[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.50,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.50,0:00:04.93,Default,,0000,0000,0000,,PROFESSOR: We will pick\Nup from where we left.
Dialogue: 0,0:00:04.93,0:00:08.41,Default,,0000,0000,0000,,I hope the attendance will\Nget a little bit better today.
Dialogue: 0,0:00:08.41,0:00:13.86,Default,,0000,0000,0000,,It's not even Friday,\Nit's Thursday night.
Dialogue: 0,0:00:13.86,0:00:17.33,Default,,0000,0000,0000,,So last time we talked a\Nlittle bit about chapter 13,
Dialogue: 0,0:00:17.33,0:00:21.30,Default,,0000,0000,0000,,we started 13-1.
Dialogue: 0,0:00:21.30,0:00:27.75,Default,,0000,0000,0000,,I wanted to remind you that we\Nrevisited the notion of work.
Dialogue: 0,0:00:27.75,0:00:40.70,Default,,0000,0000,0000,,
Dialogue: 0,0:00:40.70,0:00:44.23,Default,,0000,0000,0000,,Now, if you notice\Nwhat the book does,
Dialogue: 0,0:00:44.23,0:00:45.98,Default,,0000,0000,0000,,it doesn't give\Nyou any specifics
Dialogue: 0,0:00:45.98,0:00:48.86,Default,,0000,0000,0000,,about the force field.
Dialogue: 0,0:00:48.86,0:00:50.55,Default,,0000,0000,0000,,May the force be with you.
Dialogue: 0,0:00:50.55,0:00:56.03,Default,,0000,0000,0000,,They don't say what kind\Nof animal this f is.
Dialogue: 0,0:00:56.03,0:01:02.57,Default,,0000,0000,0000,,We sort of informally\Nsaid I'm going to have
Dialogue: 0,0:01:02.57,0:01:06.29,Default,,0000,0000,0000,,some sort of path integral.
Dialogue: 0,0:01:06.29,0:01:10.88,Default,,0000,0000,0000,,And I didn't say what conditions\NI was assuming about f.
Dialogue: 0,0:01:10.88,0:01:15.03,Default,,0000,0000,0000,,And I just said that r\Nis the position vector.
Dialogue: 0,0:01:15.03,0:01:18.17,Default,,0000,0000,0000,,
Dialogue: 0,0:01:18.17,0:01:22.82,Default,,0000,0000,0000,,It's important for us to\Nimagine that is plus c1, what
Dialogue: 0,0:01:22.82,0:01:24.29,Default,,0000,0000,0000,,does that mean, c1?
Dialogue: 0,0:01:24.29,0:01:30.20,Default,,0000,0000,0000,,It means that this function,\Nlet's write it R of t, equals.
Dialogue: 0,0:01:30.20,0:01:35.36,Default,,0000,0000,0000,,Let's say we are implying not\Nin space, so we have x of t,
Dialogue: 0,0:01:35.36,0:01:40.45,Default,,0000,0000,0000,,y of t, the parametrization\Nof this position vector.
Dialogue: 0,0:01:40.45,0:01:42.90,Default,,0000,0000,0000,,Of course we wrote that\Nlast time as well, we
Dialogue: 0,0:01:42.90,0:01:46.28,Default,,0000,0000,0000,,said x is x of t, y is y of t.
Dialogue: 0,0:01:46.28,0:01:51.64,Default,,0000,0000,0000,,But why I took c1\Nand not continuous?
Dialogue: 0,0:01:51.64,0:01:53.24,Default,,0000,0000,0000,,Could anybody tell me?
Dialogue: 0,0:01:53.24,0:01:56.29,Default,,0000,0000,0000,,If I'm going to go ahead\Nand differentiate it,
Dialogue: 0,0:01:56.29,0:01:59.29,Default,,0000,0000,0000,,of course I'd like it\Nto be differentiable.
Dialogue: 0,0:01:59.29,0:02:03.52,Default,,0000,0000,0000,,And its derivatives\Nshould be continuous.
Dialogue: 0,0:02:03.52,0:02:07.66,Default,,0000,0000,0000,,But that's actually not\Nenough for my purposes.
Dialogue: 0,0:02:07.66,0:02:11.17,Default,,0000,0000,0000,,So if I want R of t\Nto be c1, that's good,
Dialogue: 0,0:02:11.17,0:02:13.15,Default,,0000,0000,0000,,I'm going to smile.
Dialogue: 0,0:02:13.15,0:02:18.08,Default,,0000,0000,0000,,But when we did that in\Nchapter-- was it chapter 10?
Dialogue: 0,0:02:18.08,0:02:20.56,Default,,0000,0000,0000,,It was chapter 10,\NErin, am I right?
Dialogue: 0,0:02:20.56,0:02:23.62,Default,,0000,0000,0000,,We assumed this was\Na regular curve.
Dialogue: 0,0:02:23.62,0:02:27.65,Default,,0000,0000,0000,,A regular curve is not\Njust as differentiable
Dialogue: 0,0:02:27.65,0:02:31.39,Default,,0000,0000,0000,,with the derivative's\Ncontinuous with respect to time.
Dialogue: 0,0:02:31.39,0:02:34.38,Default,,0000,0000,0000,,x prime of t, y prime\Nof t, both must exist
Dialogue: 0,0:02:34.38,0:02:36.03,Default,,0000,0000,0000,,and must be continuous.
Dialogue: 0,0:02:36.03,0:02:39.85,Default,,0000,0000,0000,,We wanted something\Nelse about the velocity.
Dialogue: 0,0:02:39.85,0:02:42.20,Default,,0000,0000,0000,,Do you remember the drunken bug?
Dialogue: 0,0:02:42.20,0:02:46.52,Default,,0000,0000,0000,,The drunken but was\Nfine and he was flying.
Dialogue: 0,0:02:46.52,0:02:49.82,Default,,0000,0000,0000,,As long as he was flying,\Neverything was fine.
Dialogue: 0,0:02:49.82,0:02:54.08,Default,,0000,0000,0000,,When did the drunken\Nbug have a problem?
Dialogue: 0,0:02:54.08,0:02:56.90,Default,,0000,0000,0000,,When the velocity\Nfield became 0,
Dialogue: 0,0:02:56.90,0:03:02.86,Default,,0000,0000,0000,,at the instant where the bug\Nlost his velocity, right?
Dialogue: 0,0:03:02.86,0:03:10.74,Default,,0000,0000,0000,,So we said regular means c1 and\NR prime of t at any value of t
Dialogue: 0,0:03:10.74,0:03:13.52,Default,,0000,0000,0000,,should be different from 0.
Dialogue: 0,0:03:13.52,0:03:16.62,Default,,0000,0000,0000,,We do not allow the particle\Nto stop on it's way.
Dialogue: 0,0:03:16.62,0:03:20.68,Default,,0000,0000,0000,,We don't allow it, whether it\Nis a photon, a drunken bug,
Dialogue: 0,0:03:20.68,0:03:23.69,Default,,0000,0000,0000,,an airplane, or whatever it is.
Dialogue: 0,0:03:23.69,0:03:27.57,Default,,0000,0000,0000,,We don't want it to\Nstop in it's trajectory.
Dialogue: 0,0:03:27.57,0:03:31.99,Default,,0000,0000,0000,,Is that good for\Nother reasons as well?
Dialogue: 0,0:03:31.99,0:03:35.26,Default,,0000,0000,0000,,Very good for the\Nreason that we want
Dialogue: 0,0:03:35.26,0:03:39.15,Default,,0000,0000,0000,,to think later in arc length.
Dialogue: 0,0:03:39.15,0:03:43.32,Default,,0000,0000,0000,,[INAUDIBLE] came up with\Nthis idea last time.
Dialogue: 0,0:03:43.32,0:03:46.48,Default,,0000,0000,0000,,I didn't want to tell you\Nthe truth, but he was right.
Dialogue: 0,0:03:46.48,0:03:53.01,Default,,0000,0000,0000,,One can define certain\Npath integrals with respect
Dialogue: 0,0:03:53.01,0:03:56.54,Default,,0000,0000,0000,,to s, with respect to\Narc length parameter.
Dialogue: 0,0:03:56.54,0:03:58.40,Default,,0000,0000,0000,,But as you remember\Nvery well, he
Dialogue: 0,0:03:58.40,0:04:03.26,Default,,0000,0000,0000,,had this correspondence between\Nan arbitrary parameter type t
Dialogue: 0,0:04:03.26,0:04:07.64,Default,,0000,0000,0000,,and s, and this is s of t.
Dialogue: 0,0:04:07.64,0:04:10.20,Default,,0000,0000,0000,,And also going back\Nand forth, that
Dialogue: 0,0:04:10.20,0:04:15.37,Default,,0000,0000,0000,,means from s you head back to t.
Dialogue: 0,0:04:15.37,0:04:19.82,Default,,0000,0000,0000,,So here's s of t and\Nthis is t of s, right?
Dialogue: 0,0:04:19.82,0:04:26.50,Default,,0000,0000,0000,,So we have this correspondence\Nand everything worked fine
Dialogue: 0,0:04:26.50,0:04:31.45,Default,,0000,0000,0000,,in terms of being\Nable to invert that.
Dialogue: 0,0:04:31.45,0:04:34.32,Default,,0000,0000,0000,,And having some sort\Nof equal morphisms
Dialogue: 0,0:04:34.32,0:04:40.62,Default,,0000,0000,0000,,as long as the\Nvelocity was non-0.
Dialogue: 0,0:04:40.62,0:04:43.72,Default,,0000,0000,0000,,OK, do you remember\Nwho s of t was?
Dialogue: 0,0:04:43.72,0:04:47.90,Default,,0000,0000,0000,,S of t was defined--\NIt was a long time ago.
Dialogue: 0,0:04:47.90,0:04:50.71,Default,,0000,0000,0000,,So I'm reminding you\Ns of t was integral
Dialogue: 0,0:04:50.71,0:04:55.35,Default,,0000,0000,0000,,from 0 to t-- or from t0 to t.
Dialogue: 0,0:04:55.35,0:04:59.20,Default,,0000,0000,0000,,Your favorite initial\Nmoment in time.
Dialogue: 0,0:04:59.20,0:05:03.57,Default,,0000,0000,0000,,Of the speed, uh-huh, and\Nwhat the heck was the speed?
Dialogue: 0,0:05:03.57,0:05:10.66,Default,,0000,0000,0000,,The speed was the norm or the\Nlength of the R prime of t.
Dialogue: 0,0:05:10.66,0:05:16.65,Default,,0000,0000,0000,,This is called speed, that\Nwe assume different from 0,
Dialogue: 0,0:05:16.65,0:05:19.16,Default,,0000,0000,0000,,for a good purpose.
Dialogue: 0,0:05:19.16,0:05:24.60,Default,,0000,0000,0000,,We can go back and forth\Nbetween t and s, t to s,
Dialogue: 0,0:05:24.60,0:05:30.44,Default,,0000,0000,0000,,s to t, with\Ndifferentiable functions.
Dialogue: 0,0:05:30.44,0:05:35.30,Default,,0000,0000,0000,,Good, so now we can apply the\Ninverse mapping through them.
Dialogue: 0,0:05:35.30,0:05:38.17,Default,,0000,0000,0000,,We can do all sorts\Nof stuff with that.
Dialogue: 0,0:05:38.17,0:05:42.43,Default,,0000,0000,0000,,On this one we did not\Nquite define it rigorously.
Dialogue: 0,0:05:42.43,0:05:43.69,Default,,0000,0000,0000,,What did they say is?
Dialogue: 0,0:05:43.69,0:05:46.49,Default,,0000,0000,0000,,We said f would be a\Ngood enough function,
Dialogue: 0,0:05:46.49,0:05:50.33,Default,,0000,0000,0000,,but know that I do\Nnot need f to be c1.
Dialogue: 0,0:05:50.33,0:05:54.42,Default,,0000,0000,0000,,This is too strong, too strong.
Dialogue: 0,0:05:54.42,0:05:58.87,Default,,0000,0000,0000,,So in Calc 1 when you had to\Nintegrate a function of one
Dialogue: 0,0:05:58.87,0:06:02.97,Default,,0000,0000,0000,,variable you just\Nassumed that- in Calc 1
Dialogue: 0,0:06:02.97,0:06:04.70,Default,,0000,0000,0000,,I remember- you assume\Nthat continuous.
Dialogue: 0,0:06:04.70,0:06:07.58,Default,,0000,0000,0000,,It doesn't even have\Nto be continuous
Dialogue: 0,0:06:07.58,0:06:09.86,Default,,0000,0000,0000,,but let's assume that\Nf would be continuous.
Dialogue: 0,0:06:09.86,0:06:15.65,Default,,0000,0000,0000,,
Dialogue: 0,0:06:15.65,0:06:21.10,Default,,0000,0000,0000,,OK, so you have, in one sense,\Nthat the composition with R,
Dialogue: 0,0:06:21.10,0:06:27.01,Default,,0000,0000,0000,,if you have f of x\Nof t, y of t, z of t.
Dialogue: 0,0:06:27.01,0:06:33.60,Default,,0000,0000,0000,,In terms of time will be a\Nfunctions of one variable,
Dialogue: 0,0:06:33.60,0:06:35.58,Default,,0000,0000,0000,,and this will be continuous.
Dialogue: 0,0:06:35.58,0:06:39.06,Default,,0000,0000,0000,,
Dialogue: 0,0:06:39.06,0:06:41.05,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:06:41.05,0:06:44.61,Default,,0000,0000,0000,,OK, now what if\Nit's not continuous?
Dialogue: 0,0:06:44.61,0:06:47.85,Default,,0000,0000,0000,,Can't I have a piecewise,\Ncontinuous function?
Dialogue: 0,0:06:47.85,0:06:51.42,Default,,0000,0000,0000,,Like in Calc 1, do you guys\Nremember we had some of this?
Dialogue: 0,0:06:51.42,0:06:54.17,Default,,0000,0000,0000,,And from here like that\Nand from here like this
Dialogue: 0,0:06:54.17,0:06:55.17,Default,,0000,0000,0000,,and from here like that.
Dialogue: 0,0:06:55.17,0:06:58.00,Default,,0000,0000,0000,,And we had these continuities,\Nand this was piecewise
Dialogue: 0,0:06:58.00,0:06:59.86,Default,,0000,0000,0000,,continuous.
Dialogue: 0,0:06:59.86,0:07:02.66,Default,,0000,0000,0000,,Yeah, for god sake,\NI can integrate that.
Dialogue: 0,0:07:02.66,0:07:06.53,Default,,0000,0000,0000,,Why do we assume integral\Nof a continuous function?
Dialogue: 0,0:07:06.53,0:07:08.68,Default,,0000,0000,0000,,Just to make our\Nlives easier and also
Dialogue: 0,0:07:08.68,0:07:12.50,Default,,0000,0000,0000,,because we are in freshman\Nand sophomore level Calculus.
Dialogue: 0,0:07:12.50,0:07:16.69,Default,,0000,0000,0000,,If we were in advanced\NCalculus we would say,
Dialogue: 0,0:07:16.69,0:07:21.65,Default,,0000,0000,0000,,I want this function\Nto be integrable.
Dialogue: 0,0:07:21.65,0:07:24.54,Default,,0000,0000,0000,,This is a lot weaker\Nthan continuous,
Dialogue: 0,0:07:24.54,0:07:28.36,Default,,0000,0000,0000,,maybe the set of discontinuities\Nis also very large.
Dialogue: 0,0:07:28.36,0:07:31.51,Default,,0000,0000,0000,,Who told you that you\Nhave finitely many jumps
Dialogue: 0,0:07:31.51,0:07:32.59,Default,,0000,0000,0000,,these continuities?
Dialogue: 0,0:07:32.59,0:07:34.53,Default,,0000,0000,0000,,Maybe you have a\Nmuch larger set.
Dialogue: 0,0:07:34.53,0:07:39.07,Default,,0000,0000,0000,,And this is what you learn\Nin advanced Calculus.
Dialogue: 0,0:07:39.07,0:07:41.71,Default,,0000,0000,0000,,But you are not at\Nthe level of a senior
Dialogue: 0,0:07:41.71,0:07:44.88,Default,,0000,0000,0000,,yet so we'll just assume,\Nfor the time being,
Dialogue: 0,0:07:44.88,0:07:47.07,Default,,0000,0000,0000,,that f is continuous.
Dialogue: 0,0:07:47.07,0:07:49.48,Default,,0000,0000,0000,,All right, and we say,\Nwhat is this animal?
Dialogue: 0,0:07:49.48,0:07:52.33,Default,,0000,0000,0000,,We called it w and\Nbe baptised it.
Dialogue: 0,0:07:52.33,0:07:56.88,Default,,0000,0000,0000,,We said, just give\Nit some sort of name
Dialogue: 0,0:07:56.88,0:07:58.55,Default,,0000,0000,0000,,and we say that is work.
Dialogue: 0,0:07:58.55,0:08:02.60,Default,,0000,0000,0000,,And by definition,\Nby definition,
Dialogue: 0,0:08:02.60,0:08:09.38,Default,,0000,0000,0000,,this is going to be\Nintegral from-- Now,
Dialogue: 0,0:08:09.38,0:08:14.89,Default,,0000,0000,0000,,the thing is, we define this as\Na simple integral with respect
Dialogue: 0,0:08:14.89,0:08:17.90,Default,,0000,0000,0000,,to time as a definition.
Dialogue: 0,0:08:17.90,0:08:19.83,Default,,0000,0000,0000,,That doesn't mean\Nthat I introduced
Dialogue: 0,0:08:19.83,0:08:24.66,Default,,0000,0000,0000,,the notion of path\Nintegral the way I should,
Dialogue: 0,0:08:24.66,0:08:26.34,Default,,0000,0000,0000,,I was cheating on that.
Dialogue: 0,0:08:26.34,0:08:28.55,Default,,0000,0000,0000,,So the way we\Nintroduced it was like,
Dialogue: 0,0:08:28.55,0:08:33.10,Default,,0000,0000,0000,,let f be a function of\Nthe spatial coordinates
Dialogue: 0,0:08:33.10,0:08:34.65,Default,,0000,0000,0000,,in terms of time.
Dialogue: 0,0:08:34.65,0:08:37.79,Default,,0000,0000,0000,,x, y, z are space\Ncoordinates, t is time.
Dialogue: 0,0:08:37.79,0:08:42.07,Default,,0000,0000,0000,,So I have f of R of t here.
Dialogue: 0,0:08:42.07,0:08:45.44,Default,,0000,0000,0000,,Dot Product, who\Nthe heck is the R?
Dialogue: 0,0:08:45.44,0:08:48.28,Default,,0000,0000,0000,,This is nothing but\Na vector art drawing.
Dialogue: 0,0:08:48.28,0:08:50.51,Default,,0000,0000,0000,,These are both\Nvectors, sometimes
Dialogue: 0,0:08:50.51,0:08:53.44,Default,,0000,0000,0000,,I should put them in bold\Nlike they do in the book.
Dialogue: 0,0:08:53.44,0:08:56.46,Default,,0000,0000,0000,,To make it clear I can\Nput a bar on top of them,
Dialogue: 0,0:08:56.46,0:08:57.89,Default,,0000,0000,0000,,they are free vectors.
Dialogue: 0,0:08:57.89,0:09:02.99,Default,,0000,0000,0000,,So, f of R times\NR prime of t dt.
Dialogue: 0,0:09:02.99,0:09:07.18,Default,,0000,0000,0000,,And your favorite moments of\Ntime are-- let's say on my arc
Dialogue: 0,0:09:07.18,0:09:12.27,Default,,0000,0000,0000,,that I'm describing from time,\Nt, equals a, to time equals b.
Dialogue: 0,0:09:12.27,0:09:15.22,Default,,0000,0000,0000,,Therefore, I'm going to\Ntake time for a to b.
Dialogue: 0,0:09:15.22,0:09:19.54,Default,,0000,0000,0000,,And this is how we define\Nthe work of a force.
Dialogue: 0,0:09:19.54,0:09:22.79,Default,,0000,0000,0000,,
Dialogue: 0,0:09:22.79,0:09:25.61,Default,,0000,0000,0000,,The work of a\Nforce that's acting
Dialogue: 0,0:09:25.61,0:09:32.77,Default,,0000,0000,0000,,on a particle that is moving\Nbetween time, a, and time, b,
Dialogue: 0,0:09:32.77,0:09:36.89,Default,,0000,0000,0000,,on this arc of a curve\Nwhich is called c.
Dialogue: 0,0:09:36.89,0:09:38.27,Default,,0000,0000,0000,,Do you like this c?
Dialogue: 0,0:09:38.27,0:09:41.50,Default,,0000,0000,0000,,Okay, and the\Nforce is different.
Dialogue: 0,0:09:41.50,0:09:44.06,Default,,0000,0000,0000,,So we have a force field.
Dialogue: 0,0:09:44.06,0:09:46.36,Default,,0000,0000,0000,,So I cheated, I knew\Na lot in the sense
Dialogue: 0,0:09:46.36,0:09:48.51,Default,,0000,0000,0000,,that I didn't tell\Nyou how you actually
Dialogue: 0,0:09:48.51,0:09:55.28,Default,,0000,0000,0000,,introduce the path integral.
Dialogue: 0,0:09:55.28,0:10:01.22,Default,,0000,0000,0000,,Now this is more or less\Nwhere I stop and [INAUDIBLE].
Dialogue: 0,0:10:01.22,0:10:04.65,Default,,0000,0000,0000,,But couldn't we actually\Nintroduce this integral
Dialogue: 0,0:10:04.65,0:10:11.32,Default,,0000,0000,0000,,and even define it with respect\Nto some arc length grammar?
Dialogue: 0,0:10:11.32,0:10:14.58,Default,,0000,0000,0000,,Maybe if everything goes\Nfine in terms of theory?
Dialogue: 0,0:10:14.58,0:10:16.25,Default,,0000,0000,0000,,And the answer is yes.
Dialogue: 0,0:10:16.25,0:10:20.10,Default,,0000,0000,0000,,And I'm going to show\Nyou how one can do that.
Dialogue: 0,0:10:20.10,0:10:25.67,Default,,0000,0000,0000,,I'm going to go ahead and\Nclean here a little bit.
Dialogue: 0,0:10:25.67,0:10:29.17,Default,,0000,0000,0000,,I'm going to leave this on\Nby comparison for awhile.
Dialogue: 0,0:10:29.17,0:10:33.51,Default,,0000,0000,0000,,And then I will assume\Nsomething that we have not
Dialogue: 0,0:10:33.51,0:10:40.04,Default,,0000,0000,0000,,defined whatsoever, which is\Nan animal called path integral.
Dialogue: 0,0:10:40.04,0:10:54.26,Default,,0000,0000,0000,,So the path integral of a vector\Nfield along a trajectory, c.
Dialogue: 0,0:10:54.26,0:10:55.79,Default,,0000,0000,0000,,I don't know how to draw.
Dialogue: 0,0:10:55.79,0:10:58.63,Default,,0000,0000,0000,,I will draw some skewed\Ncurve, how about that?
Dialogue: 0,0:10:58.63,0:11:04.13,Default,,0000,0000,0000,,Some pretty skewed curve, c,\Nit's not self intersecting,
Dialogue: 0,0:11:04.13,0:11:05.36,Default,,0000,0000,0000,,not necessarily.
Dialogue: 0,0:11:05.36,0:11:08.77,Default,,0000,0000,0000,,You guys have to imagine\Nthis is like the trajectory
Dialogue: 0,0:11:08.77,0:11:13.22,Default,,0000,0000,0000,,of an airplane in\Nthe sky, right?
Dialogue: 0,0:11:13.22,0:11:17.74,Default,,0000,0000,0000,,OK, and I have it on d\Nequals a, to d equals b.
Dialogue: 0,0:11:17.74,0:11:21.15,Default,,0000,0000,0000,,But I said forget\Nabout the time, t,
Dialogue: 0,0:11:21.15,0:11:24.52,Default,,0000,0000,0000,,maybe I can do everything\Nin arc length forever.
Dialogue: 0,0:11:24.52,0:11:26.32,Default,,0000,0000,0000,,So if that particle,\Nor airplane,
Dialogue: 0,0:11:26.32,0:11:31.12,Default,,0000,0000,0000,,or whatever it is has\Na continuous motion,
Dialogue: 0,0:11:31.12,0:11:32.75,Default,,0000,0000,0000,,that's also differentiable.
Dialogue: 0,0:11:32.75,0:11:35.71,Default,,0000,0000,0000,,And the velocity\Nnever becomes zero.
Dialogue: 0,0:11:35.71,0:11:38.65,Default,,0000,0000,0000,,Then I can parametrize\Nan arc length
Dialogue: 0,0:11:38.65,0:11:44.16,Default,,0000,0000,0000,,and I can say, forget about\Nit, I have integral over c.
Dialogue: 0,0:11:44.16,0:11:46.97,Default,,0000,0000,0000,,See, this is c,\Nit's not f, okay?
Dialogue: 0,0:11:46.97,0:11:54.75,Default,,0000,0000,0000,,But f of x of s, y\Nof s, z of s, okay?
Dialogue: 0,0:11:54.75,0:11:57.57,Default,,0000,0000,0000,,And this is going to be a ds.
Dialogue: 0,0:11:57.57,0:12:00.69,Default,,0000,0000,0000,,And you'll say, yes Magdelina--\Nthis is little s, I'm sorry.
Dialogue: 0,0:12:00.69,0:12:03.31,Default,,0000,0000,0000,,Yes, Magdelina, but what\Nthe heck is this animal,
Dialogue: 0,0:12:03.31,0:12:05.04,Default,,0000,0000,0000,,you've never introduced it.
Dialogue: 0,0:12:05.04,0:12:09.00,Default,,0000,0000,0000,,I have not introduced it because\NI have to discuss about it.
Dialogue: 0,0:12:09.00,0:12:14.74,Default,,0000,0000,0000,,When we introduced Riemann\Nsums, then we took the limit.
Dialogue: 0,0:12:14.74,0:12:19.89,Default,,0000,0000,0000,,We always have to think how\Nto partition our domains.
Dialogue: 0,0:12:19.89,0:12:30.62,Default,,0000,0000,0000,,So this curve can be partitioned\Nin as many as n, this is s k.
Dialogue: 0,0:12:30.62,0:12:36.22,Default,,0000,0000,0000,,S k, this is s1, and this is\Ns n, the last of the Mohicans.
Dialogue: 0,0:12:36.22,0:12:42.17,Default,,0000,0000,0000,,I have n sub intervals,\Npieces of the art.
Dialogue: 0,0:12:42.17,0:12:44.41,Default,,0000,0000,0000,,And how am I going\Nto introduce this?
Dialogue: 0,0:12:44.41,0:12:47.22,Default,,0000,0000,0000,,As the limit, if it exists.
Dialogue: 0,0:12:47.22,0:12:49.48,Default,,0000,0000,0000,,Because I can be in\Ntrouble, maybe this limit
Dialogue: 0,0:12:49.48,0:12:51.58,Default,,0000,0000,0000,,is not going to exist.
Dialogue: 0,0:12:51.58,0:12:54.22,Default,,0000,0000,0000,,The sum of what?
Dialogue: 0,0:12:54.22,0:12:58.76,Default,,0000,0000,0000,,For every [? seg ?] partition\NI will take a little arbitrary
Dialogue: 0,0:12:58.76,0:13:00.19,Default,,0000,0000,0000,,point inside the subarc.
Dialogue: 0,0:13:00.19,0:13:03.15,Default,,0000,0000,0000,,
Dialogue: 0,0:13:03.15,0:13:03.78,Default,,0000,0000,0000,,Subarc?
Dialogue: 0,0:13:03.78,0:13:04.55,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:13:04.55,0:13:06.32,Default,,0000,0000,0000,,PROFESSOR: Subarc,\Nit's a little arc.
Dialogue: 0,0:13:06.32,0:13:09.45,Default,,0000,0000,0000,,Contains a-- let's take it here.
Dialogue: 0,0:13:09.45,0:13:13.00,Default,,0000,0000,0000,,What am I going to\Ndefine in terms of wind?
Dialogue: 0,0:13:13.00,0:13:20.73,Default,,0000,0000,0000,,s k, y k, and z k, some\Npeople put a star on it
Dialogue: 0,0:13:20.73,0:13:23.89,Default,,0000,0000,0000,,to make it obvious.
Dialogue: 0,0:13:23.89,0:13:27.04,Default,,0000,0000,0000,,But I'm going to\Ngo ahead and say
Dialogue: 0,0:13:27.04,0:13:35.60,Default,,0000,0000,0000,,x star k, y star k, z star k,\Nis my arbitrary point in the k
Dialogue: 0,0:13:35.60,0:13:38.14,Default,,0000,0000,0000,,subarc.
Dialogue: 0,0:13:38.14,0:13:42.10,Default,,0000,0000,0000,,Times, what shall I multiply by?
Dialogue: 0,0:13:42.10,0:13:48.11,Default,,0000,0000,0000,,A delta sk, and then\NI take k from one to n
Dialogue: 0,0:13:48.11,0:13:52.14,Default,,0000,0000,0000,,and I press to the\Nlimit with respect n.
Dialogue: 0,0:13:52.14,0:13:57.15,Default,,0000,0000,0000,,But actually I could also\Nsay in some other ways
Dialogue: 0,0:13:57.15,0:14:05.17,Default,,0000,0000,0000,,that the partitions length goes\Nto 0, delta s goes to zero.
Dialogue: 0,0:14:05.17,0:14:07.05,Default,,0000,0000,0000,,And you say but,\Nnow wait a minute,
Dialogue: 0,0:14:07.05,0:14:11.60,Default,,0000,0000,0000,,you have s1, s2 s3, s4,\Ns k, little tiny subarc,
Dialogue: 0,0:14:11.60,0:14:13.42,Default,,0000,0000,0000,,what the heck is delta s?
Dialogue: 0,0:14:13.42,0:14:21.82,Default,,0000,0000,0000,,Delta s is the largest subarc.
Dialogue: 0,0:14:21.82,0:14:25.98,Default,,0000,0000,0000,,So the length of the largest\Nsubarc, length of the largest
Dialogue: 0,0:14:25.98,0:14:29.18,Default,,0000,0000,0000,,subarc in the partition.
Dialogue: 0,0:14:29.18,0:14:34.27,Default,,0000,0000,0000,,So the more points I take,\Nthe more I refine this.
Dialogue: 0,0:14:34.27,0:14:36.19,Default,,0000,0000,0000,,I take the points closer\Nand closer and closer
Dialogue: 0,0:14:36.19,0:14:37.60,Default,,0000,0000,0000,,in this partition.
Dialogue: 0,0:14:37.60,0:14:41.01,Default,,0000,0000,0000,,What happens to the\Nlength of this partition?
Dialogue: 0,0:14:41.01,0:14:43.85,Default,,0000,0000,0000,,It shrinks to-- it goes to 0.
Dialogue: 0,0:14:43.85,0:14:45.96,Default,,0000,0000,0000,,Assuming that this\Nwould be the largest
Dialogue: 0,0:14:45.96,0:14:48.69,Default,,0000,0000,0000,,one, well if the\Nlargest one goes to 0,
Dialogue: 0,0:14:48.69,0:14:51.83,Default,,0000,0000,0000,,everybody else goes to 0.
Dialogue: 0,0:14:51.83,0:14:55.75,Default,,0000,0000,0000,,So this is a Riemann\Nsum, can we know for sure
Dialogue: 0,0:14:55.75,0:14:57.97,Default,,0000,0000,0000,,that this limit exists?
Dialogue: 0,0:14:57.97,0:15:02.69,Default,,0000,0000,0000,,No, we hope to god\Nthat this limit exists.
Dialogue: 0,0:15:02.69,0:15:08.15,Default,,0000,0000,0000,,And if the limit exists then\NI will introduce this notion
Dialogue: 0,0:15:08.15,0:15:09.77,Default,,0000,0000,0000,,of integral around the back.
Dialogue: 0,0:15:09.77,0:15:15.45,Default,,0000,0000,0000,,
Dialogue: 0,0:15:15.45,0:15:18.10,Default,,0000,0000,0000,,And you said, OK I\Nbelieve you, but look,
Dialogue: 0,0:15:18.10,0:15:22.12,Default,,0000,0000,0000,,what is the connection\Nbetween the work- the way
Dialogue: 0,0:15:22.12,0:15:25.85,Default,,0000,0000,0000,,you introduced it as a simple\NCalculus 1 integral here-
Dialogue: 0,0:15:25.85,0:15:30.50,Default,,0000,0000,0000,,and this animal that looks like\Nan alien coming from the sky.
Dialogue: 0,0:15:30.50,0:15:33.28,Default,,0000,0000,0000,,We don't know how to look at it.
Dialogue: 0,0:15:33.28,0:15:37.83,Default,,0000,0000,0000,,Actually guys it's not so\Nbad, you do the same thing
Dialogue: 0,0:15:37.83,0:15:40.24,Default,,0000,0000,0000,,as you did before.
Dialogue: 0,0:15:40.24,0:15:44.18,Default,,0000,0000,0000,,In a sense that, s is connected\Nto any time parameter.
Dialogue: 0,0:15:44.18,0:15:47.75,Default,,0000,0000,0000,,So Mr. ds says, I'm\Nyour old friend,
Dialogue: 0,0:15:47.75,0:15:55.58,Default,,0000,0000,0000,,trust me, I know who I am.\Nds was the speed times dt.
Dialogue: 0,0:15:55.58,0:16:00.64,Default,,0000,0000,0000,,Who can tell me if we are in R\Nthree, and we are drunken bugs,
Dialogue: 0,0:16:00.64,0:16:02.88,Default,,0000,0000,0000,,ds will become what?
Dialogue: 0,0:16:02.88,0:16:07.55,Default,,0000,0000,0000,,A long square root times\Ndt, and what's inside here?
Dialogue: 0,0:16:07.55,0:16:10.22,Default,,0000,0000,0000,,I want to see if\Nyou guys are awake.
Dialogue: 0,0:16:10.22,0:16:11.65,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:16:11.65,0:16:17.18,Default,,0000,0000,0000,,PROFESSOR: Very good, x prime\Nof t squared, I'm so lazy
Dialogue: 0,0:16:17.18,0:16:20.88,Default,,0000,0000,0000,,but I'll write it down. y prime\Nof t squared plus z prime of t
Dialogue: 0,0:16:20.88,0:16:21.68,Default,,0000,0000,0000,,squared.
Dialogue: 0,0:16:21.68,0:16:24.21,Default,,0000,0000,0000,,And this is going\Nto be the speed.
Dialogue: 0,0:16:24.21,0:16:28.98,Default,,0000,0000,0000,,So I can always do\Nthat, and in this case
Dialogue: 0,0:16:28.98,0:16:33.20,Default,,0000,0000,0000,,this is going to become always\Nsome-- let's say from time, t0,
Dialogue: 0,0:16:33.20,0:16:36.24,Default,,0000,0000,0000,,to time t1.
Dialogue: 0,0:16:36.24,0:16:39.24,Default,,0000,0000,0000,,Some in the integrals\Nof-- some of the limit
Dialogue: 0,0:16:39.24,0:16:40.58,Default,,0000,0000,0000,,points for the time.
Dialogue: 0,0:16:40.58,0:16:45.46,Default,,0000,0000,0000,,I'm going to have\Nf of R of s of t,
Dialogue: 0,0:16:45.46,0:16:47.67,Default,,0000,0000,0000,,in the end everything\Nwill depend on t.
Dialogue: 0,0:16:47.67,0:16:49.90,Default,,0000,0000,0000,,And this is my face being happy.
Dialogue: 0,0:16:49.90,0:16:51.97,Default,,0000,0000,0000,,It's not part of the integral.
Dialogue: 0,0:16:51.97,0:16:52.71,Default,,0000,0000,0000,,Saying what?
Dialogue: 0,0:16:52.71,0:16:55.29,Default,,0000,0000,0000,,Saying that, guys, if\NI plug in everything
Dialogue: 0,0:16:55.29,0:16:58.56,Default,,0000,0000,0000,,back in terms of t- I'm\Nmore familiar to that type
Dialogue: 0,0:16:58.56,0:17:00.77,Default,,0000,0000,0000,,of integral- then I have what?
Dialogue: 0,0:17:00.77,0:17:04.54,Default,,0000,0000,0000,,Square root of-- that's\Nthe arc length element
Dialogue: 0,0:17:04.54,0:17:07.45,Default,,0000,0000,0000,,x prime then t squared,\Nplus y prime then t
Dialogue: 0,0:17:07.45,0:17:12.51,Default,,0000,0000,0000,,squared, plus z prime\Nthen t squared, dt.
Dialogue: 0,0:17:12.51,0:17:17.79,Default,,0000,0000,0000,,So in the end it is-- I think\Nthe video doesn't see me
Dialogue: 0,0:17:17.79,0:17:21.57,Default,,0000,0000,0000,,but it heard me, presumably.
Dialogue: 0,0:17:21.57,0:17:24.22,Default,,0000,0000,0000,,This is our old\Nfriend from Calc 1,
Dialogue: 0,0:17:24.22,0:17:29.99,Default,,0000,0000,0000,,which is the simple integral\Nwith respect to t from a to b.
Dialogue: 0,0:17:29.99,0:17:34.37,Default,,0000,0000,0000,,OK, all right, and we\Nbelieve that the work
Dialogue: 0,0:17:34.37,0:17:36.87,Default,,0000,0000,0000,,can be expressed like that.
Dialogue: 0,0:17:36.87,0:17:39.61,Default,,0000,0000,0000,,I introduced it\Nlast time, I even
Dialogue: 0,0:17:39.61,0:17:41.86,Default,,0000,0000,0000,,proved it on some\Nparticular cases
Dialogue: 0,0:17:41.86,0:17:45.47,Default,,0000,0000,0000,,last time when Alex wasn't\Nhere because, I know why.
Dialogue: 0,0:17:45.47,0:17:46.93,Default,,0000,0000,0000,,Were you sick?
Dialogue: 0,0:17:46.93,0:17:48.51,Default,,0000,0000,0000,,ALEX: I'll talk to\Nyou about it later.
Dialogue: 0,0:17:48.51,0:17:49.85,Default,,0000,0000,0000,,I'm a bad person.
Dialogue: 0,0:17:49.85,0:17:56.92,Default,,0000,0000,0000,,PROFESSOR: All right, then\NI'm dragging an object like,
Dialogue: 0,0:17:56.92,0:18:02.61,Default,,0000,0000,0000,,the f was parallel to the\Ndirection of displacement.
Dialogue: 0,0:18:02.61,0:18:04.80,Default,,0000,0000,0000,,And then I said\Nthe work would be
Dialogue: 0,0:18:04.80,0:18:09.15,Default,,0000,0000,0000,,the magnitude of f times the\Nmagnitude of the displacement.
Dialogue: 0,0:18:09.15,0:18:13.45,Default,,0000,0000,0000,,And then we proved that is\Njust a particular case of this,
Dialogue: 0,0:18:13.45,0:18:15.95,Default,,0000,0000,0000,,we proved that last time,\Nit was a piece of cake.
Dialogue: 0,0:18:15.95,0:18:17.86,Default,,0000,0000,0000,,Actually, we proved\Nthe other one.
Dialogue: 0,0:18:17.86,0:18:23.28,Default,,0000,0000,0000,,It proved that if force is going\Nto be oblique and at an angle
Dialogue: 0,0:18:23.28,0:18:27.18,Default,,0000,0000,0000,,theta with the displacement\Ndirection, then
Dialogue: 0,0:18:27.18,0:18:31.03,Default,,0000,0000,0000,,the work will be the\Nmagnitude of the force times
Dialogue: 0,0:18:31.03,0:18:35.80,Default,,0000,0000,0000,,cosine of theta, times the\Nmagnitude of displacement,
Dialogue: 0,0:18:35.80,0:18:36.73,Default,,0000,0000,0000,,all right?
Dialogue: 0,0:18:36.73,0:18:42.19,Default,,0000,0000,0000,,And that was all an application\Nof this beautiful warp formula.
Dialogue: 0,0:18:42.19,0:18:46.22,Default,,0000,0000,0000,,Let's see something more\Ninteresting from an application
Dialogue: 0,0:18:46.22,0:18:48.00,Default,,0000,0000,0000,,viewpoint.
Dialogue: 0,0:18:48.00,0:18:53.03,Default,,0000,0000,0000,,Assume that you are\Nlooking at the washer,
Dialogue: 0,0:18:53.03,0:18:56.76,Default,,0000,0000,0000,,you are just doing laundry.
Dialogue: 0,0:18:56.76,0:19:01.46,Default,,0000,0000,0000,,And you are looking at\Nthis centrifugal force.
Dialogue: 0,0:19:01.46,0:19:08.17,Default,,0000,0000,0000,,We have two forces, one is\Ncentripetal towards the center
Dialogue: 0,0:19:08.17,0:19:11.59,Default,,0000,0000,0000,,of the motion, circular\Nmotion, one is centrifugal.
Dialogue: 0,0:19:11.59,0:19:15.22,Default,,0000,0000,0000,,I will take a\Ncentrifugal force f,
Dialogue: 0,0:19:15.22,0:19:19.44,Default,,0000,0000,0000,,and I will say I want\Nto measure at the work
Dialogue: 0,0:19:19.44,0:19:25.31,Default,,0000,0000,0000,,that this force is producing\Nin the circular motion
Dialogue: 0,0:19:25.31,0:19:27.70,Default,,0000,0000,0000,,of my dryer.
Dialogue: 0,0:19:27.70,0:19:31.71,Default,,0000,0000,0000,,My poor dryer died so I\Nhad to buy another one
Dialogue: 0,0:19:31.71,0:19:33.14,Default,,0000,0000,0000,,and it cost me a lot of money.
Dialogue: 0,0:19:33.14,0:19:36.18,Default,,0000,0000,0000,,And I was thinking,\Nsuch a simple thing,
Dialogue: 0,0:19:36.18,0:19:38.63,Default,,0000,0000,0000,,you pay hundreds\Nof dollars on it
Dialogue: 0,0:19:38.63,0:19:41.57,Default,,0000,0000,0000,,but, anyway, we take\Nsome things for granted.
Dialogue: 0,0:19:41.57,0:19:45.99,Default,,0000,0000,0000,,
Dialogue: 0,0:19:45.99,0:19:54.61,Default,,0000,0000,0000,,I will take the washer because\Nthe washer is a simpler
Dialogue: 0,0:19:54.61,0:19:59.78,Default,,0000,0000,0000,,case in the sense that the\Nmotion-- I can assume it's
Dialogue: 0,0:19:59.78,0:20:04.18,Default,,0000,0000,0000,,a circular motion of\Nconstant velocity.
Dialogue: 0,0:20:04.18,0:20:07.49,Default,,0000,0000,0000,,And let's say this\Nis the washer.
Dialogue: 0,0:20:07.49,0:20:10.84,Default,,0000,0000,0000,,
Dialogue: 0,0:20:10.84,0:20:16.94,Default,,0000,0000,0000,,And centrifugal\Nforce is acting here.
Dialogue: 0,0:20:16.94,0:20:21.06,Default,,0000,0000,0000,,Let's call that--\Nwhat should it be?
Dialogue: 0,0:20:21.06,0:20:28.71,Default,,0000,0000,0000,,Well, it's continuing\Nthe position vector
Dialogue: 0,0:20:28.71,0:20:36.44,Default,,0000,0000,0000,,so let's call that lambda\Nx I, plus lambda y j.
Dialogue: 0,0:20:36.44,0:20:42.36,Default,,0000,0000,0000,,In the sense that it's\Ncollinear to the vector that
Dialogue: 0,0:20:42.36,0:20:46.58,Default,,0000,0000,0000,,starts at origin, and here\Nis got to be x of t, y.
Dialogue: 0,0:20:46.58,0:20:50.04,Default,,0000,0000,0000,,X and y are the\Nspecial components
Dialogue: 0,0:20:50.04,0:20:52.56,Default,,0000,0000,0000,,at any point on my\Ncircular motion.
Dialogue: 0,0:20:52.56,0:20:55.46,Default,,0000,0000,0000,,If it's a circular\Nmotion I have x
Dialogue: 0,0:20:55.46,0:20:57.55,Default,,0000,0000,0000,,squared plus y squared\Nequals r squared,
Dialogue: 0,0:20:57.55,0:21:01.66,Default,,0000,0000,0000,,where the radius is the\Nradius of my washer.
Dialogue: 0,0:21:01.66,0:21:06.52,Default,,0000,0000,0000,,You have to compute\Nthe work produced
Dialogue: 0,0:21:06.52,0:21:10.96,Default,,0000,0000,0000,,by the centrifugal force\Nin one full rotation.
Dialogue: 0,0:21:10.96,0:21:14.31,Default,,0000,0000,0000,,It doesn't matter, I can have\Ninfinitely many rotations.
Dialogue: 0,0:21:14.31,0:21:19.03,Default,,0000,0000,0000,,I can have a hundred rotations,\NI couldn't care less.
Dialogue: 0,0:21:19.03,0:21:23.31,Default,,0000,0000,0000,,But assume that the\Nmotion has constant speed.
Dialogue: 0,0:21:23.31,0:21:26.65,Default,,0000,0000,0000,,So if I wanted, I could\Nparametrize in our things
Dialogue: 0,0:21:26.65,0:21:30.27,Default,,0000,0000,0000,,but it doesn't\Nbring a difference.
Dialogue: 0,0:21:30.27,0:21:33.71,Default,,0000,0000,0000,,Because guys, when this\Nspeed is already constant,
Dialogue: 0,0:21:33.71,0:21:36.33,Default,,0000,0000,0000,,like for the circular motion\Nyou are familiar with.
Dialogue: 0,0:21:36.33,0:21:39.76,Default,,0000,0000,0000,,Or the helicoidal case\Nyou are familiar with,
Dialogue: 0,0:21:39.76,0:21:44.12,Default,,0000,0000,0000,,you also saw the case when\Nthe speed was constant.
Dialogue: 0,0:21:44.12,0:21:47.81,Default,,0000,0000,0000,,Practically, you were\Njust rescaling the time
Dialogue: 0,0:21:47.81,0:21:52.75,Default,,0000,0000,0000,,to get to your speed, to your\Ntime parameter s, arc length.
Dialogue: 0,0:21:52.75,0:21:54.84,Default,,0000,0000,0000,,So whether you work with\Nt, or you work with s,
Dialogue: 0,0:21:54.84,0:21:58.23,Default,,0000,0000,0000,,it's the same thing if\Nthe speed is a constant.
Dialogue: 0,0:21:58.23,0:22:02.03,Default,,0000,0000,0000,,So I'm not going to use my\Nimagination to go and do it
Dialogue: 0,0:22:02.03,0:22:02.90,Default,,0000,0000,0000,,with respect to s.
Dialogue: 0,0:22:02.90,0:22:05.50,Default,,0000,0000,0000,,I could, but I\Ncouldn't give a damn
Dialogue: 0,0:22:05.50,0:22:09.56,Default,,0000,0000,0000,,because I'm going to have\Na beautiful t that you
Dialogue: 0,0:22:09.56,0:22:13.00,Default,,0000,0000,0000,,are going to help me recover.
Dialogue: 0,0:22:13.00,0:22:15.64,Default,,0000,0000,0000,,From here, what is\Nthe parametrization
Dialogue: 0,0:22:15.64,0:22:18.70,Default,,0000,0000,0000,,that comes to mind?
Dialogue: 0,0:22:18.70,0:22:19.57,Default,,0000,0000,0000,,Can you guys help me?
Dialogue: 0,0:22:19.57,0:22:23.64,Default,,0000,0000,0000,,I know you can after all\Nthe review of chapter 10
Dialogue: 0,0:22:23.64,0:22:25.86,Default,,0000,0000,0000,,and-- this is what?
Dialogue: 0,0:22:25.86,0:22:29.47,Default,,0000,0000,0000,,
Dialogue: 0,0:22:29.47,0:22:32.97,Default,,0000,0000,0000,,R what?
Dialogue: 0,0:22:32.97,0:22:34.97,Default,,0000,0000,0000,,You should whisper cosine t.
Dialogue: 0,0:22:34.97,0:22:38.68,Default,,0000,0000,0000,,Say it out loud, be\Nproud of what you know.
Dialogue: 0,0:22:38.68,0:22:40.64,Default,,0000,0000,0000,,This is R sine t.
Dialogue: 0,0:22:40.64,0:22:43.08,Default,,0000,0000,0000,,And let's take t\Nbetween 0 and 2 pi.
Dialogue: 0,0:22:43.08,0:22:50.73,Default,,0000,0000,0000,,One, the revolution only,\Nand then I say, good.
Dialogue: 0,0:22:50.73,0:22:53.100,Default,,0000,0000,0000,,The speed is what?
Dialogue: 0,0:22:53.100,0:23:00.93,Default,,0000,0000,0000,,Speed, square root of\Nx prime the t squared
Dialogue: 0,0:23:00.93,0:23:03.43,Default,,0000,0000,0000,,plus y prime the t squared.
Dialogue: 0,0:23:03.43,0:23:08.26,Default,,0000,0000,0000,,Which is the same as writing\NR prime of the t in magnitude.
Dialogue: 0,0:23:08.26,0:23:09.48,Default,,0000,0000,0000,,Thank God we know that.
Dialogue: 0,0:23:09.48,0:23:11.55,Default,,0000,0000,0000,,How much is this?
Dialogue: 0,0:23:11.55,0:23:14.27,Default,,0000,0000,0000,,R, very good, this\Nis R, very good.
Dialogue: 0,0:23:14.27,0:23:18.79,Default,,0000,0000,0000,,So life is not so hard, it's--\Nhopefully I'll be able to do
Dialogue: 0,0:23:18.79,0:23:21.66,Default,,0000,0000,0000,,the w.
Dialogue: 0,0:23:21.66,0:23:22.41,Default,,0000,0000,0000,,What is the w?
Dialogue: 0,0:23:22.41,0:23:25.91,Default,,0000,0000,0000,,It's the path integral\Nall over the circle
Dialogue: 0,0:23:25.91,0:23:30.38,Default,,0000,0000,0000,,I have here, that I traveled\Ncounterclockwise from any point
Dialogue: 0,0:23:30.38,0:23:31.27,Default,,0000,0000,0000,,to any point.
Dialogue: 0,0:23:31.27,0:23:34.56,Default,,0000,0000,0000,,Let's say this would be\Nthe origin of my motion,
Dialogue: 0,0:23:34.56,0:23:37.34,Default,,0000,0000,0000,,then I go back.
Dialogue: 0,0:23:37.34,0:23:43.32,Default,,0000,0000,0000,,And I have this\Nforce, F, that I have
Dialogue: 0,0:23:43.32,0:23:47.54,Default,,0000,0000,0000,,to redistribute in terms\Nof R. So this notation
Dialogue: 0,0:23:47.54,0:23:49.22,Default,,0000,0000,0000,,is giving me a little\Nbit of a headache,
Dialogue: 0,0:23:49.22,0:23:51.41,Default,,0000,0000,0000,,but in reality it's\Ngoing to be very simple.
Dialogue: 0,0:23:51.41,0:23:59.74,Default,,0000,0000,0000,,This is the dot product, R\Nprime dt, which was the R.
Dialogue: 0,0:23:59.74,0:24:05.39,Default,,0000,0000,0000,,Which some other people asked\Nme, how can you write that?
Dialogue: 0,0:24:05.39,0:24:12.09,Default,,0000,0000,0000,,Well, read the\Nreview, R of x equals
Dialogue: 0,0:24:12.09,0:24:16.81,Default,,0000,0000,0000,,x of t, i plus y of t, j.
Dialogue: 0,0:24:16.81,0:24:19.99,Default,,0000,0000,0000,,Also, read the\Nnext side plus y j.
Dialogue: 0,0:24:19.99,0:24:26.59,Default,,0000,0000,0000,,It short, the dR\Ndifferential out of t,
Dialogue: 0,0:24:26.59,0:24:32.92,Default,,0000,0000,0000,,sorry, I'll put R.\NdR is dx i plus dy j.
Dialogue: 0,0:24:32.92,0:24:35.95,Default,,0000,0000,0000,,And if somebody wants\Nto be expressing
Dialogue: 0,0:24:35.95,0:24:40.39,Default,,0000,0000,0000,,this in terms of speeds,\Nwe'll say this is x prime dt,
Dialogue: 0,0:24:40.39,0:24:43.01,Default,,0000,0000,0000,,this is y prime dt.
Dialogue: 0,0:24:43.01,0:24:48.13,Default,,0000,0000,0000,,So we can rewrite this x prime\Nthen t i, plus y prime then t
Dialogue: 0,0:24:48.13,0:24:49.84,Default,,0000,0000,0000,,j, dt.
Dialogue: 0,0:24:49.84,0:24:54.62,Default,,0000,0000,0000,,
Dialogue: 0,0:24:54.62,0:24:55.80,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:24:55.80,0:25:01.35,Default,,0000,0000,0000,,OK, which is the same\Nthing as R prime of t, dt.
Dialogue: 0,0:25:01.35,0:25:04.20,Default,,0000,0000,0000,,
Dialogue: 0,0:25:04.20,0:25:04.70,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:25:04.70,0:25:07.58,Default,,0000,0000,0000,,
Dialogue: 0,0:25:07.58,0:25:11.76,Default,,0000,0000,0000,,This looks awfully theoretical.
Dialogue: 0,0:25:11.76,0:25:16.55,Default,,0000,0000,0000,,I say, I don't like it, I\Nwant to put my favorite guys
Dialogue: 0,0:25:16.55,0:25:18.29,Default,,0000,0000,0000,,in the picture.
Dialogue: 0,0:25:18.29,0:25:22.14,Default,,0000,0000,0000,,So I have to think, when\NI do the dot product
Dialogue: 0,0:25:22.14,0:25:26.38,Default,,0000,0000,0000,,I have the dot product\Nbetween the vector that
Dialogue: 0,0:25:26.38,0:25:29.89,Default,,0000,0000,0000,,has components f1 and f2.
Dialogue: 0,0:25:29.89,0:25:31.87,Default,,0000,0000,0000,,How am I going to do that?
Dialogue: 0,0:25:31.87,0:25:38.70,Default,,0000,0000,0000,,Well, if I multiply with this\Nguy, dot product, the boss guy,
Dialogue: 0,0:25:38.70,0:25:39.80,Default,,0000,0000,0000,,with this boss guy.
Dialogue: 0,0:25:39.80,0:25:43.18,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:25:43.18,0:25:44.81,Default,,0000,0000,0000,,What am I going to do?
Dialogue: 0,0:25:44.81,0:25:48.56,Default,,0000,0000,0000,,First component times\Nfirst component,
Dialogue: 0,0:25:48.56,0:25:53.74,Default,,0000,0000,0000,,plus second component times\Nsecond component of a vector.
Dialogue: 0,0:25:53.74,0:25:58.83,Default,,0000,0000,0000,,So I have to be smart and\Nunderstand how I do that.
Dialogue: 0,0:25:58.83,0:26:01.06,Default,,0000,0000,0000,,Lambda is a constant.
Dialogue: 0,0:26:01.06,0:26:04.11,Default,,0000,0000,0000,,Lambda, you're my\Nfriend, you stay there.
Dialogue: 0,0:26:04.11,0:26:09.57,Default,,0000,0000,0000,,x is x of t, x of\Nt, but I multiplied
Dialogue: 0,0:26:09.57,0:26:11.49,Default,,0000,0000,0000,,with the first\Ncomponent here, so I
Dialogue: 0,0:26:11.49,0:26:14.86,Default,,0000,0000,0000,,multiplied by x prime of t.
Dialogue: 0,0:26:14.86,0:26:19.60,Default,,0000,0000,0000,,Plus lambda times y of\Nt, times y prime of t.
Dialogue: 0,0:26:19.60,0:26:23.75,Default,,0000,0000,0000,,And who gets out of the\Npicture is dt at the end.
Dialogue: 0,0:26:23.75,0:26:27.53,Default,,0000,0000,0000,,I have integrate with\Nrespect to that dt.
Dialogue: 0,0:26:27.53,0:26:29.57,Default,,0000,0000,0000,,This would be incorrect, why?
Dialogue: 0,0:26:29.57,0:26:34.60,Default,,0000,0000,0000,,Because t has to move\Nbetween some specific limits
Dialogue: 0,0:26:34.60,0:26:38.18,Default,,0000,0000,0000,,when I specify what\Na path integral is.
Dialogue: 0,0:26:38.18,0:26:41.87,Default,,0000,0000,0000,,I cannot leave a c-- very good,\Nfrom 0 to 2 pi, excellent.
Dialogue: 0,0:26:41.87,0:26:45.30,Default,,0000,0000,0000,,
Dialogue: 0,0:26:45.30,0:26:46.24,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:26:46.24,0:26:48.67,Default,,0000,0000,0000,,No, It's going to\Nbe a piece of cake.
Dialogue: 0,0:26:48.67,0:26:50.34,Default,,0000,0000,0000,,Why is that a piece of cake?
Dialogue: 0,0:26:50.34,0:26:54.19,Default,,0000,0000,0000,,Because I can keep writing.
Dialogue: 0,0:26:54.19,0:26:58.52,Default,,0000,0000,0000,,You actually are faster than me.
Dialogue: 0,0:26:58.52,0:27:00.56,Default,,0000,0000,0000,,STUDENT: Your chain rule\Nis already done for you.
Dialogue: 0,0:27:00.56,0:27:03.93,Default,,0000,0000,0000,,PROFESSOR: Right,\Nand then lambda
Dialogue: 0,0:27:03.93,0:27:06.92,Default,,0000,0000,0000,,gets out just because--\Nwell you remember
Dialogue: 0,0:27:06.92,0:27:09.60,Default,,0000,0000,0000,,you kick the lambda out, right?
Dialogue: 0,0:27:09.60,0:27:20.14,Default,,0000,0000,0000,,And then I've put R cosine\Nt times minus R sine t.
Dialogue: 0,0:27:20.14,0:27:21.31,Default,,0000,0000,0000,,I'm done with who?
Dialogue: 0,0:27:21.31,0:27:23.62,Default,,0000,0000,0000,,I'm done with this\Nfellow and that fellow.
Dialogue: 0,0:27:23.62,0:27:27.34,Default,,0000,0000,0000,,
Dialogue: 0,0:27:27.34,0:27:37.55,Default,,0000,0000,0000,,And plus y, R sine\Nt, what is R prime?
Dialogue: 0,0:27:37.55,0:27:48.58,Default,,0000,0000,0000,,R cosine, thank you guys, dt.
Dialogue: 0,0:27:48.58,0:27:50.91,Default,,0000,0000,0000,,And now I'm going to ask\Nyou, what is this animal?
Dialogue: 0,0:27:50.91,0:27:53.66,Default,,0000,0000,0000,,
Dialogue: 0,0:27:53.66,0:27:56.02,Default,,0000,0000,0000,,Stare at that, what\Nis the integrand?
Dialogue: 0,0:27:56.02,0:27:59.13,Default,,0000,0000,0000,,Is a friend of\Nyours, he's so cute.
Dialogue: 0,0:27:59.13,0:28:01.54,Default,,0000,0000,0000,,He's staring at you and\Nsaying you are done.
Dialogue: 0,0:28:01.54,0:28:04.90,Default,,0000,0000,0000,,Why are you done?
Dialogue: 0,0:28:04.90,0:28:08.35,Default,,0000,0000,0000,,What happens to the integrand?
Dialogue: 0,0:28:08.35,0:28:12.36,Default,,0000,0000,0000,,It's zero, it's a\Nblessing, it's zero.
Dialogue: 0,0:28:12.36,0:28:14.00,Default,,0000,0000,0000,,How come it's zero?
Dialogue: 0,0:28:14.00,0:28:21.39,Default,,0000,0000,0000,,Because these two terms\Nsimplify, they cancel out.
Dialogue: 0,0:28:21.39,0:28:26.29,Default,,0000,0000,0000,,They cancel out, thank god\Nthey cancel out, I got a zero.
Dialogue: 0,0:28:26.29,0:28:28.53,Default,,0000,0000,0000,,So we discover something\Nthat a physicist
Dialogue: 0,0:28:28.53,0:28:32.15,Default,,0000,0000,0000,,or a mechanical engineer\Nwould have told you already.
Dialogue: 0,0:28:32.15,0:28:34.11,Default,,0000,0000,0000,,And do you think he would\Nhave actually plugged
Dialogue: 0,0:28:34.11,0:28:36.23,Default,,0000,0000,0000,,in the path integral?
Dialogue: 0,0:28:36.23,0:28:39.70,Default,,0000,0000,0000,,No, they wouldn't\Nthink like this.
Dialogue: 0,0:28:39.70,0:28:45.54,Default,,0000,0000,0000,,He has a simpler explanation for\Nthat because he's experienced
Dialogue: 0,0:28:45.54,0:28:47.26,Default,,0000,0000,0000,,with linear experiments.
Dialogue: 0,0:28:47.26,0:28:51.74,Default,,0000,0000,0000,,And says, if I drag this like\Nthat I know what to work with.
Dialogue: 0,0:28:51.74,0:28:54.77,Default,,0000,0000,0000,,If I drag in, like\Nat an angle, I
Dialogue: 0,0:28:54.77,0:28:57.95,Default,,0000,0000,0000,,know that I have the\Nmagnitude of this,
Dialogue: 0,0:28:57.95,0:28:59.39,Default,,0000,0000,0000,,cosine theta, the angle.
Dialogue: 0,0:28:59.39,0:29:03.74,Default,,0000,0000,0000,,So, he knows for linear\Ncases what we have.
Dialogue: 0,0:29:03.74,0:29:10.06,Default,,0000,0000,0000,,For a circular case he\Ncan smell the result
Dialogue: 0,0:29:10.06,0:29:12.88,Default,,0000,0000,0000,,without doing the path integral.
Dialogue: 0,0:29:12.88,0:29:18.08,Default,,0000,0000,0000,,So how do you think the guy,\Nif he's a mechanical engineer,
Dialogue: 0,0:29:18.08,0:29:20.04,Default,,0000,0000,0000,,would think in a second?
Dialogue: 0,0:29:20.04,0:29:24.32,Default,,0000,0000,0000,,Say well, think of\Nyour trajectory, right?
Dialogue: 0,0:29:24.32,0:29:27.36,Default,,0000,0000,0000,,It's a circle.
Dialogue: 0,0:29:27.36,0:29:31.62,Default,,0000,0000,0000,,The problem is that centrifugal\Nforce being perpendicular
Dialogue: 0,0:29:31.62,0:29:33.56,Default,,0000,0000,0000,,to the circle all the time.
Dialogue: 0,0:29:33.56,0:29:37.14,Default,,0000,0000,0000,,And you say, how can a line\Nbe perpendicular to a circle?
Dialogue: 0,0:29:37.14,0:29:39.57,Default,,0000,0000,0000,,It is, it's the\Nnormal to the circle.
Dialogue: 0,0:29:39.57,0:29:43.05,Default,,0000,0000,0000,,So when you say this\Nis normal to the circle
Dialogue: 0,0:29:43.05,0:29:45.47,Default,,0000,0000,0000,,you mean it's normal to\Nthe tangent of the circle.
Dialogue: 0,0:29:45.47,0:29:50.84,Default,,0000,0000,0000,,So if you measure the angle made\Nby the normal at every point
Dialogue: 0,0:29:50.84,0:29:53.80,Default,,0000,0000,0000,,to the trajectory of a\Ncircle, it's always lines.
Dialogue: 0,0:29:53.80,0:30:01.52,Default,,0000,0000,0000,,So he goes, gosh, I got\Ncosine of 90, that's zero.
Dialogue: 0,0:30:01.52,0:30:04.27,Default,,0000,0000,0000,,So if you have some\Nsort of work produced
Dialogue: 0,0:30:04.27,0:30:06.47,Default,,0000,0000,0000,,by something perpendicular\Nto your trajectory,
Dialogue: 0,0:30:06.47,0:30:07.80,Default,,0000,0000,0000,,that must be zero.
Dialogue: 0,0:30:07.80,0:30:12.14,Default,,0000,0000,0000,,So he or she has\Nvery good intuition.
Dialogue: 0,0:30:12.14,0:30:13.88,Default,,0000,0000,0000,,Of course, how do we prove it?
Dialogue: 0,0:30:13.88,0:30:16.54,Default,,0000,0000,0000,,We are mathematicians, we\Nprove the path integral,
Dialogue: 0,0:30:16.54,0:30:19.46,Default,,0000,0000,0000,,we got zero for the\Nwork, all right?
Dialogue: 0,0:30:19.46,0:30:25.20,Default,,0000,0000,0000,,But he could sense that kind\Nof stuff from the beginning.
Dialogue: 0,0:30:25.20,0:30:34.74,Default,,0000,0000,0000,,Now, there is another example\Nwhere maybe you don't have
Dialogue: 0,0:30:34.74,0:30:38.36,Default,,0000,0000,0000,,90 degrees for your trajectory.
Dialogue: 0,0:30:38.36,0:30:47.97,Default,,0000,0000,0000,,Well, I'm going to just take--\Nwhat if I change the force
Dialogue: 0,0:30:47.97,0:30:50.57,Default,,0000,0000,0000,,and I make a difference problem?
Dialogue: 0,0:30:50.57,0:30:52.84,Default,,0000,0000,0000,,Make it into a\Ndifferent problem.
Dialogue: 0,0:30:52.84,0:31:01.29,Default,,0000,0000,0000,,
Dialogue: 0,0:31:01.29,0:31:04.77,Default,,0000,0000,0000,,I will do that later, I\Nwon't go and erase it.
Dialogue: 0,0:31:04.77,0:31:15.71,Default,,0000,0000,0000,,
Dialogue: 0,0:31:15.71,0:31:19.62,Default,,0000,0000,0000,,Last time we did one\Nthat was, compute
Dialogue: 0,0:31:19.62,0:31:25.33,Default,,0000,0000,0000,,the work along a parabola\Nfrom something to something.
Dialogue: 0,0:31:25.33,0:31:26.83,Default,,0000,0000,0000,,Let's do that again.
Dialogue: 0,0:31:26.83,0:31:30.67,Default,,0000,0000,0000,,
Dialogue: 0,0:31:30.67,0:31:35.58,Default,,0000,0000,0000,,For some sort of\Na nice force field
Dialogue: 0,0:31:35.58,0:31:39.74,Default,,0000,0000,0000,,I'll take your vector valued\Nfunction to be nice to you.
Dialogue: 0,0:31:39.74,0:31:42.38,Default,,0000,0000,0000,,I'll change it, y i plus x j.
Dialogue: 0,0:31:42.38,0:31:45.33,Default,,0000,0000,0000,,
Dialogue: 0,0:31:45.33,0:31:48.55,Default,,0000,0000,0000,,And then we are in\Nplane and we move
Dialogue: 0,0:31:48.55,0:31:55.02,Default,,0000,0000,0000,,along this parabola between\N0, 0 and-- what is this guys?
Dialogue: 0,0:31:55.02,0:31:59.35,Default,,0000,0000,0000,,1, 1-- well let make it\Ninto a one, it's cute.
Dialogue: 0,0:31:59.35,0:32:04.69,Default,,0000,0000,0000,,And I'd like you to measure\Nthe work along the parabola
Dialogue: 0,0:32:04.69,0:32:11.41,Default,,0000,0000,0000,,and also along the arc of a--\Nalong the segment of a line
Dialogue: 0,0:32:11.41,0:32:13.44,Default,,0000,0000,0000,,between the two points.
Dialogue: 0,0:32:13.44,0:32:16.73,Default,,0000,0000,0000,,So I want you to compute\Nw1 along the parabola,
Dialogue: 0,0:32:16.73,0:32:22.99,Default,,0000,0000,0000,,and w2 along this\Nthingy, the segment.
Dialogue: 0,0:32:22.99,0:32:24.07,Default,,0000,0000,0000,,Should it be hard?
Dialogue: 0,0:32:24.07,0:32:27.23,Default,,0000,0000,0000,,No, this was old\Nsession for many finals.
Dialogue: 0,0:32:27.23,0:32:33.18,Default,,0000,0000,0000,,I remember, I think it\Nwas 2003, 2006, 2008,
Dialogue: 0,0:32:33.18,0:32:36.32,Default,,0000,0000,0000,,and very recently, I\Nthink a year and 1/2 ago.
Dialogue: 0,0:32:36.32,0:32:38.02,Default,,0000,0000,0000,,A problem like that was given.
Dialogue: 0,0:32:38.02,0:32:40.75,Default,,0000,0000,0000,,Compute the path\Nintegrals correspond
Dialogue: 0,0:32:40.75,0:32:45.00,Default,,0000,0000,0000,,to work for both parametrization\Nand compare them.
Dialogue: 0,0:32:45.00,0:32:45.65,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:32:45.65,0:32:48.86,Default,,0000,0000,0000,,I have no idea, let me think.
Dialogue: 0,0:32:48.86,0:32:53.26,Default,,0000,0000,0000,,For the first one we\Nhave parametrization
Dialogue: 0,0:32:53.26,0:32:55.60,Default,,0000,0000,0000,,that we need to distinguish\Nfrom the other one.
Dialogue: 0,0:32:55.60,0:32:58.71,Default,,0000,0000,0000,,The first parametrization\Nfor a parabola,
Dialogue: 0,0:32:58.71,0:33:01.43,Default,,0000,0000,0000,,we discussed it last\Ntime, was of course
Dialogue: 0,0:33:01.43,0:33:04.08,Default,,0000,0000,0000,,the simplest possible\None you can think of.
Dialogue: 0,0:33:04.08,0:33:05.83,Default,,0000,0000,0000,,And we did this\Nlast time but I'm
Dialogue: 0,0:33:05.83,0:33:10.36,Default,,0000,0000,0000,,repeating this because I\Ndidn't want Alex to miss that.
Dialogue: 0,0:33:10.36,0:33:18.25,Default,,0000,0000,0000,,And I'm going to say integral\Nfrom some time to some time.
Dialogue: 0,0:33:18.25,0:33:21.09,Default,,0000,0000,0000,,Now, if I'm between 0\Nand 1, time of course
Dialogue: 0,0:33:21.09,0:33:25.78,Default,,0000,0000,0000,,will be between 0 and\N1 because x is time.
Dialogue: 0,0:33:25.78,0:33:28.55,Default,,0000,0000,0000,,All right, good,\Nthat means what else?
Dialogue: 0,0:33:28.55,0:33:32.14,Default,,0000,0000,0000,,This Is f1 and this is f2.
Dialogue: 0,0:33:32.14,0:33:39.72,Default,,0000,0000,0000,,So I'm going to have f1 of t,\Nx prime of t, plus f2 of t,
Dialogue: 0,0:33:39.72,0:33:46.31,Default,,0000,0000,0000,,y prime of t, all the\N[? sausage ?] times dt.
Dialogue: 0,0:33:46.31,0:33:47.93,Default,,0000,0000,0000,,Is this going to be hard?
Dialogue: 0,0:33:47.93,0:33:52.83,Default,,0000,0000,0000,,Hopefully not, I'm going to\Nhave to identify everybody.
Dialogue: 0,0:33:52.83,0:33:55.97,Default,,0000,0000,0000,,Identify this guys prime\Nof t with respect to t
Dialogue: 0,0:33:55.97,0:33:58.11,Default,,0000,0000,0000,,is 1, piece of cake, right?
Dialogue: 0,0:33:58.11,0:34:02.85,Default,,0000,0000,0000,,This fellow is-- you told\Nme last time you got 2t
Dialogue: 0,0:34:02.85,0:34:05.36,Default,,0000,0000,0000,,and you got it right.
Dialogue: 0,0:34:05.36,0:34:08.22,Default,,0000,0000,0000,,This guy, I have to be\Na little bit careful
Dialogue: 0,0:34:08.22,0:34:10.68,Default,,0000,0000,0000,,because y is the fourth guy.
Dialogue: 0,0:34:10.68,0:34:15.92,Default,,0000,0000,0000,,This is t squared and this is t.
Dialogue: 0,0:34:15.92,0:34:19.89,Default,,0000,0000,0000,,
Dialogue: 0,0:34:19.89,0:34:22.96,Default,,0000,0000,0000,,So my integral will be a joke.
Dialogue: 0,0:34:22.96,0:34:29.63,Default,,0000,0000,0000,,0 to 1, 2t squared plus t\Nsquared equals 3t squared.
Dialogue: 0,0:34:29.63,0:34:30.98,Default,,0000,0000,0000,,Is it hard to integrate?
Dialogue: 0,0:34:30.98,0:34:33.11,Default,,0000,0000,0000,,No, for God's sake,\Nthis is integral-- this
Dialogue: 0,0:34:33.11,0:34:39.27,Default,,0000,0000,0000,,is t cubed between 0\Nand 1, right, right?
Dialogue: 0,0:34:39.27,0:34:44.08,Default,,0000,0000,0000,,So I should get\N1, and if I get, I
Dialogue: 0,0:34:44.08,0:34:48.60,Default,,0000,0000,0000,,think I did it right, if I get\Nthe other parametrization you
Dialogue: 0,0:34:48.60,0:34:52.54,Default,,0000,0000,0000,,have to help me write it again.
Dialogue: 0,0:34:52.54,0:34:57.19,Default,,0000,0000,0000,,The parametrization\Nof this straight line
Dialogue: 0,0:34:57.19,0:34:59.38,Default,,0000,0000,0000,,between 0, 0 and 1, 1.
Dialogue: 0,0:34:59.38,0:35:03.30,Default,,0000,0000,0000,,Now on the actual\Nexam, I'm never
Dialogue: 0,0:35:03.30,0:35:08.50,Default,,0000,0000,0000,,going to forgive you if you\Ndon't know how to parametrize.
Dialogue: 0,0:35:08.50,0:35:13.14,Default,,0000,0000,0000,,Now you know it but two months\Nago you didn't, many of you
Dialogue: 0,0:35:13.14,0:35:14.10,Default,,0000,0000,0000,,didn't.
Dialogue: 0,0:35:14.10,0:35:21.51,Default,,0000,0000,0000,,So if somebody gives you 2\Npoints, OK, in plane I ask you,
Dialogue: 0,0:35:21.51,0:35:27.16,Default,,0000,0000,0000,,how do you write that symmetric\Nequation of the line between?
Dialogue: 0,0:35:27.16,0:35:29.03,Default,,0000,0000,0000,,You were a little\Nbit hesitant, now
Dialogue: 0,0:35:29.03,0:35:34.03,Default,,0000,0000,0000,,you shouldn't be hesitant\Nbecause it's a serious thing.
Dialogue: 0,0:35:34.03,0:35:36.29,Default,,0000,0000,0000,,So how did we write that?
Dialogue: 0,0:35:36.29,0:35:40.47,Default,,0000,0000,0000,,We memorized it. x minus\Nx1, over x2 minus x1
Dialogue: 0,0:35:40.47,0:35:44.44,Default,,0000,0000,0000,,equals y minus y1,\Nover y2 minus y1.
Dialogue: 0,0:35:44.44,0:35:47.30,Default,,0000,0000,0000,,This can also be written\Nas-- you know, guys,
Dialogue: 0,0:35:47.30,0:35:49.84,Default,,0000,0000,0000,,that this over that\Nis the actual slope.
Dialogue: 0,0:35:49.84,0:35:52.28,Default,,0000,0000,0000,,This over that, so\Nit can be written
Dialogue: 0,0:35:52.28,0:35:53.56,Default,,0000,0000,0000,,as a [INAUDIBLE] formula.
Dialogue: 0,0:35:53.56,0:35:55.25,Default,,0000,0000,0000,,It can be written in many ways.
Dialogue: 0,0:35:55.25,0:35:59.92,Default,,0000,0000,0000,,And if we put a, t, we transform\Nit into a parametric equation.
Dialogue: 0,0:35:59.92,0:36:02.14,Default,,0000,0000,0000,,So you should be\Nable, on the final,
Dialogue: 0,0:36:02.14,0:36:05.69,Default,,0000,0000,0000,,to do that for any segment of\Na line with your eyes closed.
Dialogue: 0,0:36:05.69,0:36:07.55,Default,,0000,0000,0000,,Like, you see the\Nnumbers, you plug them in,
Dialogue: 0,0:36:07.55,0:36:10.64,Default,,0000,0000,0000,,you get the\Nparametric equations.
Dialogue: 0,0:36:10.64,0:36:14.82,Default,,0000,0000,0000,,We are nice on the exams\Nbecause we usually give you
Dialogue: 0,0:36:14.82,0:36:18.75,Default,,0000,0000,0000,,a line that's easy to write.
Dialogue: 0,0:36:18.75,0:36:25.83,Default,,0000,0000,0000,,Like in this case you would\Nhave x equals t and y equal,
Dialogue: 0,0:36:25.83,0:36:29.94,Default,,0000,0000,0000,,let's see if you\Nare asleep yet, t.
Dialogue: 0,0:36:29.94,0:36:31.50,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:36:31.50,0:36:37.29,Default,,0000,0000,0000,,Because the line that joins\N0, 0 and 1, 1 is y equals x.
Dialogue: 0,0:36:37.29,0:36:40.10,Default,,0000,0000,0000,,So y equals x is called\Nalso, first bicycle.
Dialogue: 0,0:36:40.10,0:36:43.10,Default,,0000,0000,0000,,It's the old friend of\Nyours from trigonometry,
Dialogue: 0,0:36:43.10,0:36:46.30,Default,,0000,0000,0000,,from Pre-Calc, from algebra,\NI don't know where, college
Dialogue: 0,0:36:46.30,0:36:47.82,Default,,0000,0000,0000,,algebra.
Dialogue: 0,0:36:47.82,0:36:49.16,Default,,0000,0000,0000,,Alrighty, is this hard to do?
Dialogue: 0,0:36:49.16,0:36:55.85,Default,,0000,0000,0000,,No, it's easier than before.\Nw2 is integral from 0 to 1,
Dialogue: 0,0:36:55.85,0:37:00.38,Default,,0000,0000,0000,,this is t and this is t, this\Nis t and this is t, good.
Dialogue: 0,0:37:00.38,0:37:07.99,Default,,0000,0000,0000,,So we have t times\N1 plus t times 1,
Dialogue: 0,0:37:07.99,0:37:15.36,Default,,0000,0000,0000,,it's like a funny, nice, game\Nthat's too simple, 2t, 2t.
Dialogue: 0,0:37:15.36,0:37:18.34,Default,,0000,0000,0000,,
Dialogue: 0,0:37:18.34,0:37:23.10,Default,,0000,0000,0000,,So the fundamental\Ntheorem of Calc
Dialogue: 0,0:37:23.10,0:37:27.04,Default,,0000,0000,0000,,says t squared between 0\Nand 1, the answer is 1.
Dialogue: 0,0:37:27.04,0:37:28.50,Default,,0000,0000,0000,,Am I surprised?
Dialogue: 0,0:37:28.50,0:37:31.36,Default,,0000,0000,0000,,Look at me, do I\Nlook surprised at all
Dialogue: 0,0:37:31.36,0:37:34.24,Default,,0000,0000,0000,,that I got the same answer?
Dialogue: 0,0:37:34.24,0:37:37.22,Default,,0000,0000,0000,,No, I told you a\Nsecret last time.
Dialogue: 0,0:37:37.22,0:37:39.39,Default,,0000,0000,0000,,I didn't prove it.
Dialogue: 0,0:37:39.39,0:37:43.10,Default,,0000,0000,0000,,I said that R times happy times.
Dialogue: 0,0:37:43.10,0:37:46.74,Default,,0000,0000,0000,,When depending on the\Nforce that is with you,
Dialogue: 0,0:37:46.74,0:37:51.30,Default,,0000,0000,0000,,you have the same work\Nno matter what path
Dialogue: 0,0:37:51.30,0:37:53.83,Default,,0000,0000,0000,,you are taking between a and b.
Dialogue: 0,0:37:53.83,0:37:57.72,Default,,0000,0000,0000,,Between the origin\Nand finish line.
Dialogue: 0,0:37:57.72,0:38:01.03,Default,,0000,0000,0000,,So I'm claiming that if I\Ngive you this zig-zag line
Dialogue: 0,0:38:01.03,0:38:08.01,Default,,0000,0000,0000,,and I asked you what-- look,\Nit could be any crazy path
Dialogue: 0,0:38:08.01,0:38:11.09,Default,,0000,0000,0000,,but it has to be a nice\Ndifferentiable path.
Dialogue: 0,0:38:11.09,0:38:14.63,Default,,0000,0000,0000,,Along this differentiable\Npath, no matter
Dialogue: 0,0:38:14.63,0:38:17.42,Default,,0000,0000,0000,,how you compute the work,\Nthat's your business,
Dialogue: 0,0:38:17.42,0:38:19.04,Default,,0000,0000,0000,,I claim I still get 1.
Dialogue: 0,0:38:19.04,0:38:21.84,Default,,0000,0000,0000,,
Dialogue: 0,0:38:21.84,0:38:26.44,Default,,0000,0000,0000,,Can you even think why,\Nsome of you remember maybe,
Dialogue: 0,0:38:26.44,0:38:28.99,Default,,0000,0000,0000,,the force was key?
Dialogue: 0,0:38:28.99,0:38:30.44,Default,,0000,0000,0000,,STUDENT: It's a\Nconservative force.
Dialogue: 0,0:38:30.44,0:38:32.78,Default,,0000,0000,0000,,PROFESSOR: It had to\Nbe good conservative.
Dialogue: 0,0:38:32.78,0:38:37.32,Default,,0000,0000,0000,,Now this is conservative but\Nwhy is that conservative?
Dialogue: 0,0:38:37.32,0:38:39.90,Default,,0000,0000,0000,,What the heck is a\Nconservative force?
Dialogue: 0,0:38:39.90,0:38:46.71,Default,,0000,0000,0000,,So let's write it\Ndown on the-- we
Dialogue: 0,0:38:46.71,0:39:01.65,Default,,0000,0000,0000,,say that the vector\Nvalued function, f,
Dialogue: 0,0:39:01.65,0:39:19.08,Default,,0000,0000,0000,,valued in R2 or R3, is\Nconservative if there exists
Dialogue: 0,0:39:19.08,0:39:25.46,Default,,0000,0000,0000,,a smooth function.
Dialogue: 0,0:39:25.46,0:39:28.99,Default,,0000,0000,0000,,Little f, it actually\Nhas to be just c1,
Dialogue: 0,0:39:28.99,0:39:31.66,Default,,0000,0000,0000,,called scalar potential.
Dialogue: 0,0:39:31.66,0:39:45.38,Default,,0000,0000,0000,,Called scalar potential,\Nsuch that big F as a vector
Dialogue: 0,0:39:45.38,0:39:48.10,Default,,0000,0000,0000,,field will be not little f.
Dialogue: 0,0:39:48.10,0:39:52.83,Default,,0000,0000,0000,,That means it will be the\Ngradient of the scalar
Dialogue: 0,0:39:52.83,0:39:54.84,Default,,0000,0000,0000,,potential.
Dialogue: 0,0:39:54.84,0:39:57.37,Default,,0000,0000,0000,,Definition, that\Nwas the definition,
Dialogue: 0,0:39:57.37,0:40:08.44,Default,,0000,0000,0000,,and then criterion for a f\Nin R2 to be conservative.
Dialogue: 0,0:40:08.44,0:40:13.39,Default,,0000,0000,0000,,
Dialogue: 0,0:40:13.39,0:40:21.37,Default,,0000,0000,0000,,I claim that f equals f1 i,\Nplus f two eyes, no, f2 j.
Dialogue: 0,0:40:21.37,0:40:23.37,Default,,0000,0000,0000,,I'm just making\Nsilly puns, I don't
Dialogue: 0,0:40:23.37,0:40:27.88,Default,,0000,0000,0000,,know if you guys follow me.
Dialogue: 0,0:40:27.88,0:40:33.47,Default,,0000,0000,0000,,If and only if f sub 1\Nprime, with respect to y,
Dialogue: 0,0:40:33.47,0:40:37.89,Default,,0000,0000,0000,,is f sub 2 prime,\Nwith respect to x.
Dialogue: 0,0:40:37.89,0:40:40.17,Default,,0000,0000,0000,,Can I prove this?
Dialogue: 0,0:40:40.17,0:40:45.56,Default,,0000,0000,0000,,Prove, prove Magdelina, don't\Njust stare at it, prove.
Dialogue: 0,0:40:45.56,0:40:48.03,Default,,0000,0000,0000,,Why would that be\Nnecessary and sufficient?
Dialogue: 0,0:40:48.03,0:40:52.02,Default,,0000,0000,0000,,
Dialogue: 0,0:40:52.02,0:40:58.89,Default,,0000,0000,0000,,Well, for big F\Nto be conservative
Dialogue: 0,0:40:58.89,0:41:02.18,Default,,0000,0000,0000,,it means that it has\Nto be the gradient
Dialogue: 0,0:41:02.18,0:41:06.97,Default,,0000,0000,0000,,of some little function, little\Nf, some scalar potential.
Dialogue: 0,0:41:06.97,0:41:13.20,Default,,0000,0000,0000,,Alrighty, so let me\Nwrite it down, proof.
Dialogue: 0,0:41:13.20,0:41:20.58,Default,,0000,0000,0000,,f conservative if\Nand only if there
Dialogue: 0,0:41:20.58,0:41:27.96,Default,,0000,0000,0000,,exists f, such that gradient\Nof f is F. If and only
Dialogue: 0,0:41:27.96,0:41:32.99,Default,,0000,0000,0000,,if-- what does it\Nmean about f1 and f2?
Dialogue: 0,0:41:32.99,0:41:36.54,Default,,0000,0000,0000,,f1 and f2 are the f\Nsub of x and the f sub
Dialogue: 0,0:41:36.54,0:41:39.46,Default,,0000,0000,0000,,y of some scalar potential.
Dialogue: 0,0:41:39.46,0:41:44.30,Default,,0000,0000,0000,,So if f is the gradient, that\Nmeans that the first component
Dialogue: 0,0:41:44.30,0:41:48.60,Default,,0000,0000,0000,,has to be little f sub x And\Nthe second component should
Dialogue: 0,0:41:48.60,0:41:51.91,Default,,0000,0000,0000,,have to be little f sub y.
Dialogue: 0,0:41:51.91,0:41:58.24,Default,,0000,0000,0000,,But that is if and only if f\Nsub 1 prime, with respect to y,
Dialogue: 0,0:41:58.24,0:42:01.71,Default,,0000,0000,0000,,is the same as f sub 2\Nprime, with respect to x.
Dialogue: 0,0:42:01.71,0:42:04.14,Default,,0000,0000,0000,,What is that?
Dialogue: 0,0:42:04.14,0:42:14.05,Default,,0000,0000,0000,,The red thing here is called\Na compatibility condition
Dialogue: 0,0:42:14.05,0:42:16.44,Default,,0000,0000,0000,,of this system.
Dialogue: 0,0:42:16.44,0:42:22.66,Default,,0000,0000,0000,,This is a system of two OD's.
Dialogue: 0,0:42:22.66,0:42:27.43,Default,,0000,0000,0000,,You are going to\Nstudy ODEs in 3350.
Dialogue: 0,0:42:27.43,0:42:29.73,Default,,0000,0000,0000,,And you are going\Nto remember this
Dialogue: 0,0:42:29.73,0:42:33.02,Default,,0000,0000,0000,,and say, Oh, I know that because\Nshe taught me that in Calc 3.
Dialogue: 0,0:42:33.02,0:42:36.31,Default,,0000,0000,0000,,Not all instructors will\Nteach you this in Calc 3.
Dialogue: 0,0:42:36.31,0:42:38.88,Default,,0000,0000,0000,,Some of them fool you\Nand skip this material
Dialogue: 0,0:42:38.88,0:42:44.41,Default,,0000,0000,0000,,that's very important\Nto understand in 3350.
Dialogue: 0,0:42:44.41,0:42:49.49,Default,,0000,0000,0000,,So guys, what's going to\Nhappened when you prime this
Dialogue: 0,0:42:49.49,0:42:51.58,Default,,0000,0000,0000,,with respect to y?
Dialogue: 0,0:42:51.58,0:42:55.44,Default,,0000,0000,0000,,You get f sub x prime,\Nwith respect to y.
Dialogue: 0,0:42:55.44,0:42:57.11,Default,,0000,0000,0000,,When you prime this\Nwith respect to x
Dialogue: 0,0:42:57.11,0:42:59.86,Default,,0000,0000,0000,,you get f sub y prime,\Nwith respect to x.
Dialogue: 0,0:42:59.86,0:43:01.68,Default,,0000,0000,0000,,Why are they the same thing?
Dialogue: 0,0:43:01.68,0:43:04.72,Default,,0000,0000,0000,,I'm going to remind\Nyou that they
Dialogue: 0,0:43:04.72,0:43:07.22,Default,,0000,0000,0000,,are the same thing\Nfor a smooth function.
Dialogue: 0,0:43:07.22,0:43:09.25,Default,,0000,0000,0000,,Who said that?
Dialogue: 0,0:43:09.25,0:43:19.06,Default,,0000,0000,0000,,A crazy German mathematician\Nwhose name was Schwartz.
Dialogue: 0,0:43:19.06,0:43:22.13,Default,,0000,0000,0000,,Which means black, that's\Nwhat I'm painting it in black.
Dialogue: 0,0:43:22.13,0:43:29.09,Default,,0000,0000,0000,,Because is the Schwartz\Nguy, the first criterion
Dialogue: 0,0:43:29.09,0:43:31.75,Default,,0000,0000,0000,,saying that no\Nmatter in what order
Dialogue: 0,0:43:31.75,0:43:34.32,Default,,0000,0000,0000,,you differentiate the\Nsmooth function you
Dialogue: 0,0:43:34.32,0:43:37.80,Default,,0000,0000,0000,,get the same answer for\Nthe mixed derivative.
Dialogue: 0,0:43:37.80,0:43:41.62,Default,,0000,0000,0000,,So you see we prove if\Nand only if that you
Dialogue: 0,0:43:41.62,0:43:43.90,Default,,0000,0000,0000,,have to have this\Ncriterion, otherwise
Dialogue: 0,0:43:43.90,0:43:46.21,Default,,0000,0000,0000,,it's not going to\Nbe conservative.
Dialogue: 0,0:43:46.21,0:43:49.63,Default,,0000,0000,0000,,So I'm asking you,\Nfor your old friend, f
Dialogue: 0,0:43:49.63,0:43:52.84,Default,,0000,0000,0000,,equals- example\None or example two,
Dialogue: 0,0:43:52.84,0:43:56.20,Default,,0000,0000,0000,,I don't know- y i plus x j.
Dialogue: 0,0:43:56.20,0:43:58.51,Default,,0000,0000,0000,,Is the conservative?
Dialogue: 0,0:43:58.51,0:44:00.44,Default,,0000,0000,0000,,You can prove it in two ways.
Dialogue: 0,0:44:00.44,0:44:06.68,Default,,0000,0000,0000,,Prove in two differently\Nways that it is conservative.
Dialogue: 0,0:44:06.68,0:44:13.21,Default,,0000,0000,0000,,
Dialogue: 0,0:44:13.21,0:44:16.72,Default,,0000,0000,0000,,a, find the criteria.
Dialogue: 0,0:44:16.72,0:44:20.45,Default,,0000,0000,0000,,What does this criteria say?
Dialogue: 0,0:44:20.45,0:44:25.43,Default,,0000,0000,0000,,Take your first component,\Nprime it with respect to y.
Dialogue: 0,0:44:25.43,0:44:27.78,Default,,0000,0000,0000,,So y prime with respect to y.
Dialogue: 0,0:44:27.78,0:44:32.87,Default,,0000,0000,0000,,Take your second component,\Nx, prime it with respect to x.
Dialogue: 0,0:44:32.87,0:44:34.75,Default,,0000,0000,0000,,Is this true?
Dialogue: 0,0:44:34.75,0:44:37.46,Default,,0000,0000,0000,,Yes, and this is me,\Nhappy that it's true.
Dialogue: 0,0:44:37.46,0:44:40.74,Default,,0000,0000,0000,,So this is 1 equals\N1, so it's true.
Dialogue: 0,0:44:40.74,0:44:44.78,Default,,0000,0000,0000,,So it must be conservative,\Nso it must be conservative.
Dialogue: 0,0:44:44.78,0:44:46.78,Default,,0000,0000,0000,,Could I have done\Nit another way?
Dialogue: 0,0:44:46.78,0:44:49.61,Default,,0000,0000,0000,,
Dialogue: 0,0:44:49.61,0:44:57.53,Default,,0000,0000,0000,,By definition, by definition,\Nto prove that a force field
Dialogue: 0,0:44:57.53,0:45:01.64,Default,,0000,0000,0000,,is conservative by\Ndefinition, that it a matter
Dialogue: 0,0:45:01.64,0:45:04.08,Default,,0000,0000,0000,,of the smart people.
Dialogue: 0,0:45:04.08,0:45:07.10,Default,,0000,0000,0000,,There are people\Nwho- unlike me when
Dialogue: 0,0:45:07.10,0:45:11.67,Default,,0000,0000,0000,,I was 18- are able to see\Nthe scalar potential in just
Dialogue: 0,0:45:11.67,0:45:13.85,Default,,0000,0000,0000,,about any problem I give them.
Dialogue: 0,0:45:13.85,0:45:18.07,Default,,0000,0000,0000,,I'm not going to make this\Nexperiment with a bunch of you
Dialogue: 0,0:45:18.07,0:45:22.32,Default,,0000,0000,0000,,and I'm going to reward you\Nfor the correct answers.
Dialogue: 0,0:45:22.32,0:45:25.43,Default,,0000,0000,0000,,But, could anybody\Nsee the existence
Dialogue: 0,0:45:25.43,0:45:27.71,Default,,0000,0000,0000,,of the scalar potential?
Dialogue: 0,0:45:27.71,0:45:30.75,Default,,0000,0000,0000,,So these there, this exists.
Dialogue: 0,0:45:30.75,0:45:37.59,Default,,0000,0000,0000,,That's there exists a little\Nf scalar potential such
Dialogue: 0,0:45:37.59,0:45:43.08,Default,,0000,0000,0000,,that nabla f equals F.
Dialogue: 0,0:45:43.08,0:45:47.68,Default,,0000,0000,0000,,And some of you may see\Nit and say, I see it.
Dialogue: 0,0:45:47.68,0:45:50.07,Default,,0000,0000,0000,,So, can you see\Na little function
Dialogue: 0,0:45:50.07,0:45:54.66,Default,,0000,0000,0000,,f scalar function so\Nthat f sub of x i is y
Dialogue: 0,0:45:54.66,0:46:00.81,Default,,0000,0000,0000,,and f sub- Magdelena--\Nx of y j is this x j?
Dialogue: 0,0:46:00.81,0:46:02.22,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]?
Dialogue: 0,0:46:02.22,0:46:07.33,Default,,0000,0000,0000,,PROFESSOR: No, you need to\Ndrink some coffee first.
Dialogue: 0,0:46:07.33,0:46:12.99,Default,,0000,0000,0000,,You can get this, x times what?
Dialogue: 0,0:46:12.99,0:46:14.16,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:46:14.16,0:46:17.27,Default,,0000,0000,0000,,I'll teach you how to get it.
Dialogue: 0,0:46:17.27,0:46:18.88,Default,,0000,0000,0000,,Nevertheless, there\Nare some people
Dialogue: 0,0:46:18.88,0:46:21.14,Default,,0000,0000,0000,,who can do it with\Ntheir naked eye
Dialogue: 0,0:46:21.14,0:46:26.07,Default,,0000,0000,0000,,because they have a little\Ncomputer in their head.
Dialogue: 0,0:46:26.07,0:46:28.71,Default,,0000,0000,0000,,But how did I do it?
Dialogue: 0,0:46:28.71,0:46:30.97,Default,,0000,0000,0000,,It's just a matter of\Nexperience, I said,
Dialogue: 0,0:46:30.97,0:46:36.69,Default,,0000,0000,0000,,if I take f to be x y,\NI sort of guessed it.
Dialogue: 0,0:46:36.69,0:46:42.00,Default,,0000,0000,0000,,f sub x would be y and f sub y\Nwill be x so this should be it,
Dialogue: 0,0:46:42.00,0:46:44.44,Default,,0000,0000,0000,,and this is going to do.
Dialogue: 0,0:46:44.44,0:46:49.04,Default,,0000,0000,0000,,And f is a nice function,\Npolynomial in two variables,
Dialogue: 0,0:46:49.04,0:46:50.42,Default,,0000,0000,0000,,it's a smooth function.
Dialogue: 0,0:46:50.42,0:46:55.20,Default,,0000,0000,0000,,I'm very happy I'm\Nover the domain,
Dialogue: 0,0:46:55.20,0:46:57.77,Default,,0000,0000,0000,,open this or whatever,\Nopen domain in plane.
Dialogue: 0,0:46:57.77,0:47:00.16,Default,,0000,0000,0000,,I'm very happy, I have\Nno problem with it.
Dialogue: 0,0:47:00.16,0:47:04.76,Default,,0000,0000,0000,,So I can know that this is\Nconservative in two ways.
Dialogue: 0,0:47:04.76,0:47:07.75,Default,,0000,0000,0000,,Either I get to the\Nsource of the problem
Dialogue: 0,0:47:07.75,0:47:10.38,Default,,0000,0000,0000,,and I find the little\Nscalar potential whose
Dialogue: 0,0:47:10.38,0:47:13.48,Default,,0000,0000,0000,,gradient is my force field.
Dialogue: 0,0:47:13.48,0:47:16.63,Default,,0000,0000,0000,,Or I can verify the\Ncriterion and I say,
Dialogue: 0,0:47:16.63,0:47:19.10,Default,,0000,0000,0000,,the derivative of\Nthis with respect to y
Dialogue: 0,0:47:19.10,0:47:20.97,Default,,0000,0000,0000,,is the derivative\Nof this with respect
Dialogue: 0,0:47:20.97,0:47:23.14,Default,,0000,0000,0000,,to x is-- one is the same.
Dialogue: 0,0:47:23.14,0:47:29.14,Default,,0000,0000,0000,,The same thing, you're going\Nto see it again in math 3350.
Dialogue: 0,0:47:29.14,0:47:30.71,Default,,0000,0000,0000,,All right, that I\Ntaught many times,
Dialogue: 0,0:47:30.71,0:47:33.02,Default,,0000,0000,0000,,I'm not going to teach\Nthat in the fall.
Dialogue: 0,0:47:33.02,0:47:34.82,Default,,0000,0000,0000,,But I know of some\Nvery good people
Dialogue: 0,0:47:34.82,0:47:36.06,Default,,0000,0000,0000,,who teach that in the fall.
Dialogue: 0,0:47:36.06,0:47:38.67,Default,,0000,0000,0000,,In any case, they\Nwould reteach it to you
Dialogue: 0,0:47:38.67,0:47:42.06,Default,,0000,0000,0000,,because good teachers don't\Nassume that you know much.
Dialogue: 0,0:47:42.06,0:47:45.97,Default,,0000,0000,0000,,But when you will see\Nit you'll remember me.
Dialogue: 0,0:47:45.97,0:47:49.97,Default,,0000,0000,0000,,Hopefully fondly, not cursing\Nme or anything, right?
Dialogue: 0,0:47:49.97,0:47:54.89,Default,,0000,0000,0000,,OK, how do we actually\Nget to compute f by hand
Dialogue: 0,0:47:54.89,0:47:59.15,Default,,0000,0000,0000,,if we're not experienced\Nenough to guess it like I was
Dialogue: 0,0:47:59.15,0:48:01.74,Default,,0000,0000,0000,,experienced enough to guess?
Dialogue: 0,0:48:01.74,0:48:08.24,Default,,0000,0000,0000,,So let me show you how you solve\Na system of two differential
Dialogue: 0,0:48:08.24,0:48:11.29,Default,,0000,0000,0000,,equations like that.
Dialogue: 0,0:48:11.29,0:48:15.62,Default,,0000,0000,0000,,So how I got-- how\Nyou are supposed
Dialogue: 0,0:48:15.62,0:48:23.01,Default,,0000,0000,0000,,to get the scalar potential.
Dialogue: 0,0:48:23.01,0:48:28.79,Default,,0000,0000,0000,,f sub x equals F1,\Nf sub y equals F2.
Dialogue: 0,0:48:28.79,0:48:35.36,Default,,0000,0000,0000,,So by integration, 1 and 2.
Dialogue: 0,0:48:35.36,0:48:38.74,Default,,0000,0000,0000,,
Dialogue: 0,0:48:38.74,0:48:40.91,Default,,0000,0000,0000,,And you say, what\Nyou mean 1 and 2?
Dialogue: 0,0:48:40.91,0:48:42.67,Default,,0000,0000,0000,,I'll show you in a second.
Dialogue: 0,0:48:42.67,0:48:47.82,Default,,0000,0000,0000,,So for my case,\Nexample 2, I'll take
Dialogue: 0,0:48:47.82,0:48:50.57,Default,,0000,0000,0000,,my f sub x must be y, right?
Dialogue: 0,0:48:50.57,0:48:51.07,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:48:51.07,0:48:54.02,Default,,0000,0000,0000,,My f sub y must be x, right?
Dialogue: 0,0:48:54.02,0:48:55.27,Default,,0000,0000,0000,,Right.
Dialogue: 0,0:48:55.27,0:48:58.16,Default,,0000,0000,0000,,Who is f?
Dialogue: 0,0:48:58.16,0:49:00.69,Default,,0000,0000,0000,,Solve this property.
Dialogue: 0,0:49:00.69,0:49:05.58,Default,,0000,0000,0000,,Oh, I have to start\Nintegrating from the first guy.
Dialogue: 0,0:49:05.58,0:49:08.39,Default,,0000,0000,0000,,What kind of information\Nam I going to squeeze?
Dialogue: 0,0:49:08.39,0:49:11.57,Default,,0000,0000,0000,,I'm going to say I\Nhave to go backwards,
Dialogue: 0,0:49:11.57,0:49:17.58,Default,,0000,0000,0000,,I have to get-- f\Nis going to be what?
Dialogue: 0,0:49:17.58,0:49:23.16,Default,,0000,0000,0000,,Integral of y with respect to\Nx, say it again, Magdelina.
Dialogue: 0,0:49:23.16,0:49:27.44,Default,,0000,0000,0000,,Integral of y with respect\Nto x, but attention,
Dialogue: 0,0:49:27.44,0:49:31.62,Default,,0000,0000,0000,,this may come because, for\Nme, the variable is x here,
Dialogue: 0,0:49:31.62,0:49:35.48,Default,,0000,0000,0000,,and y is like, you cannot\Nstay in this picture.
Dialogue: 0,0:49:35.48,0:49:40.59,Default,,0000,0000,0000,,So I have a constant\Nc that depends on y.
Dialogue: 0,0:49:40.59,0:49:41.51,Default,,0000,0000,0000,,Say what?
Dialogue: 0,0:49:41.51,0:49:46.29,Default,,0000,0000,0000,,Yes, because if you go backwards\Nand prime this with respect
Dialogue: 0,0:49:46.29,0:49:49.66,Default,,0000,0000,0000,,to x, what do you\Nget? f sub x will
Dialogue: 0,0:49:49.66,0:49:51.84,Default,,0000,0000,0000,,be y because this is\Nthe anti-derivative.
Dialogue: 0,0:49:51.84,0:49:55.46,Default,,0000,0000,0000,,Plus this prime with\Nrespect to x, zero.
Dialogue: 0,0:49:55.46,0:49:59.52,Default,,0000,0000,0000,,So this c of y may, a\Nlittle bit, ruin your plans.
Dialogue: 0,0:49:59.52,0:50:02.63,Default,,0000,0000,0000,,I've had students\Nwho forgot about it
Dialogue: 0,0:50:02.63,0:50:04.95,Default,,0000,0000,0000,,and then they got in trouble\Nbecause they couldn't get
Dialogue: 0,0:50:04.95,0:50:08.13,Default,,0000,0000,0000,,the scalar potential correctly.
Dialogue: 0,0:50:08.13,0:50:08.63,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:50:08.63,0:50:14.34,Default,,0000,0000,0000,,OK, so from this one you\Nsay, OK I have some--
Dialogue: 0,0:50:14.34,0:50:16.98,Default,,0000,0000,0000,,what is the integral of y dx?
Dialogue: 0,0:50:16.98,0:50:22.25,Default,,0000,0000,0000,,xy, plus some guy c\Nconstant that depends on y.
Dialogue: 0,0:50:22.25,0:50:25.61,Default,,0000,0000,0000,,From this fellow\NI go, but I have
Dialogue: 0,0:50:25.61,0:50:29.52,Default,,0000,0000,0000,,to verify the second condition,\Nif I don't I'm dead meat.
Dialogue: 0,0:50:29.52,0:50:32.32,Default,,0000,0000,0000,,There are two coupled\Nequations, these
Dialogue: 0,0:50:32.32,0:50:33.98,Default,,0000,0000,0000,,are coupled equations\Nthat have to be
Dialogue: 0,0:50:33.98,0:50:36.32,Default,,0000,0000,0000,,verified at the same time.
Dialogue: 0,0:50:36.32,0:50:45.12,Default,,0000,0000,0000,,So f sub y will be prime with\Nrespect to y. x plus prime
Dialogue: 0,0:50:45.12,0:50:52.75,Default,,0000,0000,0000,,with respect to y. c prime\Nof y, God gave me x here.
Dialogue: 0,0:50:52.75,0:50:56.76,Default,,0000,0000,0000,,So I'm really lucky in that\Nsense that c prime of y
Dialogue: 0,0:50:56.76,0:51:01.30,Default,,0000,0000,0000,,will be 0 because I have\Nan x here and an x here.
Dialogue: 0,0:51:01.30,0:51:05.54,Default,,0000,0000,0000,,So c of y will simply be\Nany constant k. c of y
Dialogue: 0,0:51:05.54,0:51:09.49,Default,,0000,0000,0000,,is just a constant k, it's\Nnot going to depend on y,
Dialogue: 0,0:51:09.49,0:51:10.98,Default,,0000,0000,0000,,it's a constant k.
Dialogue: 0,0:51:10.98,0:51:14.51,Default,,0000,0000,0000,,So my answer was not correct.
Dialogue: 0,0:51:14.51,0:51:21.32,Default,,0000,0000,0000,,The best answer would have\Nbeen f of xy must be xy plus k.
Dialogue: 0,0:51:21.32,0:51:26.65,Default,,0000,0000,0000,,But any function like xy will\Nwork, I just need one to work.
Dialogue: 0,0:51:26.65,0:51:29.33,Default,,0000,0000,0000,,I just need a scalar\Npotential, not all of them.
Dialogue: 0,0:51:29.33,0:51:33.85,Default,,0000,0000,0000,,This will work, x2, xy plus 7\Nwill work, xy plus 3 will work,
Dialogue: 0,0:51:33.85,0:51:38.86,Default,,0000,0000,0000,,xy minus 1,033,045 will work.
Dialogue: 0,0:51:38.86,0:51:43.83,Default,,0000,0000,0000,,But I only need one\Nso I'll take xy.
Dialogue: 0,0:51:43.83,0:51:46.22,Default,,0000,0000,0000,,Now that I trained\Nyour mind a little bit,
Dialogue: 0,0:51:46.22,0:51:48.37,Default,,0000,0000,0000,,maybe you don't need\Nto actually solve
Dialogue: 0,0:51:48.37,0:51:53.72,Default,,0000,0000,0000,,the system because your\Nbrain wasn't ready before.
Dialogue: 0,0:51:53.72,0:51:57.30,Default,,0000,0000,0000,,But you'd be amazed, we\Nare very trainable people.
Dialogue: 0,0:51:57.30,0:52:03.55,Default,,0000,0000,0000,,And in the process of doing\Nsomething completely new,
Dialogue: 0,0:52:03.55,0:52:05.40,Default,,0000,0000,0000,,we are learning.
Dialogue: 0,0:52:05.40,0:52:10.30,Default,,0000,0000,0000,,And your brain next,\Nwill say, I think
Dialogue: 0,0:52:10.30,0:52:15.33,Default,,0000,0000,0000,,I know how to function\Na little bit backwards.
Dialogue: 0,0:52:15.33,0:52:20.46,Default,,0000,0000,0000,,And try to integrate and\Nsee and guess a potential
Dialogue: 0,0:52:20.46,0:52:24.08,Default,,0000,0000,0000,,because it's not so hard.
Dialogue: 0,0:52:24.08,0:52:25.58,Default,,0000,0000,0000,,So let me give you example 3.
Dialogue: 0,0:52:25.58,0:52:29.65,Default,,0000,0000,0000,,
Dialogue: 0,0:52:29.65,0:52:36.24,Default,,0000,0000,0000,,Somebody give you over a\Ndomain in plane x i plus y j,
Dialogue: 0,0:52:36.24,0:52:40.60,Default,,0000,0000,0000,,and says, over D, simply\Nconnected domain in plane,
Dialogue: 0,0:52:40.60,0:52:44.00,Default,,0000,0000,0000,,open, doesn't matter.
Dialogue: 0,0:52:44.00,0:52:45.46,Default,,0000,0000,0000,,Is this conservative?
Dialogue: 0,0:52:45.46,0:52:48.37,Default,,0000,0000,0000,,
Dialogue: 0,0:52:48.37,0:52:51.88,Default,,0000,0000,0000,,Find a scalar potential.
Dialogue: 0,0:52:51.88,0:53:00.71,Default,,0000,0000,0000,,
Dialogue: 0,0:53:00.71,0:53:03.96,Default,,0000,0000,0000,,This is again, we\Ndo section 13-2,
Dialogue: 0,0:53:03.96,0:53:07.65,Default,,0000,0000,0000,,so today we did 13-1\Nand 13-2 jointly.
Dialogue: 0,0:53:07.65,0:53:10.63,Default,,0000,0000,0000,,
Dialogue: 0,0:53:10.63,0:53:11.93,Default,,0000,0000,0000,,Find the scalar potential.
Dialogue: 0,0:53:11.93,0:53:14.99,Default,,0000,0000,0000,,Do you see it now?
Dialogue: 0,0:53:14.99,0:53:16.39,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]?
Dialogue: 0,0:53:16.39,0:53:19.56,Default,,0000,0000,0000,,PROFESSOR: Excellent,\Nwe teach now, got it.
Dialogue: 0,0:53:19.56,0:53:28.45,Default,,0000,0000,0000,,He says, I know where\Nthis comes from.
Dialogue: 0,0:53:28.45,0:53:31.65,Default,,0000,0000,0000,,I've got it, x squared\Nplus y squared over 2.
Dialogue: 0,0:53:31.65,0:53:32.85,Default,,0000,0000,0000,,How did he do it?
Dialogue: 0,0:53:32.85,0:53:33.90,Default,,0000,0000,0000,,He's a genius.
Dialogue: 0,0:53:33.90,0:53:39.56,Default,,0000,0000,0000,,No he's not, he's just\Nlearning from the first time
Dialogue: 0,0:53:39.56,0:53:40.76,Default,,0000,0000,0000,,when he failed.
Dialogue: 0,0:53:40.76,0:53:44.18,Default,,0000,0000,0000,,And now he knows what he has\Nto do and his brain says,
Dialogue: 0,0:53:44.18,0:53:47.23,Default,,0000,0000,0000,,oh, I got it.
Dialogue: 0,0:53:47.23,0:53:50.11,Default,,0000,0000,0000,,Now, [INAUDIBLE] could\Nhave applied this method
Dialogue: 0,0:53:50.11,0:53:54.64,Default,,0000,0000,0000,,and solved the coupled\Nsystem and do it slowly
Dialogue: 0,0:53:54.64,0:53:56.82,Default,,0000,0000,0000,,and it would have taken\Nhim another 10 minutes.
Dialogue: 0,0:53:56.82,0:54:00.10,Default,,0000,0000,0000,,And he's in the final, he\Ndoesn't have time to spare.
Dialogue: 0,0:54:00.10,0:54:06.03,Default,,0000,0000,0000,,If he can guess the potential\Nand then verify that,
Dialogue: 0,0:54:06.03,0:54:07.72,Default,,0000,0000,0000,,it's going to be easy for him.
Dialogue: 0,0:54:07.72,0:54:08.32,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:54:08.32,0:54:15.49,Default,,0000,0000,0000,,This is going to be 2x over 2\Nx i, and this is 2y over 2 y j.
Dialogue: 0,0:54:15.49,0:54:17.84,Default,,0000,0000,0000,,So yeah, he was right.
Dialogue: 0,0:54:17.84,0:54:21.81,Default,,0000,0000,0000,,
Dialogue: 0,0:54:21.81,0:54:26.63,Default,,0000,0000,0000,,All right, let me\Ngive you another one.
Dialogue: 0,0:54:26.63,0:54:29.10,Default,,0000,0000,0000,,Let's see who gets this one.
Dialogue: 0,0:54:29.10,0:54:34.08,Default,,0000,0000,0000,,F is a vector valued\Nfunction, maybe a force field,
Dialogue: 0,0:54:34.08,0:54:35.34,Default,,0000,0000,0000,,that is this.
Dialogue: 0,0:54:35.34,0:54:38.05,Default,,0000,0000,0000,,
Dialogue: 0,0:54:38.05,0:54:41.64,Default,,0000,0000,0000,,Of course there are many\Nways-- maybe somebody's
Dialogue: 0,0:54:41.64,0:54:44.18,Default,,0000,0000,0000,,going to ask you\Nto prove this is
Dialogue: 0,0:54:44.18,0:54:49.32,Default,,0000,0000,0000,,conservative by the criterion,\Nbut they shouldn't tell you
Dialogue: 0,0:54:49.32,0:54:51.55,Default,,0000,0000,0000,,how to do it.
Dialogue: 0,0:54:51.55,0:54:53.05,Default,,0000,0000,0000,,So show this is conservative.
Dialogue: 0,0:54:53.05,0:54:56.55,Default,,0000,0000,0000,,If somebody doesn't want the\Nscalar potential because they
Dialogue: 0,0:54:56.55,0:54:57.89,Default,,0000,0000,0000,,don't need it, let's say.
Dialogue: 0,0:54:57.89,0:55:01.94,Default,,0000,0000,0000,,Well, prime f1 with respect to\Ny, I'll prime this with respect
Dialogue: 0,0:55:01.94,0:55:03.40,Default,,0000,0000,0000,,to x.
Dialogue: 0,0:55:03.40,0:55:07.56,Default,,0000,0000,0000,,f1 prime with respect to\Ny equals 2x is the same
Dialogue: 0,0:55:07.56,0:55:09.72,Default,,0000,0000,0000,,as f2 prime with respect with.
Dialogue: 0,0:55:09.72,0:55:13.60,Default,,0000,0000,0000,,Yeah, it is conservative, I\Nknow it from the criterion.
Dialogue: 0,0:55:13.60,0:55:16.57,Default,,0000,0000,0000,,But [INAUDIBLE] knows\Nthat later I will ask him
Dialogue: 0,0:55:16.57,0:55:19.15,Default,,0000,0000,0000,,for the scalar potential.
Dialogue: 0,0:55:19.15,0:55:25.61,Default,,0000,0000,0000,,And I wonder if he can find\Nit for me without computing it
Dialogue: 0,0:55:25.61,0:55:27.03,Default,,0000,0000,0000,,by solving the system.
Dialogue: 0,0:55:27.03,0:55:30.26,Default,,0000,0000,0000,,Just from his\Nmathematical intuition
Dialogue: 0,0:55:30.26,0:55:33.04,Default,,0000,0000,0000,,that is running in the\Nbackground of your--
Dialogue: 0,0:55:33.04,0:55:35.88,Default,,0000,0000,0000,,STUDENT: x squared,\Nmultiply y [INAUDIBLE].
Dialogue: 0,0:55:35.88,0:55:38.40,Default,,0000,0000,0000,,PROFESSOR: x squared\Ny, excellent.
Dialogue: 0,0:55:38.40,0:55:42.77,Default,,0000,0000,0000,,
Dialogue: 0,0:55:42.77,0:55:47.14,Default,,0000,0000,0000,,Zach came up with\Nit and anybody else?
Dialogue: 0,0:55:47.14,0:55:49.09,Default,,0000,0000,0000,,Alex?
Dialogue: 0,0:55:49.09,0:55:51.61,Default,,0000,0000,0000,,So all three of you, OK?
Dialogue: 0,0:55:51.61,0:55:54.06,Default,,0000,0000,0000,,Squared y, very good.
Dialogue: 0,0:55:54.06,0:55:56.03,Default,,0000,0000,0000,,Was it hard?
Dialogue: 0,0:55:56.03,0:55:59.48,Default,,0000,0000,0000,,Yeah, it's hard for most people.
Dialogue: 0,0:55:59.48,0:56:02.72,Default,,0000,0000,0000,,It was hard for me\Nwhen I first saw
Dialogue: 0,0:56:02.72,0:56:05.53,Default,,0000,0000,0000,,that in the first 30\Nminutes of becoming familiar
Dialogue: 0,0:56:05.53,0:56:09.28,Default,,0000,0000,0000,,with the scalar\Npotential, I was 18 or 19.
Dialogue: 0,0:56:09.28,0:56:11.93,Default,,0000,0000,0000,,But then I got it in\Nabout half an hour
Dialogue: 0,0:56:11.93,0:56:15.54,Default,,0000,0000,0000,,and I was able to\Ndo them mentally.
Dialogue: 0,0:56:15.54,0:56:20.43,Default,,0000,0000,0000,,Most of the examples I\Ngot were really nice.
Dialogue: 0,0:56:20.43,0:56:28.73,Default,,0000,0000,0000,,Were on purpose made nice for\Nus for the exam to work fast.
Dialogue: 0,0:56:28.73,0:56:35.79,Default,,0000,0000,0000,,And now let's see why\Nwould the work really not
Dialogue: 0,0:56:35.79,0:56:38.62,Default,,0000,0000,0000,,depend on the trajectory\Nyou are taking
Dialogue: 0,0:56:38.62,0:56:41.07,Default,,0000,0000,0000,,if your force is conservative.
Dialogue: 0,0:56:41.07,0:56:42.96,Default,,0000,0000,0000,,If the force is\Nconservative there
Dialogue: 0,0:56:42.96,0:56:46.44,Default,,0000,0000,0000,,is something magic\Nthat's going to happen.
Dialogue: 0,0:56:46.44,0:56:50.60,Default,,0000,0000,0000,,And we really don't\Nknow what that is,
Dialogue: 0,0:56:50.60,0:56:53.16,Default,,0000,0000,0000,,but we should be able to prove.
Dialogue: 0,0:56:53.16,0:56:57.60,Default,,0000,0000,0000,,
Dialogue: 0,0:56:57.60,0:57:19.89,Default,,0000,0000,0000,,So theorem, actually\Nthis is funny.
Dialogue: 0,0:57:19.89,0:57:25.51,Default,,0000,0000,0000,,It's called the fundamental\Ntheorem of path integrals
Dialogue: 0,0:57:25.51,0:57:28.46,Default,,0000,0000,0000,,but it's the fundamental\Ntheorem of calculus 3.
Dialogue: 0,0:57:28.46,0:57:36.31,Default,,0000,0000,0000,,I'm going to write it like\Nthis, the fundamental theorem
Dialogue: 0,0:57:36.31,0:57:45.17,Default,,0000,0000,0000,,of calc 3, path integrals.
Dialogue: 0,0:57:45.17,0:57:54.71,Default,,0000,0000,0000,,It's also called- 13.3,\Nsection- Independence of path.
Dialogue: 0,0:57:54.71,0:58:01.98,Default,,0000,0000,0000,,
Dialogue: 0,0:58:01.98,0:58:11.73,Default,,0000,0000,0000,,So remember you have a\Nwork, w, over a path, c.
Dialogue: 0,0:58:11.73,0:58:22.61,Default,,0000,0000,0000,,F dot dR where there R is the\Nregular parametrized curve
Dialogue: 0,0:58:22.61,0:58:23.11,Default,,0000,0000,0000,,overseen.
Dialogue: 0,0:58:23.11,0:58:33.85,Default,,0000,0000,0000,,
Dialogue: 0,0:58:33.85,0:58:36.85,Default,,0000,0000,0000,,This is called a\Nsupposition vector.
Dialogue: 0,0:58:36.85,0:58:40.85,Default,,0000,0000,0000,,
Dialogue: 0,0:58:40.85,0:58:45.73,Default,,0000,0000,0000,,Regular meaning c1, and\Nnever vanishing speed,
Dialogue: 0,0:58:45.73,0:58:49.40,Default,,0000,0000,0000,,the velocity never vanishes.
Dialogue: 0,0:58:49.40,0:58:55.02,Default,,0000,0000,0000,,Velocity times 0 such\Nthat f is continuous,
Dialogue: 0,0:58:55.02,0:59:01.58,Default,,0000,0000,0000,,or a nice enough integral.
Dialogue: 0,0:59:01.58,0:59:07.95,Default,,0000,0000,0000,,
Dialogue: 0,0:59:07.95,0:59:25.92,Default,,0000,0000,0000,,If F is conservative of\Nscalar potential, little f,
Dialogue: 0,0:59:25.92,0:59:39.43,Default,,0000,0000,0000,,then the work, w, equals\Nlittle f at the endpoint
Dialogue: 0,0:59:39.43,0:59:41.56,Default,,0000,0000,0000,,minus little f at the origin.
Dialogue: 0,0:59:41.56,0:59:44.73,Default,,0000,0000,0000,,
Dialogue: 0,0:59:44.73,0:59:56.68,Default,,0000,0000,0000,,Where, by origin and endpoint\Nare those for the path,
Dialogue: 0,0:59:56.68,1:00:00.62,Default,,0000,0000,0000,,are those for the arc, are\Nthose for the curve, c.
Dialogue: 0,1:00:00.62,1:00:05.06,Default,,0000,0000,0000,,
Dialogue: 0,1:00:05.06,1:00:09.58,Default,,0000,0000,0000,,So the work, the w, will\Nbe independent of time.
Dialogue: 0,1:00:09.58,1:00:18.72,Default,,0000,0000,0000,,So w will be independent of f.
Dialogue: 0,1:00:18.72,1:00:21.12,Default,,0000,0000,0000,,And you saw an\Nexample when I took
Dialogue: 0,1:00:21.12,1:00:24.26,Default,,0000,0000,0000,,a conservative function\Nthat was really nice,
Dialogue: 0,1:00:24.26,1:00:27.62,Default,,0000,0000,0000,,y times i plus x times j.
Dialogue: 0,1:00:27.62,1:00:29.44,Default,,0000,0000,0000,,That was the force field.
Dialogue: 0,1:00:29.44,1:00:33.47,Default,,0000,0000,0000,,Because that was\Nconservative, we got w being 1
Dialogue: 0,1:00:33.47,1:00:34.78,Default,,0000,0000,0000,,no matter what path we took.
Dialogue: 0,1:00:34.78,1:00:37.97,Default,,0000,0000,0000,,We took a parabola, we\Ntook a straight line,
Dialogue: 0,1:00:37.97,1:00:39.77,Default,,0000,0000,0000,,and we could have\Ntaken a zig-zag
Dialogue: 0,1:00:39.77,1:00:41.86,Default,,0000,0000,0000,,and we still get w equals 1.
Dialogue: 0,1:00:41.86,1:00:44.55,Default,,0000,0000,0000,,So no matter what\Npath you are taking.
Dialogue: 0,1:00:44.55,1:00:46.57,Default,,0000,0000,0000,,Can we prove this?
Dialogue: 0,1:00:46.57,1:00:53.00,Default,,0000,0000,0000,,Well, regular classes don't\Nprove anything, almost nothing.
Dialogue: 0,1:00:53.00,1:00:56.90,Default,,0000,0000,0000,,But we are honor students\Nso lets see what we can do.
Dialogue: 0,1:00:56.90,1:00:59.21,Default,,0000,0000,0000,,We have to understand\Nwhat's going on.
Dialogue: 0,1:00:59.21,1:01:06.52,Default,,0000,0000,0000,,Why do we have this fundamental\Ntheorem of calculus 3?
Dialogue: 0,1:01:06.52,1:01:15.61,Default,,0000,0000,0000,,The work, w, can be expressed--\Nassume f is conservative
Dialogue: 0,1:01:15.61,1:01:19.70,Default,,0000,0000,0000,,which means it's going to come\Nfrom a potential little f.
Dialogue: 0,1:01:19.70,1:01:29.08,Default,,0000,0000,0000,,Where f is [INAUDIBLE] scalar\Nfunction over my domain, omega.
Dialogue: 0,1:01:29.08,1:01:33.28,Default,,0000,0000,0000,,
Dialogue: 0,1:01:33.28,1:01:36.67,Default,,0000,0000,0000,,Now, the curve, c,\Nis part of this omega
Dialogue: 0,1:01:36.67,1:01:40.49,Default,,0000,0000,0000,,so I don't have any\Nproblems on the curve.
Dialogue: 0,1:01:40.49,1:01:44.68,Default,,0000,0000,0000,,w will be rewritten beautifully.
Dialogue: 0,1:01:44.68,1:01:46.62,Default,,0000,0000,0000,,So I'm giving you a\Nsketch of a proof.
Dialogue: 0,1:01:46.62,1:01:50.34,Default,,0000,0000,0000,,But you would be able to do\Nthis, maybe even better than me
Dialogue: 0,1:01:50.34,1:01:56.31,Default,,0000,0000,0000,,because I have taught\Nyou what you need to do.
Dialogue: 0,1:01:56.31,1:02:03.58,Default,,0000,0000,0000,,So this is going to\Nbe f1 i, plus f2 j.
Dialogue: 0,1:02:03.58,1:02:09.75,Default,,0000,0000,0000,,And I'm going to write it.\Nf1 times-- what is this guys?
Dialogue: 0,1:02:09.75,1:02:14.42,Default,,0000,0000,0000,,dR, I taught you, you taught\Nme, x prime of t, right?
Dialogue: 0,1:02:14.42,1:02:23.01,Default,,0000,0000,0000,,Plus f2 times y prime of t,\Nall dt, and time from t0 to t1.
Dialogue: 0,1:02:23.01,1:02:27.48,Default,,0000,0000,0000,,I start my motion along\Nthe curve at t equals t0
Dialogue: 0,1:02:27.48,1:02:30.73,Default,,0000,0000,0000,,and I finished my\Nmotion at t equals t1.
Dialogue: 0,1:02:30.73,1:02:33.62,Default,,0000,0000,0000,,
Dialogue: 0,1:02:33.62,1:02:36.77,Default,,0000,0000,0000,,Do I know where f1 and f2 are?
Dialogue: 0,1:02:36.77,1:02:38.86,Default,,0000,0000,0000,,This is the point,\Nthat's the whole point,
Dialogue: 0,1:02:38.86,1:02:41.78,Default,,0000,0000,0000,,I know who they are, thank God.
Dialogue: 0,1:02:41.78,1:02:47.31,Default,,0000,0000,0000,,And now I have to again\Napply some magical think,
Dialogue: 0,1:02:47.31,1:02:49.65,Default,,0000,0000,0000,,I'll ask you in a\Nminute what that is.
Dialogue: 0,1:02:49.65,1:02:53.51,Default,,0000,0000,0000,,So, what is f1?
Dialogue: 0,1:02:53.51,1:02:56.10,Default,,0000,0000,0000,,df dx, or f sub base.
Dialogue: 0,1:02:56.10,1:02:59.26,Default,,0000,0000,0000,,If you don't like f sub base,\Nif you don't like my notation,
Dialogue: 0,1:02:59.26,1:03:00.60,Default,,0000,0000,0000,,you put f sub x, right?
Dialogue: 0,1:03:00.60,1:03:02.55,Default,,0000,0000,0000,,And this df dy.
Dialogue: 0,1:03:02.55,1:03:03.05,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:03:03.05,1:03:05.34,Default,,0000,0000,0000,,Because it's\Nconservative and that
Dialogue: 0,1:03:05.34,1:03:08.08,Default,,0000,0000,0000,,was the gradient of little f.
Dialogue: 0,1:03:08.08,1:03:11.06,Default,,0000,0000,0000,,Of course I'm using the fact\Nthat the first component would
Dialogue: 0,1:03:11.06,1:03:13.61,Default,,0000,0000,0000,,be the partial of little\Nf with respect to x.
Dialogue: 0,1:03:13.61,1:03:16.03,Default,,0000,0000,0000,,The second component would\Nbe the partial of little
Dialogue: 0,1:03:16.03,1:03:16.86,Default,,0000,0000,0000,,f with respect to y.
Dialogue: 0,1:03:16.86,1:03:19.27,Default,,0000,0000,0000,,Have you seen this\Nformula before?
Dialogue: 0,1:03:19.27,1:03:23.60,Default,,0000,0000,0000,,What in the world\Nis this formula?
Dialogue: 0,1:03:23.60,1:03:25.50,Default,,0000,0000,0000,,STUDENT: It's the chain rule?
Dialogue: 0,1:03:25.50,1:03:26.83,Default,,0000,0000,0000,,PROFESSOR: It's the chain rule.
Dialogue: 0,1:03:26.83,1:03:30.63,Default,,0000,0000,0000,,I don't have a dollar but I\Nwill give you a dollar, OK?
Dialogue: 0,1:03:30.63,1:03:33.96,Default,,0000,0000,0000,,Imagine a virtual dollar.
Dialogue: 0,1:03:33.96,1:03:35.19,Default,,0000,0000,0000,,This is the chain rule.
Dialogue: 0,1:03:35.19,1:03:38.11,Default,,0000,0000,0000,,So by the chain\Nrule we can write
Dialogue: 0,1:03:38.11,1:03:41.35,Default,,0000,0000,0000,,this to be the\Nderivative with respect
Dialogue: 0,1:03:41.35,1:03:48.84,Default,,0000,0000,0000,,to t of little f of\Nx of t, and y of t.
Dialogue: 0,1:03:48.84,1:03:52.63,Default,,0000,0000,0000,,Alrighty, so I know\Nwhat I'm doing.
Dialogue: 0,1:03:52.63,1:03:57.86,Default,,0000,0000,0000,,I know that by chain rule I had\Nlittle f evaluated at x of t,
Dialogue: 0,1:03:57.86,1:04:01.64,Default,,0000,0000,0000,,y of t, and time t,\Nprime with respect to t.
Dialogue: 0,1:04:01.64,1:04:07.75,Default,,0000,0000,0000,,Now when we take the fundamental\Ntheorem of calculus, FTC.
Dialogue: 0,1:04:07.75,1:04:13.54,Default,,0000,0000,0000,,That reminds me, I was\Nteaching calc 1 a few years ago
Dialogue: 0,1:04:13.54,1:04:17.39,Default,,0000,0000,0000,,and I said, that's the\NFederal Trade Commission.
Dialogue: 0,1:04:17.39,1:04:20.76,Default,,0000,0000,0000,,Federal Trade Commission,\Nfundamental theorem
Dialogue: 0,1:04:20.76,1:04:22.45,Default,,0000,0000,0000,,of calculus.
Dialogue: 0,1:04:22.45,1:04:27.99,Default,,0000,0000,0000,,So coming back to\Nwhat I have, I prove
Dialogue: 0,1:04:27.99,1:04:36.89,Default,,0000,0000,0000,,that w is the Federal\NTrade Commission, no.
Dialogue: 0,1:04:36.89,1:04:43.45,Default,,0000,0000,0000,,w is the application\Nof something
Dialogue: 0,1:04:43.45,1:04:49.72,Default,,0000,0000,0000,,that we knew from calc\N1, which is beautiful.
Dialogue: 0,1:04:49.72,1:05:00.46,Default,,0000,0000,0000,,f of xt, y of t dt, this\Nis nothing but what?
Dialogue: 0,1:05:00.46,1:05:05.53,Default,,0000,0000,0000,,Little f evaluated at-- I'm\Ngoing to have to write it down,
Dialogue: 0,1:05:05.53,1:05:07.45,Default,,0000,0000,0000,,this whole sausage.
Dialogue: 0,1:05:07.45,1:05:13.70,Default,,0000,0000,0000,,f of x of t1, y of t1,\Nminus f of x of t0, y of t0.
Dialogue: 0,1:05:13.70,1:05:17.24,Default,,0000,0000,0000,,For somebody as lazy as\NI am, that they effort.
Dialogue: 0,1:05:17.24,1:05:19.49,Default,,0000,0000,0000,,How can I write It better?
Dialogue: 0,1:05:19.49,1:05:26.06,Default,,0000,0000,0000,,f at the endpoint\Nminus f at the origin.
Dialogue: 0,1:05:26.06,1:05:29.45,Default,,0000,0000,0000,,And of course, we are trying to\Nbe quite rigorous in the book.
Dialogue: 0,1:05:29.45,1:05:31.48,Default,,0000,0000,0000,,We would never say\Nthat in the book.
Dialogue: 0,1:05:31.48,1:05:34.65,Default,,0000,0000,0000,,We actually denote\Nthe first point
Dialogue: 0,1:05:34.65,1:05:37.69,Default,,0000,0000,0000,,with p, the origin, and\Nthe endpoint with q.
Dialogue: 0,1:05:37.69,1:05:41.81,Default,,0000,0000,0000,,So we say, f of q minus f of p.
Dialogue: 0,1:05:41.81,1:05:49.92,Default,,0000,0000,0000,,And we proved q e d, we\Nproved the fundamental theorem
Dialogue: 0,1:05:49.92,1:05:53.00,Default,,0000,0000,0000,,of path integrals, the\Nindependence of that.
Dialogue: 0,1:05:53.00,1:06:00.20,Default,,0000,0000,0000,,So that means the work\Nis independent of path
Dialogue: 0,1:06:00.20,1:06:03.00,Default,,0000,0000,0000,,when the force is conservative.
Dialogue: 0,1:06:03.00,1:06:07.36,Default,,0000,0000,0000,,Now attention, if f is not\Nconservative you are dead meat.
Dialogue: 0,1:06:07.36,1:06:12.50,Default,,0000,0000,0000,,You cannot say what I just said.
Dialogue: 0,1:06:12.50,1:06:16.39,Default,,0000,0000,0000,,So I'll give you two\Nseparate examples
Dialogue: 0,1:06:16.39,1:06:19.69,Default,,0000,0000,0000,,and let's see how we\Nsolve each of them.
Dialogue: 0,1:06:19.69,1:06:27.40,Default,,0000,0000,0000,,
Dialogue: 0,1:06:27.40,1:06:30.40,Default,,0000,0000,0000,,A final exam type of\Nproblem-- every final exam
Dialogue: 0,1:06:30.40,1:06:36.10,Default,,0000,0000,0000,,contains an\Napplication like that.
Dialogue: 0,1:06:36.10,1:06:41.94,Default,,0000,0000,0000,,Even the force field, f,\Nor the vector value, f.
Dialogue: 0,1:06:41.94,1:06:44.06,Default,,0000,0000,0000,,Is it conservative?
Dialogue: 0,1:06:44.06,1:06:51.60,Default,,0000,0000,0000,,Prove what is proved and\Nafter that-- so the path
Dialogue: 0,1:06:51.60,1:06:54.64,Default,,0000,0000,0000,,integral in any way you can.
Dialogue: 0,1:06:54.64,1:06:58.02,Default,,0000,0000,0000,,If it's conservative you're\Nreally lucky because you're
Dialogue: 0,1:06:58.02,1:06:58.52,Default,,0000,0000,0000,,in business.
Dialogue: 0,1:06:58.52,1:06:59.90,Default,,0000,0000,0000,,You don't have to do any work.
Dialogue: 0,1:06:59.90,1:07:03.58,Default,,0000,0000,0000,,You just find the little\Nscalar potential evaluated
Dialogue: 0,1:07:03.58,1:07:08.50,Default,,0000,0000,0000,,at the endpoints and subtract,\Nand that's your answer.
Dialogue: 0,1:07:08.50,1:07:13.07,Default,,0000,0000,0000,,So I'm going to give you an\Nexample of a final exam problem
Dialogue: 0,1:07:13.07,1:07:15.54,Default,,0000,0000,0000,,that happened in the past year.
Dialogue: 0,1:07:15.54,1:07:18.52,Default,,0000,0000,0000,,So, a final type exam problem.
Dialogue: 0,1:07:18.52,1:07:25.44,Default,,0000,0000,0000,,
Dialogue: 0,1:07:25.44,1:07:37.24,Default,,0000,0000,0000,,f of xy equals 2xy\Ni plus x squared j,
Dialogue: 0,1:07:37.24,1:07:43.50,Default,,0000,0000,0000,,over r over r squared.
Dialogue: 0,1:07:43.50,1:07:45.75,Default,,0000,0000,0000,,STUDENT: Didn't we just\Ndo [INAUDIBLE] that?
Dialogue: 0,1:07:45.75,1:07:49.15,Default,,0000,0000,0000,,PROFESSOR: Well, I just did\Nthat but I changed the problem.
Dialogue: 0,1:07:49.15,1:07:51.24,Default,,0000,0000,0000,,I wanted to keep the\Nsame force field.
Dialogue: 0,1:07:51.24,1:07:51.95,Default,,0000,0000,0000,,STUDENT: Alright.
Dialogue: 0,1:07:51.95,1:07:52.81,Default,,0000,0000,0000,,PROFESSOR: OK.
Dialogue: 0,1:07:52.81,1:08:18.32,Default,,0000,0000,0000,,Compute the work, w,\Nperformed by f along the arc
Dialogue: 0,1:08:18.32,1:08:23.72,Default,,0000,0000,0000,,of the circle in the picture.
Dialogue: 0,1:08:23.72,1:08:26.71,Default,,0000,0000,0000,,
Dialogue: 0,1:08:26.71,1:08:27.93,Default,,0000,0000,0000,,And they draw a picture.
Dialogue: 0,1:08:27.93,1:08:31.65,Default,,0000,0000,0000,,And they do a picture for you,\Nand you stare at this picture
Dialogue: 0,1:08:31.65,1:08:49.29,Default,,0000,0000,0000,,and-- So you say,\Noh my God, if I
Dialogue: 0,1:08:49.29,1:08:53.05,Default,,0000,0000,0000,,were to parametrize it\Nwould be a little bit of-- I
Dialogue: 0,1:08:53.05,1:08:56.26,Default,,0000,0000,0000,,could, but it would be\Na little bit of work.
Dialogue: 0,1:08:56.26,1:09:00.34,Default,,0000,0000,0000,,I would have x equals 2\Ncosine t, y equals sine t.
Dialogue: 0,1:09:00.34,1:09:05.03,Default,,0000,0000,0000,,I would have to plug in and\Ndo that whole work, definition
Dialogue: 0,1:09:05.03,1:09:06.27,Default,,0000,0000,0000,,with parametrization.
Dialogue: 0,1:09:06.27,1:09:08.21,Default,,0000,0000,0000,,Do you have to parametrize?
Dialogue: 0,1:09:08.21,1:09:10.15,Default,,0000,0000,0000,,Not in this case, why?
Dialogue: 0,1:09:10.15,1:09:12.73,Default,,0000,0000,0000,,Because the f is conservative.
Dialogue: 0,1:09:12.73,1:09:19.47,Default,,0000,0000,0000,,If they ask you- some professors\Ngive hints, most of them
Dialogue: 0,1:09:19.47,1:09:22.56,Default,,0000,0000,0000,,are nice and give hints-\Nshow f is conservative.
Dialogue: 0,1:09:22.56,1:09:28.86,Default,,0000,0000,0000,,
Dialogue: 0,1:09:28.86,1:09:30.73,Default,,0000,0000,0000,,So that's a big\Nhint in the sense
Dialogue: 0,1:09:30.73,1:09:33.10,Default,,0000,0000,0000,,that you see it\Nimmediately, how you do it.
Dialogue: 0,1:09:33.10,1:09:36.15,Default,,0000,0000,0000,,You have 2xy prime\Nwith respect to y,
Dialogue: 0,1:09:36.15,1:09:39.71,Default,,0000,0000,0000,,is 2x, which is x squared\Nprime with respect to x.
Dialogue: 0,1:09:39.71,1:09:41.70,Default,,0000,0000,0000,,So it is conservative.
Dialogue: 0,1:09:41.70,1:09:44.35,Default,,0000,0000,0000,,But he or she told you more.
Dialogue: 0,1:09:44.35,1:09:46.86,Default,,0000,0000,0000,,He said, I'm selling\Nyou something here,
Dialogue: 0,1:09:46.86,1:09:50.08,Default,,0000,0000,0000,,you have to get your\Nown scalar potential.
Dialogue: 0,1:09:50.08,1:09:56.92,Default,,0000,0000,0000,,And you did, and\Nyou got x squared y.
Dialogue: 0,1:09:56.92,1:09:58.97,Default,,0000,0000,0000,,Now, most of the\Nscalar potentials
Dialogue: 0,1:09:58.97,1:10:01.62,Default,,0000,0000,0000,,that we are giving\Nyou on the exam
Dialogue: 0,1:10:01.62,1:10:04.20,Default,,0000,0000,0000,,can be seen with naked eyes.
Dialogue: 0,1:10:04.20,1:10:06.77,Default,,0000,0000,0000,,You wouldn't have to do\Nall the integration of that
Dialogue: 0,1:10:06.77,1:10:09.54,Default,,0000,0000,0000,,coupled system with respect\Nto x, with respect to y,
Dialogue: 0,1:10:09.54,1:10:14.16,Default,,0000,0000,0000,,integrate backwards,\Nand things like that.
Dialogue: 0,1:10:14.16,1:10:16.70,Default,,0000,0000,0000,,What do I need to do\Nin that case guys?
Dialogue: 0,1:10:16.70,1:10:19.02,Default,,0000,0000,0000,,Say in words.
Dialogue: 0,1:10:19.02,1:10:22.48,Default,,0000,0000,0000,,Since the force is\Nconservative, just two lines.
Dialogue: 0,1:10:22.48,1:10:25.80,Default,,0000,0000,0000,,I'm applying the fundamental\Ntheorem of calculus.
Dialogue: 0,1:10:25.80,1:10:29.65,Default,,0000,0000,0000,,I'm applying the fundamental\Ntheorem of path integrals.
Dialogue: 0,1:10:29.65,1:10:33.05,Default,,0000,0000,0000,,I know the work is\Nindependent of that.
Dialogue: 0,1:10:33.05,1:10:38.75,Default,,0000,0000,0000,,So w, in this case\Nis already there,
Dialogue: 0,1:10:38.75,1:10:43.23,Default,,0000,0000,0000,,is going to be f at the point.
Dialogue: 0,1:10:43.23,1:10:48.01,Default,,0000,0000,0000,,I didn't say how I'm going to\Ntravel it, in what direction.
Dialogue: 0,1:10:48.01,1:10:52.03,Default,,0000,0000,0000,,f of q minus f of p.
Dialogue: 0,1:10:52.03,1:10:57.36,Default,,0000,0000,0000,,
Dialogue: 0,1:10:57.36,1:11:00.63,Default,,0000,0000,0000,,And you'll say, well\Nwhat does it mean?
Dialogue: 0,1:11:00.63,1:11:02.49,Default,,0000,0000,0000,,How do we do that?
Dialogue: 0,1:11:02.49,1:11:05.89,Default,,0000,0000,0000,,That means x squared\Ny, evaluated at q,
Dialogue: 0,1:11:05.89,1:11:09.94,Default,,0000,0000,0000,,who the heck is q?
Dialogue: 0,1:11:09.94,1:11:13.71,Default,,0000,0000,0000,,Attention, negative 2,\N0, Matt, you got it?
Dialogue: 0,1:11:13.71,1:11:14.92,Default,,0000,0000,0000,,OK, right?
Dialogue: 0,1:11:14.92,1:11:17.77,Default,,0000,0000,0000,,So, are you guys with me?
Dialogue: 0,1:11:17.77,1:11:19.55,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:11:19.55,1:11:22.52,Default,,0000,0000,0000,,And p is 2, 0.
Dialogue: 0,1:11:22.52,1:11:31.69,Default,,0000,0000,0000,,So at negative 2, 0,\Nminus x squared y at 2, 0.
Dialogue: 0,1:11:31.69,1:11:35.17,Default,,0000,0000,0000,,So, if you set 0, big, as\Nyou knew that you got 0,
Dialogue: 0,1:11:35.17,1:11:38.24,Default,,0000,0000,0000,,that is the answer.
Dialogue: 0,1:11:38.24,1:11:43.83,Default,,0000,0000,0000,,Now if somebody would give you\Na wiggly look that this guy's
Dialogue: 0,1:11:43.83,1:11:49.64,Default,,0000,0000,0000,,wrong, then he took this past.
Dialogue: 0,1:11:49.64,1:11:52.74,Default,,0000,0000,0000,,He's going to do\Nexactly the same work
Dialogue: 0,1:11:52.74,1:11:57.80,Default,,0000,0000,0000,,if he's under influence the\Nsame conservative force.
Dialogue: 0,1:11:57.80,1:12:01.25,Default,,0000,0000,0000,,If the force acting\Non it is the same.
Dialogue: 0,1:12:01.25,1:12:04.26,Default,,0000,0000,0000,,No matter what path you\Nare taking-- yes, sir?
Dialogue: 0,1:12:04.26,1:12:07.32,Default,,0000,0000,0000,,STUDENT: Is it still working\Nfor your self-intersecting.
Dialogue: 0,1:12:07.32,1:12:11.42,Default,,0000,0000,0000,,PROFESSOR: Yeah, because\Nyou're not stopping,
Dialogue: 0,1:12:11.42,1:12:13.12,Default,,0000,0000,0000,,it works for any\Nparametrization.
Dialogue: 0,1:12:13.12,1:12:16.39,Default,,0000,0000,0000,,So if you're able to parametrize\Nthat as a differentiable
Dialogue: 0,1:12:16.39,1:12:20.64,Default,,0000,0000,0000,,function so that the\Nderivative would never vanish,
Dialogue: 0,1:12:20.64,1:12:23.09,Default,,0000,0000,0000,,it's going to work, right?
Dialogue: 0,1:12:23.09,1:12:27.41,Default,,0000,0000,0000,,All right, it can work also\Nfor this piecewise contour
Dialogue: 0,1:12:27.41,1:12:29.30,Default,,0000,0000,0000,,or any other path point.
Dialogue: 0,1:12:29.30,1:12:32.72,Default,,0000,0000,0000,,As long as it starts and\Nit ends at the same point
Dialogue: 0,1:12:32.72,1:12:38.02,Default,,0000,0000,0000,,and as long as your\Nconservative force is the same.
Dialogue: 0,1:12:38.02,1:12:39.89,Default,,0000,0000,0000,,So that force is very\N[INAUDIBLE], yes?
Dialogue: 0,1:12:39.89,1:12:42.54,Default,,0000,0000,0000,,STUDENT: Do [INAUDIBLE] graphs\Nbut the endpoints the same.
Dialogue: 0,1:12:42.54,1:12:44.84,Default,,0000,0000,0000,,And if they are conservative\Nthen [INTERPOSING VOICES].
Dialogue: 0,1:12:44.84,1:12:46.51,Default,,0000,0000,0000,,PROFESSOR: If everything\Nis conservative
Dialogue: 0,1:12:46.51,1:12:49.55,Default,,0000,0000,0000,,the work along this\Npath is the same.
Dialogue: 0,1:12:49.55,1:12:52.08,Default,,0000,0000,0000,,There were people\Nwho played games
Dialogue: 0,1:12:52.08,1:12:54.25,Default,,0000,0000,0000,,like that to catch if\Nthe student knows what
Dialogue: 0,1:12:54.25,1:12:55.42,Default,,0000,0000,0000,,they are talking-- yes, sir?
Dialogue: 0,1:12:55.42,1:12:59.07,Default,,0000,0000,0000,,STUDENT: So, in a\Nproblem, if they wanted
Dialogue: 0,1:12:59.07,1:13:00.73,Default,,0000,0000,0000,,to find the work\Ncouldn't we simplify it
Dialogue: 0,1:13:00.73,1:13:02.54,Default,,0000,0000,0000,,by just saying, we\Nneed to find-- because,
Dialogue: 0,1:13:02.54,1:13:05.08,Default,,0000,0000,0000,,since we're in R2, couldn't we\Njust say it's a straight line?
Dialogue: 0,1:13:05.08,1:13:07.13,Default,,0000,0000,0000,,Because that's the--\NLike, instead of a curve
Dialogue: 0,1:13:07.13,1:13:08.66,Default,,0000,0000,0000,,we could just set the straight\Nline [INTERPOSING VOICES].
Dialogue: 0,1:13:08.66,1:13:10.26,Default,,0000,0000,0000,,PROFESSOR: If it\Nwere moving from here
Dialogue: 0,1:13:10.26,1:13:12.27,Default,,0000,0000,0000,,to here in a straight\Nline you would still
Dialogue: 0,1:13:12.27,1:13:13.94,Default,,0000,0000,0000,,get the same answer.
Dialogue: 0,1:13:13.94,1:13:17.81,Default,,0000,0000,0000,,And you could have-- if\Nyou did that, actually,
Dialogue: 0,1:13:17.81,1:13:21.17,Default,,0000,0000,0000,,if you were to compute this\Nkind of work on a straight line
Dialogue: 0,1:13:21.17,1:13:26.12,Default,,0000,0000,0000,,it has-- let me\Nshow you something.
Dialogue: 0,1:13:26.12,1:13:32.47,Default,,0000,0000,0000,,You see, when you compute\Ndx y dx, plus x squared dy.
Dialogue: 0,1:13:32.47,1:13:38.46,Default,,0000,0000,0000,,If y is 0 like Matthew said,\NI'm walking on-- I'm not drunk.
Dialogue: 0,1:13:38.46,1:13:41.35,Default,,0000,0000,0000,,I'm walking straight.
Dialogue: 0,1:13:41.35,1:13:46.81,Default,,0000,0000,0000,,y will be 0 here, and 0 here,\Nand the integral will be 0.
Dialogue: 0,1:13:46.81,1:13:50.58,Default,,0000,0000,0000,,So he would have noticed\Nthat from the beginning.
Dialogue: 0,1:13:50.58,1:13:54.24,Default,,0000,0000,0000,,But unless you know the\Nforce is conservative,
Dialogue: 0,1:13:54.24,1:13:57.89,Default,,0000,0000,0000,,there is no guarantee\Nthat on another path
Dialogue: 0,1:13:57.89,1:14:01.49,Default,,0000,0000,0000,,you don't have a\Ndifferent answer, right?
Dialogue: 0,1:14:01.49,1:14:05.23,Default,,0000,0000,0000,,So, let me give you another\Nexample because now Matthew
Dialogue: 0,1:14:05.23,1:14:06.61,Default,,0000,0000,0000,,brought this up.
Dialogue: 0,1:14:06.61,1:14:09.23,Default,,0000,0000,0000,,
Dialogue: 0,1:14:09.23,1:14:13.62,Default,,0000,0000,0000,,A catchy example\Nthat a professor
Dialogue: 0,1:14:13.62,1:14:18.47,Default,,0000,0000,0000,,gave just to make the\Nstudents life miserable,
Dialogue: 0,1:14:18.47,1:14:22.74,Default,,0000,0000,0000,,and I'll show it\Nto you in a second.
Dialogue: 0,1:14:22.74,1:14:29.06,Default,,0000,0000,0000,,
Dialogue: 0,1:14:29.06,1:14:36.86,Default,,0000,0000,0000,,He said, for the picture--\Nvery similar to this one,
Dialogue: 0,1:14:36.86,1:14:39.92,Default,,0000,0000,0000,,just to make people confused.
Dialogue: 0,1:14:39.92,1:14:43.27,Default,,0000,0000,0000,,Somebody gives you\Nthis arc of a circle
Dialogue: 0,1:14:43.27,1:14:45.66,Default,,0000,0000,0000,,and you travel from a to b.
Dialogue: 0,1:14:45.66,1:14:49.50,Default,,0000,0000,0000,,And this the thing.
Dialogue: 0,1:14:49.50,1:15:01.85,Default,,0000,0000,0000,,And he says, compute w for\Nthe force given by y i plus j.
Dialogue: 0,1:15:01.85,1:15:07.76,Default,,0000,0000,0000,,And the students\Nsaid OK, I guess
Dialogue: 0,1:15:07.76,1:15:12.44,Default,,0000,0000,0000,,I'm going to get 0 because\NI'm going to get something
Dialogue: 0,1:15:12.44,1:15:16.61,Default,,0000,0000,0000,,like y dx plus x dy.
Dialogue: 0,1:15:16.61,1:15:21.70,Default,,0000,0000,0000,,And if y is 0, I get 0, and\Nthat way you wouldn't be 0
Dialogue: 0,1:15:21.70,1:15:22.24,Default,,0000,0000,0000,,and I'm done.
Dialogue: 0,1:15:22.24,1:15:27.49,Default,,0000,0000,0000,,No, it's not how you\Nthink because this is not
Dialogue: 0,1:15:27.49,1:15:30.21,Default,,0000,0000,0000,,conservative.
Dialogue: 0,1:15:30.21,1:15:33.80,Default,,0000,0000,0000,,So you cannot say I can change\Nmy path and it's still going
Dialogue: 0,1:15:33.80,1:15:35.82,Default,,0000,0000,0000,,to be the same.
Dialogue: 0,1:15:35.82,1:15:38.26,Default,,0000,0000,0000,,No, why is this\Nnot conservative?
Dialogue: 0,1:15:38.26,1:15:39.98,Default,,0000,0000,0000,,Quickly, this prime\Nwith respect to y
Dialogue: 0,1:15:39.98,1:15:42.58,Default,,0000,0000,0000,,is 1, this prime with\Nrespect to x is 0.
Dialogue: 0,1:15:42.58,1:15:46.06,Default,,0000,0000,0000,,So 1 different from\N0 so, oh my God, no.
Dialogue: 0,1:15:46.06,1:15:48.16,Default,,0000,0000,0000,,In that case, why do we do?
Dialogue: 0,1:15:48.16,1:15:52.48,Default,,0000,0000,0000,,We have no other choice but\Nsay, x equals 2 cosine t,
Dialogue: 0,1:15:52.48,1:15:57.77,Default,,0000,0000,0000,,y equals 2 sine t, and\Nt between 0 and pi.
Dialogue: 0,1:15:57.77,1:16:04.11,Default,,0000,0000,0000,,And then I get integral of\Nf1, y, what the heck is y?
Dialogue: 0,1:16:04.11,1:16:08.95,Default,,0000,0000,0000,,2 sine t, times x prime of t.
Dialogue: 0,1:16:08.95,1:16:15.36,Default,,0000,0000,0000,,
Dialogue: 0,1:16:15.36,1:16:20.85,Default,,0000,0000,0000,,Yeah, minus 2 sine\Nt, this is x prime.
Dialogue: 0,1:16:20.85,1:16:24.65,Default,,0000,0000,0000,,Plus 1, are you guys with me?
Dialogue: 0,1:16:24.65,1:16:30.14,Default,,0000,0000,0000,,Times y prime, which\Nis 2 cosine t, dt.
Dialogue: 0,1:16:30.14,1:16:33.37,Default,,0000,0000,0000,,And t between 0 and pi.
Dialogue: 0,1:16:33.37,1:16:36.19,Default,,0000,0000,0000,,
Dialogue: 0,1:16:36.19,1:16:42.24,Default,,0000,0000,0000,,And you get something\Nugly, you get 0 to pi.
Dialogue: 0,1:16:42.24,1:16:43.66,Default,,0000,0000,0000,,What is the nice thing?
Dialogue: 0,1:16:43.66,1:16:48.84,Default,,0000,0000,0000,,When you integrate this with\Nrespect to t, you get sine t.
Dialogue: 0,1:16:48.84,1:16:53.48,Default,,0000,0000,0000,,And thank God, sine, whether you\Nare at 0 or if pi is still 0.
Dialogue: 0,1:16:53.48,1:16:55.60,Default,,0000,0000,0000,,So this part will disappear.
Dialogue: 0,1:16:55.60,1:17:01.80,Default,,0000,0000,0000,,So all you have left is\Nminus 4 sine squared dt.
Dialogue: 0,1:17:01.80,1:17:04.76,Default,,0000,0000,0000,,But you are not done\Nso compute this at home
Dialogue: 0,1:17:04.76,1:17:07.44,Default,,0000,0000,0000,,because we are out of time.
Dialogue: 0,1:17:07.44,1:17:11.08,Default,,0000,0000,0000,,So don't jump to\Nconclusions unless you know
Dialogue: 0,1:17:11.08,1:17:12.89,Default,,0000,0000,0000,,that the force is conservative.
Dialogue: 0,1:17:12.89,1:17:14.88,Default,,0000,0000,0000,,If your force is\Nnot conservative
Dialogue: 0,1:17:14.88,1:17:17.69,Default,,0000,0000,0000,,then things are going\Nto look very ugly
Dialogue: 0,1:17:17.69,1:17:20.83,Default,,0000,0000,0000,,and your only chance is to go\Nback to the parametrization,
Dialogue: 0,1:17:20.83,1:17:24.30,Default,,0000,0000,0000,,to the basics.
Dialogue: 0,1:17:24.30,1:17:27.31,Default,,0000,0000,0000,,So we are practically\Ndone with 13.3
Dialogue: 0,1:17:27.31,1:17:31.32,Default,,0000,0000,0000,,but I want to watch\Nmore examples next time.
Dialogue: 0,1:17:31.32,1:17:33.12,Default,,0000,0000,0000,,And I'll send you the homework.
Dialogue: 0,1:17:33.12,1:17:38.22,Default,,0000,0000,0000,,Over the weekend you would be\Nable to start doing homework.
Dialogue: 0,1:17:38.22,1:17:40.62,Default,,0000,0000,0000,,Now, when shall I\Ngrab the homework?
Dialogue: 0,1:17:40.62,1:17:43.32,Default,,0000,0000,0000,,What if I closed it right\Nbefore the final, is that?
Dialogue: 0,1:17:43.32,1:17:44.22,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,1:17:44.22,1:17:46.07,Default,,0000,0000,0000,,PROFESSOR: Yeah?
Dialogue: 0,1:17:46.07,2:39:17.40,Default,,0000,0000,0000,,