54.97,1:36:17.04,Default,,0000,0000,0000,,
Dialogue: 0,1:36:17.04,1:36:19.17,Default,,0000,0000,0000,,A typical exam problem.
Dialogue: 0,1:36:19.17,1:36:22.98,Default,,0000,0000,0000,,And I saw it at\NTexas A&M, as well.
Dialogue: 0,1:36:22.98,1:36:27.61,Default,,0000,0000,0000,,So maybe some people like this\Nkind of a, b, c, d problem.
Dialogue: 0,1:36:27.61,1:36:28.84,Default,,0000,0000,0000,,Is f conservative?
Dialogue: 0,1:36:28.84,1:36:37.22,Default,,0000,0000,0000,,
Dialogue: 0,1:36:37.22,1:36:38.40,Default,,0000,0000,0000,,STUDENT: Yep
Dialogue: 0,1:36:38.40,1:36:39.86,Default,,0000,0000,0000,,MAGDALENA TODA:\NYou already did it?
Dialogue: 0,1:36:39.86,1:36:41.16,Default,,0000,0000,0000,,Good for you guys.
Dialogue: 0,1:36:41.16,1:36:45.34,Default,,0000,0000,0000,,So if I gave you one that\Nhas three components what
Dialogue: 0,1:36:45.34,1:36:47.48,Default,,0000,0000,0000,,did you have to do?
Dialogue: 0,1:36:47.48,1:36:49.86,Default,,0000,0000,0000,,Compute the curl.
Dialogue: 0,1:36:49.86,1:36:52.46,Default,,0000,0000,0000,,You can, of course, compute\Nthe curl also on this one
Dialogue: 0,1:36:52.46,1:36:55.12,Default,,0000,0000,0000,,and have 0 for the\Nthird component.
Dialogue: 0,1:36:55.12,1:37:01.82,Default,,0000,0000,0000,,But the simplest thing\Nis to do f1 and f2.
Dialogue: 0,1:37:01.82,1:37:07.04,Default,,0000,0000,0000,,f1 prime with respect to y\Nequals f2 prime with respect
Dialogue: 0,1:37:07.04,1:37:08.09,Default,,0000,0000,0000,,to x.
Dialogue: 0,1:37:08.09,1:37:12.87,Default,,0000,0000,0000,,So I'm going to\Nmake a smile here.
Dialogue: 0,1:37:12.87,1:37:16.31,Default,,0000,0000,0000,,And you realize that the authors\Nof such a problem, whether they
Dialogue: 0,1:37:16.31,1:37:21.27,Default,,0000,0000,0000,,are at Tech or at Texas\NA&M. They do that on purpose
Dialogue: 0,1:37:21.27,1:37:28.39,Default,,0000,0000,0000,,so that you can use this\Nresult to the next level.
Dialogue: 0,1:37:28.39,1:38:04.07,Default,,0000,0000,0000,,And they're saying compute\Nthe happy u over the curve
Dialogue: 0,1:38:04.07,1:38:23.07,Default,,0000,0000,0000,,x cubed and y cubed equals 8 on\Nthe path that connects points
Dialogue: 0,1:38:23.07,1:38:29.38,Default,,0000,0000,0000,,2, 1 and 1, 2 in [INAUDIBLE].
Dialogue: 0,1:38:29.38,1:38:39.28,Default,,0000,0000,0000,,
Dialogue: 0,1:38:39.28,1:38:46.20,Default,,0000,0000,0000,,Does this integral depend on f?
Dialogue: 0,1:38:46.20,1:38:51.14,Default,,0000,0000,0000,,
Dialogue: 0,1:38:51.14,1:38:52.11,Default,,0000,0000,0000,,State why.
Dialogue: 0,1:38:52.11,1:38:58.26,Default,,0000,0000,0000,,
Dialogue: 0,1:38:58.26,1:39:04.54,Default,,0000,0000,0000,,And you see, they don't tell\Nyou find the scalar potential.
Dialogue: 0,1:39:04.54,1:39:06.92,Default,,0000,0000,0000,,Which is bad, and\Nmany of you will
Dialogue: 0,1:39:06.92,1:39:09.12,Default,,0000,0000,0000,,be able to see it\Nbecause you have
Dialogue: 0,1:39:09.12,1:39:14.00,Default,,0000,0000,0000,,good mathematical intuition,\Nand a computer process
Dialogue: 0,1:39:14.00,1:39:17.29,Default,,0000,0000,0000,,planning in the background\Nover all the other processes.
Dialogue: 0,1:39:17.29,1:39:18.90,Default,,0000,0000,0000,,We are very visual people.
Dialogue: 0,1:39:18.90,1:39:22.31,Default,,0000,0000,0000,,If you realize that every time\Njust there with each other
Dialogue: 0,1:39:22.31,1:39:25.72,Default,,0000,0000,0000,,through the classroom, there\Nare hundreds of distractions.
Dialogue: 0,1:39:25.72,1:39:28.16,Default,,0000,0000,0000,,There's the screen,\Nthere is somebody
Dialogue: 0,1:39:28.16,1:39:31.21,Default,,0000,0000,0000,,who's next to you\Nwho's sneezing,
Dialogue: 0,1:39:31.21,1:39:34.57,Default,,0000,0000,0000,,all sorts of distractions.
Dialogue: 0,1:39:34.57,1:39:37.86,Default,,0000,0000,0000,,Still, your computer\Nunit can still
Dialogue: 0,1:39:37.86,1:39:41.20,Default,,0000,0000,0000,,function, trying to\Nintegrate and find the scalar
Dialogue: 0,1:39:41.20,1:39:42.45,Default,,0000,0000,0000,,potential, which is a miracle.
Dialogue: 0,1:39:42.45,1:39:46.25,Default,,0000,0000,0000,,I don't know how we managed\Nto do that after all.
Dialogue: 0,1:39:46.25,1:39:49.84,Default,,0000,0000,0000,,If you don't manage to do that,\Nwhat do you have to set up?
Dialogue: 0,1:39:49.84,1:39:54.87,Default,,0000,0000,0000,,You have to say, find is\Nthere-- well, you know there is.
Dialogue: 0,1:39:54.87,1:39:59.90,Default,,0000,0000,0000,,So you're not going to question\Nthe existence of the scalar
Dialogue: 0,1:39:59.90,1:40:03.68,Default,,0000,0000,0000,,potential You know it exists,\Nbut you don't know what it is.
Dialogue: 0,1:40:03.68,1:40:09.75,Default,,0000,0000,0000,,What is f such that f sub\Nx would be 6xy plus 1,
Dialogue: 0,1:40:09.75,1:40:14.49,Default,,0000,0000,0000,,and m sub y will be 3x squared?
Dialogue: 0,1:40:14.49,1:40:18.32,Default,,0000,0000,0000,,And normally, you would\Nhave to integrate backwards.
Dialogue: 0,1:40:18.32,1:40:21.35,Default,,0000,0000,0000,,Now, I'll give you 10 seconds.
Dialogue: 0,1:40:21.35,1:40:25.19,Default,,0000,0000,0000,,If in 10 seconds, you don't\Nfind me a scalar potential,
Dialogue: 0,1:40:25.19,1:40:27.12,Default,,0000,0000,0000,,I'm going to make you\Nintegrate backwards.
Dialogue: 0,1:40:27.12,1:40:31.36,Default,,0000,0000,0000,,So this is finding the scalar\Npotential by integration.
Dialogue: 0,1:40:31.36,1:40:34.46,Default,,0000,0000,0000,,The way you should, if\Nyou weren't very smart.
Dialogue: 0,1:40:34.46,1:40:37.97,Default,,0000,0000,0000,,But I think you're\Nsmart enough to smell
Dialogue: 0,1:40:37.97,1:40:42.41,Default,,0000,0000,0000,,the potential-- Very good.
Dialogue: 0,1:40:42.41,1:40:44.03,Default,,0000,0000,0000,,But what if you don't?
Dialogue: 0,1:40:44.03,1:40:46.05,Default,,0000,0000,0000,,OK I'm asking.
Dialogue: 0,1:40:46.05,1:40:50.48,Default,,0000,0000,0000,,So we had one or two\Nstudent who figured it out.
Dialogue: 0,1:40:50.48,1:40:51.22,Default,,0000,0000,0000,,What if you don't?
Dialogue: 0,1:40:51.22,1:40:55.24,Default,,0000,0000,0000,,If you don't, you can still do\Nperfectly fine on this problem.
Dialogue: 0,1:40:55.24,1:41:01.12,Default,,0000,0000,0000,,Let's see how we do it\Nwithout seeing or guessing.
Dialogue: 0,1:41:01.12,1:41:03.31,Default,,0000,0000,0000,,His brain was running\Nin the background.
Dialogue: 0,1:41:03.31,1:41:05.06,Default,,0000,0000,0000,,He came up with the answer.
Dialogue: 0,1:41:05.06,1:41:05.66,Default,,0000,0000,0000,,He's happy.
Dialogue: 0,1:41:05.66,1:41:09.64,Default,,0000,0000,0000,,He can move on to\Nthe next level.
Dialogue: 0,1:41:09.64,1:41:12.27,Default,,0000,0000,0000,,STUDENT: Integrate both\Nsides with respect to r.
Dialogue: 0,1:41:12.27,1:41:15.97,Default,,0000,0000,0000,,MAGDALENA TODA: Right, and\Nthen mix and match them.
Dialogue: 0,1:41:15.97,1:41:17.66,Default,,0000,0000,0000,,Make them in work.
Dialogue: 0,1:41:17.66,1:41:20.55,Default,,0000,0000,0000,,So try to integrate\Nwith respect to x.
Dialogue: 0,1:41:20.55,1:41:25.97,Default,,0000,0000,0000,,6y-- or plus 1, I'm sorry guys.
Dialogue: 0,1:41:25.97,1:41:28.97,Default,,0000,0000,0000,,And once you get it,\Nyou're going to get--
Dialogue: 0,1:41:28.97,1:41:32.13,Default,,0000,0000,0000,,STUDENT: 3x squared y plus x.
Dialogue: 0,1:41:32.13,1:41:34.69,Default,,0000,0000,0000,,MAGDALENA TODA: And\Nplus a c of what?
Dialogue: 0,1:41:34.69,1:41:37.97,Default,,0000,0000,0000,,And then take this fellow and\Nprime it with respect to y.
Dialogue: 0,1:41:37.97,1:41:41.48,Default,,0000,0000,0000,,And you're going to\Nget-- it's not hard.
Dialogue: 0,1:41:41.48,1:41:44.26,Default,,0000,0000,0000,,You're going to get dx\Nsquared plus nothing,
Dialogue: 0,1:41:44.26,1:41:50.17,Default,,0000,0000,0000,,plus c from the y, and it's\Ngood because I gave you
Dialogue: 0,1:41:50.17,1:41:51.33,Default,,0000,0000,0000,,a simple one.
Dialogue: 0,1:41:51.33,1:41:54.39,Default,,0000,0000,0000,,So sometimes you can\Nhave something here,
Dialogue: 0,1:41:54.39,1:41:57.26,Default,,0000,0000,0000,,but in this case, it was just 0.
Dialogue: 0,1:41:57.26,1:42:00.40,Default,,0000,0000,0000,,So c is kappa as a constant.
Dialogue: 0,1:42:00.40,1:42:04.77,Default,,0000,0000,0000,,So instead of why we teach\Nfound with a plus kappa here,
Dialogue: 0,1:42:04.77,1:42:07.94,Default,,0000,0000,0000,,and it still does it.
Dialogue: 0,1:42:07.94,1:42:12.81,Default,,0000,0000,0000,,So on such a problem,\NI don't know,
Dialogue: 0,1:42:12.81,1:42:18.07,Default,,0000,0000,0000,,but I think I would give equal\Nweights to it, B and C. Compute
Dialogue: 0,1:42:18.07,1:42:20.87,Default,,0000,0000,0000,,the path integral\Nover the curve.
Dialogue: 0,1:42:20.87,1:42:24.20,Default,,0000,0000,0000,,This is horrible, as\Nan increasing curve.
Dialogue: 0,1:42:24.20,1:42:26.62,Default,,0000,0000,0000,,But I know that\Nthere is a path that
Dialogue: 0,1:42:26.62,1:42:28.68,Default,,0000,0000,0000,,connects the points 2, 1 and 1.
Dialogue: 0,1:42:28.68,1:42:30.43,Default,,0000,0000,0000,,What I have to pay\Nattention to in my mind
Dialogue: 0,1:42:30.43,1:42:33.39,Default,,0000,0000,0000,,is that these points\Nactually are on the curve.
Dialogue: 0,1:42:33.39,1:42:36.17,Default,,0000,0000,0000,,And they are, because I\Nhave 8 times 1 equals 8,
Dialogue: 0,1:42:36.17,1:42:37.81,Default,,0000,0000,0000,,1 times 8 equals 8.
Dialogue: 0,1:42:37.81,1:42:41.36,Default,,0000,0000,0000,,So while I was writing it,\NI had to think a little bit
Dialogue: 0,1:42:41.36,1:42:42.58,Default,,0000,0000,0000,,on the problem.
Dialogue: 0,1:42:42.58,1:42:45.44,Default,,0000,0000,0000,,If you were to\Ndraw-- well that's
Dialogue: 0,1:42:45.44,1:42:48.45,Default,,0000,0000,0000,,for you have to find\Nout when you go home.
Dialogue: 0,1:42:48.45,1:42:52.16,Default,,0000,0000,0000,,What do you think\Nthis is going to be?
Dialogue: 0,1:42:52.16,1:42:55.08,Default,,0000,0000,0000,,
Dialogue: 0,1:42:55.08,1:42:58.00,Default,,0000,0000,0000,,Actually, we have to\Ndo it now, because it's
Dialogue: 0,1:42:58.00,1:43:01.77,Default,,0000,0000,0000,,a lot simpler than\Nyou think it is.
Dialogue: 0,1:43:01.77,1:43:06.51,Default,,0000,0000,0000,,x and y will be positive,\NI can also restrict that.
Dialogue: 0,1:43:06.51,1:43:09.06,Default,,0000,0000,0000,,It looks horrible, but\Nit's actually much easier
Dialogue: 0,1:43:09.06,1:43:09.99,Default,,0000,0000,0000,,than you think.
Dialogue: 0,1:43:09.99,1:43:15.77,Default,,0000,0000,0000,,So how do I compute that path\Nintegral that makes the points?
Dialogue: 0,1:43:15.77,1:43:19.26,Default,,0000,0000,0000,,I'm going to have\Nfundamental there.
Dialogue: 0,1:43:19.26,1:43:22.76,Default,,0000,0000,0000,,
Dialogue: 0,1:43:22.76,1:43:27.35,Default,,0000,0000,0000,,Which has f of x at q\Nminus f, with p, which
Dialogue: 0,1:43:27.35,1:43:31.28,Default,,0000,0000,0000,,says that little f is here.
Dialogue: 0,1:43:31.28,1:43:41.12,Default,,0000,0000,0000,,3x squared y plus\Nx at 2, 1 minus 3x
Dialogue: 0,1:43:41.12,1:43:47.30,Default,,0000,0000,0000,,squared y plus x at 1, 2.
Dialogue: 0,1:43:47.30,1:43:53.45,Default,,0000,0000,0000,,So all I have to do is\Ngo ahead and-- do you
Dialogue: 0,1:43:53.45,1:43:56.98,Default,,0000,0000,0000,,see what I'm actually doing?
Dialogue: 0,1:43:56.98,1:43:57.79,Default,,0000,0000,0000,,It's funny.
Dialogue: 0,1:43:57.79,1:44:02.46,Default,,0000,0000,0000,,Which one is the origin, and\Nwhich one is the endpoint?
Dialogue: 0,1:44:02.46,1:44:03.83,Default,,0000,0000,0000,,The problem doesn't tell you.
Dialogue: 0,1:44:03.83,1:44:07.30,Default,,0000,0000,0000,,It tells you only you are\Nconnecting the two points.
Dialogue: 0,1:44:07.30,1:44:10.16,Default,,0000,0000,0000,,But which one is the alpha,\Nand which one is the omega?
Dialogue: 0,1:44:10.16,1:44:10.96,Default,,0000,0000,0000,,Where do you start?
Dialogue: 0,1:44:10.96,1:44:12.55,Default,,0000,0000,0000,,You start here or\Nyou start here?
Dialogue: 0,1:44:12.55,1:44:16.46,Default,,0000,0000,0000,,
Dialogue: 0,1:44:16.46,1:44:17.07,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:44:17.07,1:44:19.15,Default,,0000,0000,0000,,Sort of arbitrary.
Dialogue: 0,1:44:19.15,1:44:22.23,Default,,0000,0000,0000,,How do you handle this problem?
Dialogue: 0,1:44:22.23,1:44:26.15,Default,,0000,0000,0000,,Depending on the direction--\Npick one direction you move on
Dialogue: 0,1:44:26.15,1:44:28.53,Default,,0000,0000,0000,,along the r, it's up to you.
Dialogue: 0,1:44:28.53,1:44:31.69,Default,,0000,0000,0000,,And then you get an answer, and\Nif you change the direction,
Dialogue: 0,1:44:31.69,1:44:34.25,Default,,0000,0000,0000,,what's going to happen\Nto the integral?
Dialogue: 0,1:44:34.25,1:44:37.97,Default,,0000,0000,0000,,It's just change the\Nsign and that's all.
Dialogue: 0,1:44:37.97,1:44:43.27,Default,,0000,0000,0000,,3 times 4, times 1, plus 2--\Nguys, keep an eye on my algebra
Dialogue: 0,1:44:43.27,1:44:48.10,Default,,0000,0000,0000,,please, because I\Ndon't want to mess up.
Dialogue: 0,1:44:48.10,1:44:49.88,Default,,0000,0000,0000,,Am I right, here?
Dialogue: 0,1:44:49.88,1:44:50.42,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,1:44:50.42,1:44:52.30,Default,,0000,0000,0000,,MAGDALENA TODA: So how much?
Dialogue: 0,1:44:52.30,1:44:53.99,Default,,0000,0000,0000,,14, is it?
Dialogue: 0,1:44:53.99,1:44:56.32,Default,,0000,0000,0000,,STUDENT: It's 7.
Dialogue: 0,1:44:56.32,1:44:57.32,Default,,0000,0000,0000,,MAGDALENA TODA: Minus 7.
Dialogue: 0,1:44:57.32,1:45:06.49,Default,,0000,0000,0000,,
Dialogue: 0,1:45:06.49,1:45:06.99,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:45:06.99,1:45:07.88,Default,,0000,0000,0000,,Wonderful.
Dialogue: 0,1:45:07.88,1:45:11.67,Default,,0000,0000,0000,,So we know what to get,\Nand we know this does not
Dialogue: 0,1:45:11.67,1:45:13.18,Default,,0000,0000,0000,,depend on the fact.
Dialogue: 0,1:45:13.18,1:45:16.77,Default,,0000,0000,0000,,How much blah, blah,\Nblah does the instructor
Dialogue: 0,1:45:16.77,1:45:20.89,Default,,0000,0000,0000,,expect for you to get full\Ncredit on the problem?
Dialogue: 0,1:45:20.89,1:45:22.22,Default,,0000,0000,0000,,STUDENT: Just enough to explain.
Dialogue: 0,1:45:22.22,1:45:23.84,Default,,0000,0000,0000,,MAGDALENA TODA: Just\Nenough to explain.
Dialogue: 0,1:45:23.84,1:45:29.81,Default,,0000,0000,0000,,About 2 lines or 1 line saying\Nyou can say anything really.
Dialogue: 0,1:45:29.81,1:45:34.47,Default,,0000,0000,0000,,You can say this is the theorem\Nthat either shows independence
Dialogue: 0,1:45:34.47,1:45:35.86,Default,,0000,0000,0000,,of that integral.
Dialogue: 0,1:45:35.86,1:45:43.31,Default,,0000,0000,0000,,If the force F vector value\Nfunction is conservative,
Dialogue: 0,1:45:43.31,1:45:46.75,Default,,0000,0000,0000,,then this is what\Nyou have to write.
Dialogue: 0,1:45:46.75,1:45:49.25,Default,,0000,0000,0000,,This doesn't depend\Non the path c.
Dialogue: 0,1:45:49.25,1:45:51.45,Default,,0000,0000,0000,,And you apply the\Nfundamental theorem
Dialogue: 0,1:45:51.45,1:45:54.29,Default,,0000,0000,0000,,of path integrals for\Nthe scalar potential.
Dialogue: 0,1:45:54.29,1:45:57.76,Default,,0000,0000,0000,,And that scalar potential\Ndepends on the endpoints
Dialogue: 0,1:45:57.76,1:45:59.66,Default,,0000,0000,0000,,that you're taking.
Dialogue: 0,1:45:59.66,1:46:02.15,Default,,0000,0000,0000,,And the value of\Nthe work depends--
Dialogue: 0,1:46:02.15,1:46:06.29,Default,,0000,0000,0000,,the work depends only on the\Nscalar potential and the two
Dialogue: 0,1:46:06.29,1:46:07.51,Default,,0000,0000,0000,,points.
Dialogue: 0,1:46:07.51,1:46:08.32,Default,,0000,0000,0000,,That's enough.
Dialogue: 0,1:46:08.32,1:46:09.78,Default,,0000,0000,0000,,That's more than enough.
Dialogue: 0,1:46:09.78,1:46:13.82,Default,,0000,0000,0000,,What if somebody's\Nnot good with wording?
Dialogue: 0,1:46:13.82,1:46:17.17,Default,,0000,0000,0000,,I'm not going to write\Nher all that explanation.
Dialogue: 0,1:46:17.17,1:46:21.65,Default,,0000,0000,0000,,I'm just going to say whatever.
Dialogue: 0,1:46:21.65,1:46:25.10,Default,,0000,0000,0000,,I'm going to give\Nher the theorem
Dialogue: 0,1:46:25.10,1:46:27.21,Default,,0000,0000,0000,,in mathematical compressed way.
Dialogue: 0,1:46:27.21,1:46:30.96,Default,,0000,0000,0000,,And I don't care if she\Nunderstands it or not.
Dialogue: 0,1:46:30.96,1:46:34.62,Default,,0000,0000,0000,,Even if you write this\Nformula with not much wording,
Dialogue: 0,1:46:34.62,1:46:36.87,Default,,0000,0000,0000,,I still give you credit.
Dialogue: 0,1:46:36.87,1:46:38.56,Default,,0000,0000,0000,,But I would prefer\Nthat you give me
Dialogue: 0,1:46:38.56,1:46:41.64,Default,,0000,0000,0000,,some sort of-- some\Nsort of explanation.
Dialogue: 0,1:46:41.64,1:46:42.80,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:46:42.80,1:46:44.09,Default,,0000,0000,0000,,STUDENT: You said answer was 0.
Dialogue: 0,1:46:44.09,1:46:45.80,Default,,0000,0000,0000,,Then it would have\Nbeen path independent?
Dialogue: 0,1:46:45.80,1:46:50.86,Default,,0000,0000,0000,,
Dialogue: 0,1:46:50.86,1:46:53.63,Default,,0000,0000,0000,,MAGDALENA TODA: No, the\Nanswer would not be for sure 0
Dialogue: 0,1:46:53.63,1:46:56.77,Default,,0000,0000,0000,,if it was a longer loop.
Dialogue: 0,1:46:56.77,1:46:58.87,Default,,0000,0000,0000,,If it were a longer\Nclosed curve,
Dialogue: 0,1:46:58.87,1:47:03.57,Default,,0000,0000,0000,,that way where it\Nstarts, it ends.
Dialogue: 0,1:47:03.57,1:47:07.47,Default,,0000,0000,0000,,Even if I take a weekly\Nroad between the two points,
Dialogue: 0,1:47:07.47,1:47:09.17,Default,,0000,0000,0000,,I still get 7, right?
Dialogue: 0,1:47:09.17,1:47:11.29,Default,,0000,0000,0000,,That's the whole idea.
Dialogue: 0,1:47:11.29,1:47:12.53,Default,,0000,0000,0000,,Am I clear about that?
Dialogue: 0,1:47:12.53,1:47:14.71,Default,,0000,0000,0000,,Are we clear about that?
Dialogue: 0,1:47:14.71,1:47:21.36,Default,,0000,0000,0000,,Let me ask you though,\Nhow do you find out?
Dialogue: 0,1:47:21.36,1:47:25.76,Default,,0000,0000,0000,,Because I don't know how\Nmany of you figured out
Dialogue: 0,1:47:25.76,1:47:28.53,Default,,0000,0000,0000,,what kind of curve that is.
Dialogue: 0,1:47:28.53,1:47:32.60,Default,,0000,0000,0000,,And it looks like an enemy\Nto you, but there is a catch.
Dialogue: 0,1:47:32.60,1:47:38.52,Default,,0000,0000,0000,,It's an old friend of\Nyours and you don't see it.
Dialogue: 0,1:47:38.52,1:47:40.38,Default,,0000,0000,0000,,So what is the curve?
Dialogue: 0,1:47:40.38,1:47:41.26,Default,,0000,0000,0000,,What is the curve?
Dialogue: 0,1:47:41.26,1:47:46.65,Default,,0000,0000,0000,,And what is this arc of a\Ncurve between 2, 1 and 1, 2?
Dialogue: 0,1:47:46.65,1:47:48.23,Default,,0000,0000,0000,,Can we find out what that is?
Dialogue: 0,1:47:48.23,1:47:49.35,Default,,0000,0000,0000,,Of course, or cubic.
Dialogue: 0,1:47:49.35,1:47:50.45,Default,,0000,0000,0000,,It's a fake cubic.
Dialogue: 0,1:47:50.45,1:47:53.78,Default,,0000,0000,0000,,It's a fake cubic--
Dialogue: 0,1:47:53.78,1:47:56.22,Default,,0000,0000,0000,,STUDENT: To function together?
Dialogue: 0,1:47:56.22,1:47:58.15,Default,,0000,0000,0000,,MAGDALENA TODA: Let's\Nsee what this is.
Dialogue: 0,1:47:58.15,1:48:03.06,Default,,0000,0000,0000,,xy cubed minus 2 cubed equals 0.
Dialogue: 0,1:48:03.06,1:48:06.32,Default,,0000,0000,0000,,We were in fourth grade--\Nwell, our teachers--
Dialogue: 0,1:48:06.32,1:48:13.64,Default,,0000,0000,0000,,I think our teachers teach us\Nwhen we were little that this,
Dialogue: 0,1:48:13.64,1:48:16.66,Default,,0000,0000,0000,,if you divided by a\Nminus- I wasn't little.
Dialogue: 0,1:48:16.66,1:48:19.24,Default,,0000,0000,0000,,I was in high school.
Dialogue: 0,1:48:19.24,1:48:21.13,Default,,0000,0000,0000,,Well, 14-year-old.
Dialogue: 0,1:48:21.13,1:48:22.08,Default,,0000,0000,0000,,STUDENT: A cubed.
Dialogue: 0,1:48:22.08,1:48:23.28,Default,,0000,0000,0000,,STUDENT: A squared.
Dialogue: 0,1:48:23.28,1:48:24.37,Default,,0000,0000,0000,,MAGDALENA TODA: A squared.
Dialogue: 0,1:48:24.37,1:48:26.01,Default,,0000,0000,0000,,STUDENT: Minus 2AB.
Dialogue: 0,1:48:26.01,1:48:27.08,Default,,0000,0000,0000,,Plus 2AB.
Dialogue: 0,1:48:27.08,1:48:28.96,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,1:48:28.96,1:48:30.87,Default,,0000,0000,0000,,Plus AB, not 2AB.
Dialogue: 0,1:48:30.87,1:48:31.63,Default,,0000,0000,0000,,STUDENT: Oh, darn.
Dialogue: 0,1:48:31.63,1:48:33.75,Default,,0000,0000,0000,,MAGDALENA TODA: Plus B squared.
Dialogue: 0,1:48:33.75,1:48:34.89,Default,,0000,0000,0000,,Suppose you don't believe.
Dialogue: 0,1:48:34.89,1:48:36.73,Default,,0000,0000,0000,,That proves this.
Dialogue: 0,1:48:36.73,1:48:38.06,Default,,0000,0000,0000,,Let's multiply.
Dialogue: 0,1:48:38.06,1:48:41.66,Default,,0000,0000,0000,,A cubed plus A squared\NB plus AB squared.
Dialogue: 0,1:48:41.66,1:48:44.02,Default,,0000,0000,0000,,I'm done with the\Nfirst multiplication.
Dialogue: 0,1:48:44.02,1:48:50.37,Default,,0000,0000,0000,,Minus BA squared minus\NAB squared minus B cubed.
Dialogue: 0,1:48:50.37,1:48:52.29,Default,,0000,0000,0000,,Do they cancel out?
Dialogue: 0,1:48:52.29,1:48:53.73,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:48:53.73,1:48:55.18,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:48:55.18,1:48:57.60,Default,,0000,0000,0000,,Cancel out.
Dialogue: 0,1:48:57.60,1:49:00.04,Default,,0000,0000,0000,,And cancel out.
Dialogue: 0,1:49:00.04,1:49:01.56,Default,,0000,0000,0000,,Out, poof.
Dialogue: 0,1:49:01.56,1:49:02.85,Default,,0000,0000,0000,,We've proved it, why?
Dialogue: 0,1:49:02.85,1:49:08.70,Default,,0000,0000,0000,,Because maybe some of you--\Nnobody gave it to proof before.
Dialogue: 0,1:49:08.70,1:49:11.59,Default,,0000,0000,0000,,
Dialogue: 0,1:49:11.59,1:49:17.51,Default,,0000,0000,0000,,So as an application,\Nwhat is this?
Dialogue: 0,1:49:17.51,1:49:18.01,Default,,0000,0000,0000,,There.
Dialogue: 0,1:49:18.01,1:49:19.10,Default,,0000,0000,0000,,Who is A and who is B?
Dialogue: 0,1:49:19.10,1:49:23.79,Default,,0000,0000,0000,,A is xy, B is 2.
Dialogue: 0,1:49:23.79,1:49:34.44,Default,,0000,0000,0000,,So you have xy minus 2 times\Nall this fluffy guy, xy
Dialogue: 0,1:49:34.44,1:49:42.38,Default,,0000,0000,0000,,squared plus 2xy plus--
Dialogue: 0,1:49:42.38,1:49:44.51,Default,,0000,0000,0000,,STUDENT: 4.
Dialogue: 0,1:49:44.51,1:49:45.26,Default,,0000,0000,0000,,MAGDALENA TODA: 4.
Dialogue: 0,1:49:45.26,1:49:49.29,Default,,0000,0000,0000,,And I also said, because\NI was sneaky, that's why.
Dialogue: 0,1:49:49.29,1:49:54.51,Default,,0000,0000,0000,,To make your life easier\Nor harder. xy is positive.
Dialogue: 0,1:49:54.51,1:49:57.78,Default,,0000,0000,0000,,When I said xy was positive,\Nwhat was I intending?
Dialogue: 0,1:49:57.78,1:50:03.32,Default,,0000,0000,0000,,I was intending for you to see\Nthat this cannot be 0 ever.
Dialogue: 0,1:50:03.32,1:50:07.68,Default,,0000,0000,0000,,So the only possible\Nfor you to have 0 here
Dialogue: 0,1:50:07.68,1:50:10.07,Default,,0000,0000,0000,,is when xy equals 2.
Dialogue: 0,1:50:10.07,1:50:14.20,Default,,0000,0000,0000,,And xy equals 2 is a\Nmuch simpler curve.
Dialogue: 0,1:50:14.20,1:50:17.72,Default,,0000,0000,0000,,And I want to know\Nif you realize
Dialogue: 0,1:50:17.72,1:50:22.16,Default,,0000,0000,0000,,that this will have the points\N2,1 and 1, 2 staring at you.
Dialogue: 0,1:50:22.16,1:50:23.37,Default,,0000,0000,0000,,Have a nice day today.
Dialogue: 0,1:50:23.37,1:50:25.20,Default,,0000,0000,0000,,Take care.
Dialogue: 0,1:50:25.20,1:50:26.63,Default,,0000,0000,0000,,And good luck.
Dialogue: 0,1:50:26.63,1:50:31.88,Default,,0000,0000,0000,,
Dialogue: 0,1:50:31.88,1:50:34.08,Default,,0000,0000,0000,,What is it?
Dialogue: 0,1:50:34.08,1:50:34.95,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:50:34.95,1:50:36.76,Default,,0000,0000,0000,,MAGDALENA TODA:\NSome sort of animal.
Dialogue: 0,1:50:36.76,1:50:38.38,Default,,0000,0000,0000,,It's a curve, a linear curve.
Dialogue: 0,1:50:38.38,1:50:42.28,Default,,0000,0000,0000,,It's not a line.
Dialogue: 0,1:50:42.28,1:50:43.25,Default,,0000,0000,0000,,What is it?
Dialogue: 0,1:50:43.25,1:50:47.63,Default,,0000,0000,0000,,Talking about conics because\NI was talking a little bit
Dialogue: 0,1:50:47.63,1:50:49.58,Default,,0000,0000,0000,,with Casey about conics.
Dialogue: 0,1:50:49.58,1:50:52.50,Default,,0000,0000,0000,,Is this a conic?
Dialogue: 0,1:50:52.50,1:50:53.48,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,1:50:53.48,1:50:55.43,Default,,0000,0000,0000,,What is a conic?
Dialogue: 0,1:50:55.43,1:50:59.97,Default,,0000,0000,0000,,A conic is any kind of\Ncurve that looks like this.
Dialogue: 0,1:50:59.97,1:51:04.53,Default,,0000,0000,0000,,In general form--\Noh my god, ABCD.
Dialogue: 0,1:51:04.53,1:51:08.19,Default,,0000,0000,0000,,Now I got my ABC\Nplus f equals 0.
Dialogue: 0,1:51:08.19,1:51:10.03,Default,,0000,0000,0000,,This is a conic in plane.
Dialogue: 0,1:51:10.03,1:51:13.65,Default,,0000,0000,0000,,My conic is missing\Neverything else.
Dialogue: 0,1:51:13.65,1:51:16.02,Default,,0000,0000,0000,,And B is 0.
Dialogue: 0,1:51:16.02,1:51:18.83,Default,,0000,0000,0000,,And there is a way where\Nyou-- I showed you how you
Dialogue: 0,1:51:18.83,1:51:21.73,Default,,0000,0000,0000,,know what kind of conic it is.
Dialogue: 0,1:51:21.73,1:51:28.13,Default,,0000,0000,0000,,A, A, B, B, C. A is\Npositive is-- no, A is 0,
Dialogue: 0,1:51:28.13,1:51:31.74,Default,,0000,0000,0000,,B is-- it should be 2 here.
Dialogue: 0,1:51:31.74,1:51:34.16,Default,,0000,0000,0000,,So you split this in half.
Dialogue: 0,1:51:34.16,1:51:37.11,Default,,0000,0000,0000,,1/2, 1/2, and 0.
Dialogue: 0,1:51:37.11,1:51:41.09,Default,,0000,0000,0000,,The determinant of this is\Nnegative, the discriminant.
Dialogue: 0,1:51:41.09,1:51:43.65,Default,,0000,0000,0000,,That's why we call it\Ndiscriminant about the conic.
Dialogue: 0,1:51:43.65,1:51:44.90,Default,,0000,0000,0000,,So it cannot be an ellipse.
Dialogue: 0,1:51:44.90,1:51:46.49,Default,,0000,0000,0000,,So what the heck is it?
Dialogue: 0,1:51:46.49,1:51:47.36,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:51:47.36,1:51:48.70,Default,,0000,0000,0000,,MAGDALENA TODA: Well, I'm silly.
Dialogue: 0,1:51:48.70,1:51:50.18,Default,,0000,0000,0000,,I should have pulled out for y.
Dialogue: 0,1:51:50.18,1:51:53.04,Default,,0000,0000,0000,,
Dialogue: 0,1:51:53.04,1:51:56.72,Default,,0000,0000,0000,,And I knew that it\Ngoes down like 1/x.
Dialogue: 0,1:51:56.72,1:52:00.59,Default,,0000,0000,0000,,But I'm asking you, why in\Nthe world is that a conic?
Dialogue: 0,1:52:00.59,1:52:01.67,Default,,0000,0000,0000,,Because you say, wait.
Dialogue: 0,1:52:01.67,1:52:03.23,Default,,0000,0000,0000,,Wait a minute.
Dialogue: 0,1:52:03.23,1:52:09.90,Default,,0000,0000,0000,,I know this curve since I was\Nfive year old in kindergarten.
Dialogue: 0,1:52:09.90,1:52:13.15,Default,,0000,0000,0000,,And this is the point 2, 1.
Dialogue: 0,1:52:13.15,1:52:16.51,Default,,0000,0000,0000,,
Dialogue: 0,1:52:16.51,1:52:17.24,Default,,0000,0000,0000,,It's on it.
Dialogue: 0,1:52:17.24,1:52:22.93,Default,,0000,0000,0000,,And there is a symmetric\Npoint for your pleasure here.
Dialogue: 0,1:52:22.93,1:52:25.12,Default,,0000,0000,0000,,1, 2.
Dialogue: 0,1:52:25.12,1:52:26.64,Default,,0000,0000,0000,,And between the\Ntwo points, there
Dialogue: 0,1:52:26.64,1:52:31.53,Default,,0000,0000,0000,,is just one arc of a curve.
Dialogue: 0,1:52:31.53,1:52:34.12,Default,,0000,0000,0000,,And this is the path that\Nyou are dragging some object
Dialogue: 0,1:52:34.12,1:52:35.39,Default,,0000,0000,0000,,with force f.
Dialogue: 0,1:52:35.39,1:52:37.98,Default,,0000,0000,0000,,You are computing\Nthe work of a-- maybe
Dialogue: 0,1:52:37.98,1:52:40.99,Default,,0000,0000,0000,,you're computing the work of\Na neutron between those two
Dialogue: 0,1:52:40.99,1:52:42.93,Default,,0000,0000,0000,,locations.
Dialogue: 0,1:52:42.93,1:52:43.79,Default,,0000,0000,0000,,It's a--
Dialogue: 0,1:52:43.79,1:52:44.64,Default,,0000,0000,0000,,STUDENT: Hyperbola?
Dialogue: 0,1:52:44.64,1:52:46.00,Default,,0000,0000,0000,,MAGDALENA TODA: Hyperbola.
Dialogue: 0,1:52:46.00,1:52:47.03,Default,,0000,0000,0000,,Why Nitish?
Dialogue: 0,1:52:47.03,1:52:47.77,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:52:47.77,1:52:49.52,Default,,0000,0000,0000,,STUDENT: I was just\Nwondering, couldn't we
Dialogue: 0,1:52:49.52,1:52:51.70,Default,,0000,0000,0000,,have gone to xy equals 2 plane?
Dialogue: 0,1:52:51.70,1:52:52.82,Default,,0000,0000,0000,,STUDENT: Yeah, way quicker.
Dialogue: 0,1:52:52.82,1:52:55.26,Default,,0000,0000,0000,,STUDENT: x cubed, y\Ncubed equals 2 cubed.
Dialogue: 0,1:52:55.26,1:52:56.55,Default,,0000,0000,0000,,Then you'd just do both sides--
Dialogue: 0,1:52:56.55,1:52:57.25,Default,,0000,0000,0000,,MAGDALENA TODA:\NThat's what I did.
Dialogue: 0,1:52:57.25,1:52:57.86,Default,,0000,0000,0000,,STUDENT: The cubed root.
Dialogue: 0,1:52:57.86,1:52:59.40,Default,,0000,0000,0000,,MAGDALENA TODA:\NDidn't I do that?
Dialogue: 0,1:52:59.40,1:53:02.84,Default,,0000,0000,0000,,No, because in\Ngeneral, it's not--
Dialogue: 0,1:53:02.84,1:53:07.48,Default,,0000,0000,0000,,you cannot say if and only\Nif xy equals 2 in general.
Dialogue: 0,1:53:07.48,1:53:10.60,Default,,0000,0000,0000,,You have to write to\Ndecompose the polynomial.
Dialogue: 0,1:53:10.60,1:53:12.45,Default,,0000,0000,0000,,You were lucky\Nthis was positive.
Dialogue: 0,1:53:12.45,1:53:15.05,Default,,0000,0000,0000,,STUDENT: Well, because\Nwe divided by x cubed.
Dialogue: 0,1:53:15.05,1:53:16.59,Default,,0000,0000,0000,,We could have just\Ndivided everything
Dialogue: 0,1:53:16.59,1:53:18.61,Default,,0000,0000,0000,,by x cubed, and then taken\Nthe cube root of both sides.
Dialogue: 0,1:53:18.61,1:53:20.40,Default,,0000,0000,0000,,MAGDALENA TODA: He's\Nsaying the same thing.
Dialogue: 0,1:53:20.40,1:53:23.93,Default,,0000,0000,0000,,But in mathematics, we don't--\Nlet me show you something.
Dialogue: 0,1:53:23.93,1:53:25.47,Default,,0000,0000,0000,,STUDENT: It would\Nwork for this case,
Dialogue: 0,1:53:25.47,1:53:26.86,Default,,0000,0000,0000,,but not necessarily\Nfor all cases?
Dialogue: 0,1:53:26.86,1:53:27.74,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:53:27.74,1:53:39.14,Default,,0000,0000,0000,,Let me show you some other\Nexample where you just-- how
Dialogue: 0,1:53:39.14,1:53:41.13,Default,,0000,0000,0000,,do you solve this equation?
Dialogue: 0,1:53:41.13,1:53:46.02,Default,,0000,0000,0000,,By the way, a math\Nfield test is coming.
Dialogue: 0,1:53:46.02,1:53:48.27,Default,,0000,0000,0000,,No, only if you're a math major.
Dialogue: 0,1:53:48.27,1:53:50.51,Default,,0000,0000,0000,,Sorry, junior or senior.
Dialogue: 0,1:53:50.51,1:53:53.36,Default,,0000,0000,0000,,In one math field test,\Nyou don't have to take it.
Dialogue: 0,1:53:53.36,1:53:56.88,Default,,0000,0000,0000,,But some people who\Ngo to graduate school,
Dialogue: 0,1:53:56.88,1:54:01.50,Default,,0000,0000,0000,,if they take the math field\Ntest, that replaces the GRE,
Dialogue: 0,1:54:01.50,1:54:03.48,Default,,0000,0000,0000,,if the school agrees.
Dialogue: 0,1:54:03.48,1:54:06.45,Default,,0000,0000,0000,,So there was this questions,\Nhow many roots does it have
Dialogue: 0,1:54:06.45,1:54:07.77,Default,,0000,0000,0000,,and what kind?
Dialogue: 0,1:54:07.77,1:54:11.38,Default,,0000,0000,0000,,Two are imaginary\Nand one is real.
Dialogue: 0,1:54:11.38,1:54:14.74,Default,,0000,0000,0000,,But everybody said\Nit only had one root.
Dialogue: 0,1:54:14.74,1:54:17.03,Default,,0000,0000,0000,,How can it have one root\Nif it's a cubic equation?
Dialogue: 0,1:54:17.03,1:54:18.58,Default,,0000,0000,0000,,So one root.
Dialogue: 0,1:54:18.58,1:54:20.60,Default,,0000,0000,0000,,x1 is 1.
Dialogue: 0,1:54:20.60,1:54:23.08,Default,,0000,0000,0000,,The other two are imaginary.
Dialogue: 0,1:54:23.08,1:54:24.33,Default,,0000,0000,0000,,This is the case in this also.
Dialogue: 0,1:54:24.33,1:54:26.45,Default,,0000,0000,0000,,You have some imaginary roots.
Dialogue: 0,1:54:26.45,1:54:31.44,Default,,0000,0000,0000,,So those roots\Nare funny, but you
Dialogue: 0,1:54:31.44,1:54:35.74,Default,,0000,0000,0000,,would have to\Nsolve this equation
Dialogue: 0,1:54:35.74,1:54:42.10,Default,,0000,0000,0000,,because this is x minus 1\Ntimes x squared plus x plus 1.
Dialogue: 0,1:54:42.10,1:54:45.58,Default,,0000,0000,0000,,So the roots are minus\N1, plus minus square root
Dialogue: 0,1:54:45.58,1:54:52.23,Default,,0000,0000,0000,,of b squared minus 4ac\Nover 2, which are minus 1
Dialogue: 0,1:54:52.23,1:54:57.29,Default,,0000,0000,0000,,plus minus square\Nroot of 3i over 2.
Dialogue: 0,1:54:57.29,1:55:01.23,Default,,0000,0000,0000,,Do you guys know\Nhow they are called?
Dialogue: 0,1:55:01.23,1:55:05.60,Default,,0000,0000,0000,,You know them because in\Nsome countries we learn them.
Dialogue: 0,1:55:05.60,1:55:07.98,Default,,0000,0000,0000,,But do you know the notations?
Dialogue: 0,1:55:07.98,1:55:09.19,Default,,0000,0000,0000,,STUDENT: What they call them?
Dialogue: 0,1:55:09.19,1:55:10.06,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:55:10.06,1:55:12.19,Default,,0000,0000,0000,,
Dialogue: 0,1:55:12.19,1:55:14.15,Default,,0000,0000,0000,,There is a Greek letter.
Dialogue: 0,1:55:14.15,1:55:15.87,Default,,0000,0000,0000,,STUDENT: Iota.
Dialogue: 0,1:55:15.87,1:55:17.33,Default,,0000,0000,0000,,MAGDALENA TODA: In\NIndia, probably.
Dialogue: 0,1:55:17.33,1:55:19.06,Default,,0000,0000,0000,,In my country, it was omega.
Dialogue: 0,1:55:19.06,1:55:19.87,Default,,0000,0000,0000,,But I don't think--
Dialogue: 0,1:55:19.87,1:55:20.87,Default,,0000,0000,0000,,STUDENT: In India, iota.
Dialogue: 0,1:55:20.87,1:55:22.88,Default,,0000,0000,0000,,
Dialogue: 0,1:55:22.88,1:55:25.81,Default,,0000,0000,0000,,MAGDALENA TODA: But we call\Nthem omega and omega squared.
Dialogue: 0,1:55:25.81,1:55:28.64,Default,,0000,0000,0000,,Because one is the\Nsquare of the other.
Dialogue: 0,1:55:28.64,1:55:30.14,Default,,0000,0000,0000,,They are, of course,\Nboth imaginary.
Dialogue: 0,1:55:30.14,1:55:35.66,Default,,0000,0000,0000,,And we call this the\Ncubic roots of unity.
Dialogue: 0,1:55:35.66,1:55:39.11,Default,,0000,0000,0000,,
Dialogue: 0,1:55:39.11,1:55:41.97,Default,,0000,0000,0000,,You say Magdalena, why would\Nyou talk about imaginary numbers
Dialogue: 0,1:55:41.97,1:55:43.88,Default,,0000,0000,0000,,when everything is real?
Dialogue: 0,1:55:43.88,1:55:44.50,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:55:44.50,1:55:48.14,Default,,0000,0000,0000,,It's real for the time being\Nwhile you are still with me.
Dialogue: 0,1:55:48.14,1:55:50.33,Default,,0000,0000,0000,,The moment you're going\Nto say goodbye to me
Dialogue: 0,1:55:50.33,1:55:55.12,Default,,0000,0000,0000,,and you know in 3350 your\Nlife is going to change.
Dialogue: 0,1:55:55.12,1:55:57.34,Default,,0000,0000,0000,,In that course,\Nthey will ask you
Dialogue: 0,1:55:57.34,1:56:02.57,Default,,0000,0000,0000,,to solve this equation just like\Nwe asked all our 3350 students.
Dialogue: 0,1:56:02.57,1:56:05.05,Default,,0000,0000,0000,,To our surprise,\Nthe students don't
Dialogue: 0,1:56:05.05,1:56:06.88,Default,,0000,0000,0000,,know what imaginary roots are.
Dialogue: 0,1:56:06.88,1:56:07.94,Default,,0000,0000,0000,,Many, you know.
Dialogue: 0,1:56:07.94,1:56:10.38,Default,,0000,0000,0000,,You will refresh your memory.
Dialogue: 0,1:56:10.38,1:56:12.00,Default,,0000,0000,0000,,But the majority of\Nthe students didn't
Dialogue: 0,1:56:12.00,1:56:15.14,Default,,0000,0000,0000,,know how to get to\Nthose imaginary numbers.
Dialogue: 0,1:56:15.14,1:56:20.36,Default,,0000,0000,0000,,You're going to need to not\Nonly use them, but also express
Dialogue: 0,1:56:20.36,1:56:22.56,Default,,0000,0000,0000,,these in terms of trigonometry.
Dialogue: 0,1:56:22.56,1:56:25.54,Default,,0000,0000,0000,,
Dialogue: 0,1:56:25.54,1:56:31.73,Default,,0000,0000,0000,,So just out of curiosity, since\NI am already talking to you,
Dialogue: 0,1:56:31.73,1:56:34.88,Default,,0000,0000,0000,,and since I've preparing you a\Nlittle bit for the differential
Dialogue: 0,1:56:34.88,1:56:38.66,Default,,0000,0000,0000,,equations class where you\Nhave lots of electric circuits
Dialogue: 0,1:56:38.66,1:56:41.43,Default,,0000,0000,0000,,and applications\Nof trigonometry,
Dialogue: 0,1:56:41.43,1:56:45.78,Default,,0000,0000,0000,,these imaginary numbers\Ncan also be put-- they
Dialogue: 0,1:56:45.78,1:56:50.73,Default,,0000,0000,0000,,are in general of the form\Na plus ib. a plus minus ib.
Dialogue: 0,1:56:50.73,1:56:54.87,Default,,0000,0000,0000,,And we agree that in\N3350 you have to do that.
Dialogue: 0,1:56:54.87,1:56:56.94,Default,,0000,0000,0000,,Out of curiosity,\Nis there anybody
Dialogue: 0,1:56:56.94,1:57:02.93,Default,,0000,0000,0000,,who knows the trigonometric\Nform of these complex numbers?
Dialogue: 0,1:57:02.93,1:57:06.30,Default,,0000,0000,0000,,STUDENT: Isn't it r e to the j--
Dialogue: 0,1:57:06.30,1:57:10.17,Default,,0000,0000,0000,,
Dialogue: 0,1:57:10.17,1:57:14.24,Default,,0000,0000,0000,,MAGDALENA TODA: So you would\Nhave exactly what he says here.
Dialogue: 0,1:57:14.24,1:57:18.48,Default,,0000,0000,0000,,This number will\Nbe-- if it's plus.
Dialogue: 0,1:57:18.48,1:57:20.95,Default,,0000,0000,0000,,r e to the i theta.
Dialogue: 0,1:57:20.95,1:57:26.22,Default,,0000,0000,0000,,He knows a little bit\Nmore than most students.
Dialogue: 0,1:57:26.22,1:57:34.28,Default,,0000,0000,0000,,And that is cosine\Ntheta plus i sine theta.
Dialogue: 0,1:57:34.28,1:57:36.81,Default,,0000,0000,0000,,Can you find me the\Nangle theta if I
Dialogue: 0,1:57:36.81,1:57:42.87,Default,,0000,0000,0000,,want to write cosine theta\Nplus i sine theta or cosine
Dialogue: 0,1:57:42.87,1:57:46.39,Default,,0000,0000,0000,,theta minus i sine theta?
Dialogue: 0,1:57:46.39,1:57:50.21,Default,,0000,0000,0000,,Can you find me\Nthe angle of theta?
Dialogue: 0,1:57:50.21,1:57:50.96,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:57:50.96,1:57:53.29,Default,,0000,0000,0000,,Is it easy?
Dialogue: 0,1:57:53.29,1:57:54.99,Default,,0000,0000,0000,,What in the world is it?
Dialogue: 0,1:57:54.99,1:57:59.81,Default,,0000,0000,0000,,
Dialogue: 0,1:57:59.81,1:58:01.40,Default,,0000,0000,0000,,Think like this.
Dialogue: 0,1:58:01.40,1:58:04.44,Default,,0000,0000,0000,,We are done with this\Nexample, but I'm just
Dialogue: 0,1:58:04.44,1:58:08.17,Default,,0000,0000,0000,,saying some things that\Nwill help you in 3350.
Dialogue: 0,1:58:08.17,1:58:11.86,Default,,0000,0000,0000,,If you want cosine\Ntheta to be minus 1/2
Dialogue: 0,1:58:11.86,1:58:20.55,Default,,0000,0000,0000,,and you want sine theta to be\Nroot 3 over 2, which quadrant?
Dialogue: 0,1:58:20.55,1:58:22.90,Default,,0000,0000,0000,,Which quadrant are you in?
Dialogue: 0,1:58:22.90,1:58:24.12,Default,,0000,0000,0000,,STUDENT: Second.
Dialogue: 0,1:58:24.12,1:58:25.62,Default,,0000,0000,0000,,MAGDALENA TODA: The\Nsecond quadrant.
Dialogue: 0,1:58:25.62,1:58:27.03,Default,,0000,0000,0000,,Very good.
Dialogue: 0,1:58:27.03,1:58:28.03,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:58:28.03,1:58:31.46,Default,,0000,0000,0000,,So think cosine.
Dialogue: 0,1:58:31.46,1:58:36.42,Default,,0000,0000,0000,,If cosine would be a half and\Nsine would be root 3 over 2,
Dialogue: 0,1:58:36.42,1:58:38.39,Default,,0000,0000,0000,,it would be in first quadrant.
Dialogue: 0,1:58:38.39,1:58:40.42,Default,,0000,0000,0000,,And what angle would that be?
Dialogue: 0,1:58:40.42,1:58:40.92,Default,,0000,0000,0000,,STUDENT: 60.
Dialogue: 0,1:58:40.92,1:58:41.76,Default,,0000,0000,0000,,STUDENT: That's 60--
Dialogue: 0,1:58:41.76,1:58:45.72,Default,,0000,0000,0000,,MAGDALENA TODA: 60 degrees,\Nwhich is pi over 3, right?
Dialogue: 0,1:58:45.72,1:58:52.90,Default,,0000,0000,0000,,But pi over 3 is your\Nfriend, so he's happy.
Dialogue: 0,1:58:52.90,1:58:54.90,Default,,0000,0000,0000,,Well, he is there somewhere.
Dialogue: 0,1:58:54.90,1:58:58.90,Default,,0000,0000,0000,,
Dialogue: 0,1:58:58.90,1:59:00.77,Default,,0000,0000,0000,,STUDENT: 120.
Dialogue: 0,1:59:00.77,1:59:05.30,Default,,0000,0000,0000,,MAGDALENA TODA: Where you\Nare here, you are at what?
Dialogue: 0,1:59:05.30,1:59:07.20,Default,,0000,0000,0000,,How much is 120-- very good.
Dialogue: 0,1:59:07.20,1:59:09.21,Default,,0000,0000,0000,,How much is 120 pi?
Dialogue: 0,1:59:09.21,1:59:10.77,Default,,0000,0000,0000,,STUDENT: 4 pi?
Dialogue: 0,1:59:10.77,1:59:11.56,Default,,0000,0000,0000,,MAGDALENA TODA: No.
Dialogue: 0,1:59:11.56,1:59:11.85,Default,,0000,0000,0000,,STUDENT: 2 pi over 3.
Dialogue: 0,1:59:11.85,1:59:13.16,Default,,0000,0000,0000,,MAGDALENA TODA: 2 pi over 3.
Dialogue: 0,1:59:13.16,1:59:14.46,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,1:59:14.46,1:59:16.84,Default,,0000,0000,0000,,So 2 pi over 3.
Dialogue: 0,1:59:16.84,1:59:21.13,Default,,0000,0000,0000,,This would be if you\Nwere to think about it--
Dialogue: 0,1:59:21.13,1:59:22.39,Default,,0000,0000,0000,,this is in radians.
Dialogue: 0,1:59:22.39,1:59:24.32,Default,,0000,0000,0000,,Let me write radians.
Dialogue: 0,1:59:24.32,1:59:28.47,Default,,0000,0000,0000,,In degrees, that's 120 degrees.
Dialogue: 0,1:59:28.47,1:59:38.82,Default,,0000,0000,0000,,So to conclude my detour\Nto introduction to 3350.
Dialogue: 0,1:59:38.82,1:59:47.10,Default,,0000,0000,0000,,When they will ask you to solve\Nthis equation, x cubed minus 1,
Dialogue: 0,1:59:47.10,1:59:49.58,Default,,0000,0000,0000,,you have to tell them like that.
Dialogue: 0,1:59:49.58,1:59:53.14,Default,,0000,0000,0000,,They will ask you to put\Nit in trigonometric form.
Dialogue: 0,1:59:53.14,2:00:04.14,Default,,0000,0000,0000,,x1 is 1, x2 is cosine of 2 pi\Nover 3 plus i sine 2 pi over 3.
Dialogue: 0,2:00:04.14,2:00:07.33,Default,,0000,0000,0000,,And the other one\Nis x3 equals cosine
Dialogue: 0,2:00:07.33,2:00:15.68,Default,,0000,0000,0000,,of 2 pi over 3 minus\Ni sine of 2 pi over 3.
Dialogue: 0,2:00:15.68,2:00:16.84,Default,,0000,0000,0000,,The last thing.
Dialogue: 0,2:00:16.84,2:00:18.66,Default,,0000,0000,0000,,Because I should let you go.
Dialogue: 0,2:00:18.66,2:00:20.02,Default,,0000,0000,0000,,There was no break.
Dialogue: 0,2:00:20.02,2:00:23.44,Default,,0000,0000,0000,,I squeezed your brains\Nreally bad today.
Dialogue: 0,2:00:23.44,2:00:26.37,Default,,0000,0000,0000,,We still have like 150 minutes.
Dialogue: 0,2:00:26.37,2:00:29.19,Default,,0000,0000,0000,,I stole from you--\Nno, I stole really big
Dialogue: 0,2:00:29.19,2:00:33.33,Default,,0000,0000,0000,,because we would have-- yeah,\Nwe still have 15 minutes.
Dialogue: 0,2:00:33.33,2:00:37.29,Default,,0000,0000,0000,,But the break was 10 minutes,\Nso I didn't give you a break.
Dialogue: 0,2:00:37.29,2:00:40.03,Default,,0000,0000,0000,,What would this be if\Nyou wanted to express it
Dialogue: 0,2:00:40.03,2:00:43.03,Default,,0000,0000,0000,,in terms of another angle?
Dialogue: 0,2:00:43.03,2:00:47.14,Default,,0000,0000,0000,,That's the last thing\NI'm asking of you.
Dialogue: 0,2:00:47.14,2:00:48.60,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,2:00:48.60,2:00:50.55,Default,,0000,0000,0000,,MAGDALENA TODA: Not minus.
Dialogue: 0,2:00:50.55,2:00:53.17,Default,,0000,0000,0000,,Like cosine of an angle\Nplus i sine of an angle.
Dialogue: 0,2:00:53.17,2:00:55.94,Default,,0000,0000,0000,,You would need to go to\Nanother quadrant, right?
Dialogue: 0,2:00:55.94,2:00:57.57,Default,,0000,0000,0000,,And which quadrant?
Dialogue: 0,2:00:57.57,2:00:58.41,Default,,0000,0000,0000,,STUDENT: 4.
Dialogue: 0,2:00:58.41,2:00:59.99,Default,,0000,0000,0000,,MAGDALENA TODA:\NYou've said it before.
Dialogue: 0,2:00:59.99,2:01:02.57,Default,,0000,0000,0000,,That would be 4 pi over 3.
Dialogue: 0,2:01:02.57,2:01:05.23,Default,,0000,0000,0000,,And 4 pi over 3.
Dialogue: 0,2:01:05.23,2:01:10.16,Default,,0000,0000,0000,,Keep in mind these things\Nwith imaginary numbers because
Dialogue: 0,2:01:10.16,2:01:13.54,Default,,0000,0000,0000,,in 3350, they will rely on\Nyou knowing these things.
Dialogue: 0,2:01:13.54,2:01:15.71,Default,,0000,0000,0000,,
Dialogue: 0,2:01:15.71,2:01:17.75,Default,,0000,0000,0000,,STUDENT: Then you apply\NEuler's formula up there.
Dialogue: 0,2:01:17.75,2:01:21.43,Default,,0000,0000,0000,,
Dialogue: 0,2:01:21.43,2:01:22.47,Default,,0000,0000,0000,,MAGDALENA TODA: Oh, yeah.
Dialogue: 0,2:01:22.47,2:01:24.26,Default,,0000,0000,0000,,By the way, this is\Ncalled Euler's formula.
Dialogue: 0,2:01:24.26,2:01:27.54,Default,,0000,0000,0000,,
Dialogue: 0,2:01:27.54,2:01:30.93,Default,,0000,0000,0000,,STUDENT: In middle\Nschool, they teach you,
Dialogue: 0,2:01:30.93,2:01:33.84,Default,,0000,0000,0000,,and they tell you when\Ndiscriminant is small,
Dialogue: 0,2:01:33.84,2:01:35.88,Default,,0000,0000,0000,,there's no solutions.
Dialogue: 0,2:01:35.88,2:01:36.76,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,2:01:36.76,2:01:37.89,Default,,0000,0000,0000,,STUDENT: And you\Ngo to [INAUDIBLE].
Dialogue: 0,2:01:37.89,2:01:40.39,Default,,0000,0000,0000,,MAGDALENA TODA: When the\Ndiscriminant is less than 0,
Dialogue: 0,2:01:40.39,2:01:42.36,Default,,0000,0000,0000,,there are no real solutions.
Dialogue: 0,2:01:42.36,2:01:44.32,Default,,0000,0000,0000,,But you have in pairs\Nimaginary solutions.
Dialogue: 0,2:01:44.32,2:01:46.28,Default,,0000,0000,0000,,They always come in pairs.
Dialogue: 0,2:01:46.28,2:01:50.23,Default,,0000,0000,0000,,
Dialogue: 0,2:01:50.23,2:01:52.50,Default,,0000,0000,0000,,Do you want me to\Nshow you probably
Dialogue: 0,2:01:52.50,2:01:55.24,Default,,0000,0000,0000,,the most important problem\Nin 3350 in 2 minutes,
Dialogue: 0,2:01:55.24,2:01:58.57,Default,,0000,0000,0000,,and then I'll let you go?
Dialogue: 0,2:01:58.57,2:02:01.00,Default,,0000,0000,0000,,STUDENT: Sure.
Dialogue: 0,2:02:01.00,2:02:07.69,Default,,0000,0000,0000,,MAGDALENA TODA: So somebody\Ngives you the equation
Dialogue: 0,2:02:07.69,2:02:10.39,Default,,0000,0000,0000,,of the harmonic oscillator.
Dialogue: 0,2:02:10.39,2:02:12.01,Default,,0000,0000,0000,,And you say, what\Nthe heck is that?
Dialogue: 0,2:02:12.01,2:02:17.42,Default,,0000,0000,0000,,You have a little spring\Nand you pull that spring.
Dialogue: 0,2:02:17.42,2:02:19.36,Default,,0000,0000,0000,,And it's going to come back.
Dialogue: 0,2:02:19.36,2:02:21.86,Default,,0000,0000,0000,,You displace it, it comes back.
Dialogue: 0,2:02:21.86,2:02:24.36,Default,,0000,0000,0000,,It oscillates back and forth,\Noscillates back and forth.
Dialogue: 0,2:02:24.36,2:02:27.95,Default,,0000,0000,0000,,If you were to write the\Nsolutions of the harmonic
Dialogue: 0,2:02:27.95,2:02:29.80,Default,,0000,0000,0000,,oscillator in electric\Ncircuits, there
Dialogue: 0,2:02:29.80,2:02:31.30,Default,,0000,0000,0000,,would be oscillating functions.
Dialogue: 0,2:02:31.30,2:02:36.53,Default,,0000,0000,0000,,So it has to do with sine and\Ncosine, so they must be trig.
Dialogue: 0,2:02:36.53,2:02:38.91,Default,,0000,0000,0000,,If somebody gives\Nyou this equation,
Dialogue: 0,2:02:38.91,2:02:57.06,Default,,0000,0000,0000,,let's say ax squared-- y\Ndouble prime of x minus b.
Dialogue: 0,2:02:57.06,2:02:59.04,Default,,0000,0000,0000,,Plus.
Dialogue: 0,2:02:59.04,2:03:04.00,Default,,0000,0000,0000,,Equals to 0.
Dialogue: 0,2:03:04.00,2:03:09.47,Default,,0000,0000,0000,,And here is a y equals 0.
Dialogue: 0,2:03:09.47,2:03:12.50,Default,,0000,0000,0000,,Why would that\Nshow up like that?
Dialogue: 0,2:03:12.50,2:03:19.37,Default,,0000,0000,0000,,Well, Hooke's law tells\Nyou that there is a force.
Dialogue: 0,2:03:19.37,2:03:21.68,Default,,0000,0000,0000,,And there is a\Nforce and the force
Dialogue: 0,2:03:21.68,2:03:23.43,Default,,0000,0000,0000,,is mass times acceleration.
Dialogue: 0,2:03:23.43,2:03:27.48,Default,,0000,0000,0000,,And acceleration is like this\Ntype of second derivative
Dialogue: 0,2:03:27.48,2:03:30.83,Default,,0000,0000,0000,,of the displacement.
Dialogue: 0,2:03:30.83,2:03:37.58,Default,,0000,0000,0000,,And F and the displacement\Nare proportional,
Dialogue: 0,2:03:37.58,2:03:41.23,Default,,0000,0000,0000,,when you write F\Nequals displacement,
Dialogue: 0,2:03:41.23,2:03:43.98,Default,,0000,0000,0000,,let's call it y of x.
Dialogue: 0,2:03:43.98,2:03:47.54,Default,,0000,0000,0000,,When you have y of x, x is time.
Dialogue: 0,2:03:47.54,2:03:49.30,Default,,0000,0000,0000,,That's the displacement.
Dialogue: 0,2:03:49.30,2:03:50.00,Default,,0000,0000,0000,,That's the force.
Dialogue: 0,2:03:50.00,2:03:50.63,Default,,0000,0000,0000,,That's the k.
Dialogue: 0,2:03:50.63,2:03:53.06,Default,,0000,0000,0000,,So you have a certain\NHooke's constant.
Dialogue: 0,2:03:53.06,2:03:54.88,Default,,0000,0000,0000,,Hooke's law constant.
Dialogue: 0,2:03:54.88,2:03:56.87,Default,,0000,0000,0000,,So when you write\Nthis, Hooke's law
Dialogue: 0,2:03:56.87,2:03:58.59,Default,,0000,0000,0000,,is going to become like that.
Dialogue: 0,2:03:58.59,2:04:04.93,Default,,0000,0000,0000,,Mass times y double prime of\Nx equals-- this is the force.
Dialogue: 0,2:04:04.93,2:04:06.72,Default,,0000,0000,0000,,k times y of x.
Dialogue: 0,2:04:06.72,2:04:09.99,Default,,0000,0000,0000,,
Dialogue: 0,2:04:09.99,2:04:16.54,Default,,0000,0000,0000,,But it depends because\Nyou can have plus minus.
Dialogue: 0,2:04:16.54,2:04:18.58,Default,,0000,0000,0000,,So you can have plus or minus.
Dialogue: 0,2:04:18.58,2:04:20.15,Default,,0000,0000,0000,,And these are\Npositive functions.
Dialogue: 0,2:04:20.15,2:04:27.50,Default,,0000,0000,0000,,
Dialogue: 0,2:04:27.50,2:04:31.92,Default,,0000,0000,0000,,You have two equations\Nin that case.
Dialogue: 0,2:04:31.92,2:04:39.64,Default,,0000,0000,0000,,One equation is the form y\Ndouble prime plus-- give me
Dialogue: 0,2:04:39.64,2:04:40.47,Default,,0000,0000,0000,,a number.
Dialogue: 0,2:04:40.47,2:04:43.40,Default,,0000,0000,0000,,Cy equals 0.
Dialogue: 0,2:04:43.40,2:04:49.57,Default,,0000,0000,0000,,And the other one would be y\Ndouble prime minus cy equals 0.
Dialogue: 0,2:04:49.57,2:04:51.37,Default,,0000,0000,0000,,All right.
Dialogue: 0,2:04:51.37,2:04:54.98,Default,,0000,0000,0000,,Now, how hard is to\Nguess your solutions?
Dialogue: 0,2:04:54.98,2:04:59.92,Default,,0000,0000,0000,,
Dialogue: 0,2:04:59.92,2:05:02.17,Default,,0000,0000,0000,,Can you guess the\Nsolutions with naked eyes?
Dialogue: 0,2:05:02.17,2:05:04.76,Default,,0000,0000,0000,,
Dialogue: 0,2:05:04.76,2:05:05.64,Default,,0000,0000,0000,,STUDENT: e to the x--
Dialogue: 0,2:05:05.64,2:05:09.08,Default,,0000,0000,0000,,
Dialogue: 0,2:05:09.08,2:05:13.92,Default,,0000,0000,0000,,MAGDALENA TODA: So if you have--\Nyou have e to the something.
Dialogue: 0,2:05:13.92,2:05:17.78,Default,,0000,0000,0000,,If you didn't have a c, it\Nwould make your life easier.
Dialogue: 0,2:05:17.78,2:05:18.70,Default,,0000,0000,0000,,Forget about the c.
Dialogue: 0,2:05:18.70,2:05:20.91,Default,,0000,0000,0000,,The c will act the\Nsame in the end.
Dialogue: 0,2:05:20.91,2:05:25.80,Default,,0000,0000,0000,,So here, what are the\Npossible solutions?
Dialogue: 0,2:05:25.80,2:05:27.01,Default,,0000,0000,0000,,STUDENT: e to the--
Dialogue: 0,2:05:27.01,2:05:29.22,Default,,0000,0000,0000,,MAGDALENA TODA: e to the t\Nis one of them. e to the x
Dialogue: 0,2:05:29.22,2:05:32.75,Default,,0000,0000,0000,,is one of them, right?
Dialogue: 0,2:05:32.75,2:05:35.54,Default,,0000,0000,0000,,So in the end, to\Nsolve such a problem
Dialogue: 0,2:05:35.54,2:05:37.30,Default,,0000,0000,0000,,they teach you the method.
Dialogue: 0,2:05:37.30,2:05:39.66,Default,,0000,0000,0000,,You take the equation.
Dialogue: 0,2:05:39.66,2:05:42.08,Default,,0000,0000,0000,,And for that, you associate\Nthe so-called characteristic
Dialogue: 0,2:05:42.08,2:05:44.48,Default,,0000,0000,0000,,equation.
Dialogue: 0,2:05:44.48,2:05:47.25,Default,,0000,0000,0000,,For power 2, you put r squared.
Dialogue: 0,2:05:47.25,2:05:51.47,Default,,0000,0000,0000,,Then you minus n for-- this\Nis how many times is it prime?
Dialogue: 0,2:05:51.47,2:05:52.25,Default,,0000,0000,0000,,No times.
Dialogue: 0,2:05:52.25,2:05:53.08,Default,,0000,0000,0000,,0 times.
Dialogue: 0,2:05:53.08,2:05:55.25,Default,,0000,0000,0000,,So you put a 1.
Dialogue: 0,2:05:55.25,2:05:58.01,Default,,0000,0000,0000,,If it's prime one times,\Ny prime is missing.
Dialogue: 0,2:05:58.01,2:06:01.77,Default,,0000,0000,0000,,It's prime 1 time,\Nyou would put minus r.
Dialogue: 0,2:06:01.77,2:06:02.88,Default,,0000,0000,0000,,Equals 0.
Dialogue: 0,2:06:02.88,2:06:06.95,Default,,0000,0000,0000,,And then you look at\Nthe two roots of that.
Dialogue: 0,2:06:06.95,2:06:07.82,Default,,0000,0000,0000,,And what are they?
Dialogue: 0,2:06:07.82,2:06:08.71,Default,,0000,0000,0000,,Plus minus 1.
Dialogue: 0,2:06:08.71,2:06:11.49,Default,,0000,0000,0000,,So r1 is 1, r2 is 2.
Dialogue: 0,2:06:11.49,2:06:13.37,Default,,0000,0000,0000,,And there is a\Ntheorem that says--
Dialogue: 0,2:06:13.37,2:06:15.26,Default,,0000,0000,0000,,STUDENT: r2 is minus 1.
Dialogue: 0,2:06:15.26,2:06:17.12,Default,,0000,0000,0000,,MAGDALENA TODA: r2 is minus 1.
Dialogue: 0,2:06:17.12,2:06:19.62,Default,,0000,0000,0000,,Excuse me.
Dialogue: 0,2:06:19.62,2:06:23.97,Default,,0000,0000,0000,,OK, there's a theorem that\Nsays all the solutions
Dialogue: 0,2:06:23.97,2:06:28.16,Default,,0000,0000,0000,,of this equation come as\Nlinear combinations of e
Dialogue: 0,2:06:28.16,2:06:31.23,Default,,0000,0000,0000,,to the r1t and e to the r2t.
Dialogue: 0,2:06:31.23,2:06:33.11,Default,,0000,0000,0000,,So linear combination\Nmeans you can
Dialogue: 0,2:06:33.11,2:06:39.15,Default,,0000,0000,0000,,take any number a and any\Nnumber b, or c1 and c2, anything
Dialogue: 0,2:06:39.15,2:06:40.14,Default,,0000,0000,0000,,like that.
Dialogue: 0,2:06:40.14,2:06:44.65,Default,,0000,0000,0000,,So all the solutions of\Nthis will look like e
Dialogue: 0,2:06:44.65,2:06:48.23,Default,,0000,0000,0000,,to the t with an a in\Nfront plus e to the minus
Dialogue: 0,2:06:48.23,2:06:50.06,Default,,0000,0000,0000,,t with a b in front.
Dialogue: 0,2:06:50.06,2:06:53.30,Default,,0000,0000,0000,,Could you have seen\Nthat with naked eye?
Dialogue: 0,2:06:53.30,2:06:54.35,Default,,0000,0000,0000,,Well, yeah.
Dialogue: 0,2:06:54.35,2:06:57.27,Default,,0000,0000,0000,,I mean, you are smart\Nand you guessed one.
Dialogue: 0,2:06:57.27,2:06:59.25,Default,,0000,0000,0000,,An you said e to\Nthe t satisfied.
Dialogue: 0,2:06:59.25,2:07:02.32,Default,,0000,0000,0000,,Because if you put e to the\Np and prime it as many times
Dialogue: 0,2:07:02.32,2:07:04.87,Default,,0000,0000,0000,,as you want, you\Nstill get e to the t.
Dialogue: 0,2:07:04.87,2:07:06.05,Default,,0000,0000,0000,,So you get 0.
Dialogue: 0,2:07:06.05,2:07:09.18,Default,,0000,0000,0000,,But nobody thought of-- or maybe\Nsome people thought about e
Dialogue: 0,2:07:09.18,2:07:10.48,Default,,0000,0000,0000,,to the minus t.
Dialogue: 0,2:07:10.48,2:07:11.07,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:07:11.07,2:07:11.88,Default,,0000,0000,0000,,I was about to go\Nthrough that one.
Dialogue: 0,2:07:11.88,2:07:13.03,Default,,0000,0000,0000,,MAGDALENA TODA: You were about.
Dialogue: 0,2:07:13.03,2:07:13.80,Default,,0000,0000,0000,,STUDENT: That's for a selection.
Dialogue: 0,2:07:13.80,2:07:16.01,Default,,0000,0000,0000,,MAGDALENA TODA: So even if\Nyou take e to the minus t,
Dialogue: 0,2:07:16.01,2:07:17.24,Default,,0000,0000,0000,,you get the same answer.
Dialogue: 0,2:07:17.24,2:07:19.79,Default,,0000,0000,0000,,And you get this thing.
Dialogue: 0,2:07:19.79,2:07:24.04,Default,,0000,0000,0000,,All right, all the combinations\Nwill satisfy the same equation
Dialogue: 0,2:07:24.04,2:07:24.66,Default,,0000,0000,0000,,as well.
Dialogue: 0,2:07:24.66,2:07:26.63,Default,,0000,0000,0000,,This is a superposition\Nprinciple.
Dialogue: 0,2:07:26.63,2:07:28.62,Default,,0000,0000,0000,,With this, it was easy.
Dialogue: 0,2:07:28.62,2:07:31.61,Default,,0000,0000,0000,,But this is the so-called\Nharmonic oscillator equation.
Dialogue: 0,2:07:31.61,2:07:36.10,Default,,0000,0000,0000,,
Dialogue: 0,2:07:36.10,2:07:40.29,Default,,0000,0000,0000,,So either you have it simplified\Ny double prime plus y equals 0,
Dialogue: 0,2:07:40.29,2:07:46.25,Default,,0000,0000,0000,,or you have some constant c.
Dialogue: 0,2:07:46.25,2:07:48.70,Default,,0000,0000,0000,,Well, what do you\Ndo in that case?
Dialogue: 0,2:07:48.70,2:07:50.77,Default,,0000,0000,0000,,Let's assume you have 1.
Dialogue: 0,2:07:50.77,2:07:53.95,Default,,0000,0000,0000,,Who can guess the solutions?
Dialogue: 0,2:07:53.95,2:07:55.92,Default,,0000,0000,0000,,STUDENT: 0 and cosine--
Dialogue: 0,2:07:55.92,2:07:57.98,Default,,0000,0000,0000,,MAGDALENA TODA: No, 0\Nis the trivial solution
Dialogue: 0,2:07:57.98,2:07:59.16,Default,,0000,0000,0000,,and it's not going to count.
Dialogue: 0,2:07:59.16,2:08:03.86,Default,,0000,0000,0000,,You can get it from the\Ncombination of the--
Dialogue: 0,2:08:03.86,2:08:05.35,Default,,0000,0000,0000,,STUDENT: y equals sine t.
Dialogue: 0,2:08:05.35,2:08:06.85,Default,,0000,0000,0000,,MAGDALENA TODA:\NSine t is a solution
Dialogue: 0,2:08:06.85,2:08:11.66,Default,,0000,0000,0000,,because sine t prime is cosine.
Dialogue: 0,2:08:11.66,2:08:13.77,Default,,0000,0000,0000,,When you prime it\Nagain, it's minus sine.
Dialogue: 0,2:08:13.77,2:08:16.68,Default,,0000,0000,0000,,When you add sine and\Nminus sine, you get 0.
Dialogue: 0,2:08:16.68,2:08:19.41,Default,,0000,0000,0000,,So you just guessed\N1 and you're right.
Dialogue: 0,2:08:19.41,2:08:20.67,Default,,0000,0000,0000,,Make a face.
Dialogue: 0,2:08:20.67,2:08:21.79,Default,,0000,0000,0000,,Do you see another one?
Dialogue: 0,2:08:21.79,2:08:22.77,Default,,0000,0000,0000,,STUDENT: Cosine t.
Dialogue: 0,2:08:22.77,2:08:23.73,Default,,0000,0000,0000,,MAGDALENA TODA: Cosine.
Dialogue: 0,2:08:23.73,2:08:26.16,Default,,0000,0000,0000,,
Dialogue: 0,2:08:26.16,2:08:28.60,Default,,0000,0000,0000,,They are independent,\Nlinear independent.
Dialogue: 0,2:08:28.60,2:08:31.22,Default,,0000,0000,0000,,And so the multitude\Nof solutions
Dialogue: 0,2:08:31.22,2:08:34.27,Default,,0000,0000,0000,,for that-- I taught you\Na whole chapter in 3350.
Dialogue: 0,2:08:34.27,2:08:37.14,Default,,0000,0000,0000,,Now you don't have\Nto take it anymore--
Dialogue: 0,2:08:37.14,2:08:39.72,Default,,0000,0000,0000,,is going to be a equals sine t--
Dialogue: 0,2:08:39.72,2:08:41.05,Default,,0000,0000,0000,,STUDENT: How about e to the i t?
Dialogue: 0,2:08:41.05,2:08:42.30,Default,,0000,0000,0000,,MAGDALENA TODA: Plus b sine t.
Dialogue: 0,2:08:42.30,2:08:43.44,Default,,0000,0000,0000,,I tell you in a second.
Dialogue: 0,2:08:43.44,2:08:46.53,Default,,0000,0000,0000,,All right, we have to\Ndo an e to the i t.
Dialogue: 0,2:08:46.53,2:08:47.52,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:08:47.52,2:08:51.35,Default,,0000,0000,0000,,So you guessed that all the\Nsolutions will be combinations
Dialogue: 0,2:08:51.35,2:08:56.02,Default,,0000,0000,0000,,like-- on the monitor when you\Nhave cosine and sine, if you
Dialogue: 0,2:08:56.02,2:08:58.87,Default,,0000,0000,0000,,add them up-- multiply\Nand add them up,
Dialogue: 0,2:08:58.87,2:09:02.39,Default,,0000,0000,0000,,you get something like the\Nmonitor thing at the hospital.
Dialogue: 0,2:09:02.39,2:09:04.48,Default,,0000,0000,0000,,So any kind of\Noscillation like that
Dialogue: 0,2:09:04.48,2:09:07.69,Default,,0000,0000,0000,,is a combination of this kind.
Dialogue: 0,2:09:07.69,2:09:13.05,Default,,0000,0000,0000,,Maybe with some different\Nphases and amplitudes.
Dialogue: 0,2:09:13.05,2:09:16.83,Default,,0000,0000,0000,,You have cosine of 70 or\Ncosine of 5t or something.
Dialogue: 0,2:09:16.83,2:09:18.65,Default,,0000,0000,0000,,But let me show\Nyou what they are
Dialogue: 0,2:09:18.65,2:09:24.64,Default,,0000,0000,0000,,going to show you [INAUDIBLE]\Nfor the harmonic oscillator
Dialogue: 0,2:09:24.64,2:09:27.27,Default,,0000,0000,0000,,equation how the method goes.
Dialogue: 0,2:09:27.27,2:09:29.25,Default,,0000,0000,0000,,You solve for the\Ncharacteristic equation.
Dialogue: 0,2:09:29.25,2:09:34.60,Default,,0000,0000,0000,,So you have r squared\Nplus 1 equals 0.
Dialogue: 0,2:09:34.60,2:09:38.86,Default,,0000,0000,0000,,Now, here's where most of\Nthe students in 3350 fail.
Dialogue: 0,2:09:38.86,2:09:40.47,Default,,0000,0000,0000,,They understand that.
Dialogue: 0,2:09:40.47,2:09:43.88,Default,,0000,0000,0000,,And some of them say, OK,\Nthis has no solutions.
Dialogue: 0,2:09:43.88,2:09:46.99,Default,,0000,0000,0000,,Some of them even say this\Nhas solutions plus minus 1.
Dialogue: 0,2:09:46.99,2:09:48.87,Default,,0000,0000,0000,,I mean, crazy stuff.
Dialogue: 0,2:09:48.87,2:09:51.49,Default,,0000,0000,0000,,Now, what are the\Nsolutions of that?
Dialogue: 0,2:09:51.49,2:09:53.46,Default,,0000,0000,0000,,Because the theory\Nin this case says
Dialogue: 0,2:09:53.46,2:09:57.33,Default,,0000,0000,0000,,if your solutions are\Nimaginary, then y1
Dialogue: 0,2:09:57.33,2:10:00.98,Default,,0000,0000,0000,,would be e to the ax cosine bx.
Dialogue: 0,2:10:00.98,2:10:05.45,Default,,0000,0000,0000,,And y2 will be e\Nto the ax sine bx
Dialogue: 0,2:10:05.45,2:10:09.45,Default,,0000,0000,0000,,where your imaginary\Nsolutions are a plus minus ib.
Dialogue: 0,2:10:09.45,2:10:13.99,Default,,0000,0000,0000,,It has a lot to do with\NEuler's formula in a way.
Dialogue: 0,2:10:13.99,2:10:21.00,Default,,0000,0000,0000,,So if you knew the theory in\N3350 and not be just very smart
Dialogue: 0,2:10:21.00,2:10:24.13,Default,,0000,0000,0000,,and get these by yourselves\Nby guessing them,
Dialogue: 0,2:10:24.13,2:10:26.52,Default,,0000,0000,0000,,how are you supposed\Nto know that?
Dialogue: 0,2:10:26.52,2:10:30.58,Default,,0000,0000,0000,,Well, r squared\Nequals minus 1, right?
Dialogue: 0,2:10:30.58,2:10:34.04,Default,,0000,0000,0000,,The square root of minus 1 is i.
Dialogue: 0,2:10:34.04,2:10:34.91,Default,,0000,0000,0000,,STUDENT: Or negative.
Dialogue: 0,2:10:34.91,2:10:36.16,Default,,0000,0000,0000,,MAGDALENA TODA: Or negative i.
Dialogue: 0,2:10:36.16,2:10:40.60,Default,,0000,0000,0000,,So r1 is 0 plus minus i.
Dialogue: 0,2:10:40.60,2:10:42.46,Default,,0000,0000,0000,,So who is a?
Dialogue: 0,2:10:42.46,2:10:44.18,Default,,0000,0000,0000,,a is 0.
Dialogue: 0,2:10:44.18,2:10:45.71,Default,,0000,0000,0000,,Who is b?
Dialogue: 0,2:10:45.71,2:10:46.72,Default,,0000,0000,0000,,b is 1.
Dialogue: 0,2:10:46.72,2:10:53.01,Default,,0000,0000,0000,,So the solutions are e to\Nthe 0x equal cosine 1x and e
Dialogue: 0,2:10:53.01,2:10:59.04,Default,,0000,0000,0000,,to the 0x sine 1x, which\Nis cosine x, sine x.
Dialogue: 0,2:10:59.04,2:11:03.21,Default,,0000,0000,0000,,Now you know why you can\Ndo everything formalized
Dialogue: 0,2:11:03.21,2:11:06.17,Default,,0000,0000,0000,,and you get all these\Nsolutions from a method.
Dialogue: 0,2:11:06.17,2:11:10.29,Default,,0000,0000,0000,,This method is an\Nentire chapter.
Dialogue: 0,2:11:10.29,2:11:12.40,Default,,0000,0000,0000,,It's so much easier than in 350.
Dialogue: 0,2:11:12.40,2:11:14.67,Default,,0000,0000,0000,,So much easier than Calculus 3.
Dialogue: 0,2:11:14.67,2:11:16.42,Default,,0000,0000,0000,,You will say this is easy.
Dialogue: 0,2:11:16.42,2:11:17.72,Default,,0000,0000,0000,,It's a pleasure.
Dialogue: 0,2:11:17.72,2:11:22.79,Default,,0000,0000,0000,,You spend about one\Nfourth of the semester
Dialogue: 0,2:11:22.79,2:11:24.56,Default,,0000,0000,0000,,just on this method.
Dialogue: 0,2:11:24.56,2:11:26.27,Default,,0000,0000,0000,,So now you don't have\Nto take it anymore.
Dialogue: 0,2:11:26.27,2:11:29.08,Default,,0000,0000,0000,,You can learn it all by\Nyourself and you're going
Dialogue: 0,2:11:29.08,2:11:33.06,Default,,0000,0000,0000,,to be ready for the next thing.
Dialogue: 0,2:11:33.06,2:11:34.76,Default,,0000,0000,0000,,So I'm just giving you courage.
Dialogue: 0,2:11:34.76,2:11:40.35,Default,,0000,0000,0000,,If you do really, really well in\NCalc 3, 3350 will be a breeze.
Dialogue: 0,2:11:40.35,2:11:42.29,Default,,0000,0000,0000,,You can breeze through that.
Dialogue: 0,2:11:42.29,2:11:46.17,Default,,0000,0000,0000,,You only have the probability\Nin stats for most engineers
Dialogue: 0,2:11:46.17,2:11:48.60,Default,,0000,0000,0000,,to take.
Dialogue: 0,2:11:48.60,2:11:54.21,Default,,0000,0000,0000,,Math is not so complicated.