[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.71,Default,,0000,0000,0000,,MAGDALENA TODA: I'm\Nstarting early, am I?
Dialogue: 0,0:00:01.71,0:00:04.00,Default,,0000,0000,0000,,It's exactly 12:30.
Dialogue: 0,0:00:04.00,0:00:07.00,Default,,0000,0000,0000,,The weather is getting\Nbetter, hopefully,
Dialogue: 0,0:00:07.00,0:00:14.00,Default,,0000,0000,0000,,and not too many people\Nshould miss class today.
Dialogue: 0,0:00:14.00,0:00:18.00,Default,,0000,0000,0000,,Can you start an attendance\Nsheet for me [INAUDIBLE]?
Dialogue: 0,0:00:18.00,0:00:22.00,Default,,0000,0000,0000,,I know I can count on you.
Dialogue: 0,0:00:22.00,0:00:22.50,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:00:22.50,0:00:25.50,Default,,0000,0000,0000,,I have good markers today.
Dialogue: 0,0:00:25.50,0:00:31.30,Default,,0000,0000,0000,,I'm going to go ahead\Nand talk about 12.3,
Dialogue: 0,0:00:31.30,0:00:34.28,Default,,0000,0000,0000,,double integrals in\Npolar coordinates.
Dialogue: 0,0:00:34.28,0:00:36.26,Default,,0000,0000,0000,,These are all friends of yours.
Dialogue: 0,0:00:36.26,0:00:55.11,Default,,0000,0000,0000,,
Dialogue: 0,0:00:55.11,0:00:58.80,Default,,0000,0000,0000,,You've seen until now\Nonly double integrals that
Dialogue: 0,0:00:58.80,0:01:05.58,Default,,0000,0000,0000,,involve the rectangles, either\Na rectangle, we saw [INAUDIBLE],
Dialogue: 0,0:01:05.58,0:01:10.61,Default,,0000,0000,0000,,and we saw some type\Nof double integrals,
Dialogue: 0,0:01:10.61,0:01:17.60,Default,,0000,0000,0000,,of course that involved\Nx and y, so-called type
Dialogue: 0,0:01:17.60,0:01:21.21,Default,,0000,0000,0000,,1 and type 2\Nregions, which were--
Dialogue: 0,0:01:21.21,0:01:24.80,Default,,0000,0000,0000,,so we saw the rectangular case.
Dialogue: 0,0:01:24.80,0:01:29.77,Default,,0000,0000,0000,,You have ab plus\Ncd, a rectangle.
Dialogue: 0,0:01:29.77,0:01:33.25,Default,,0000,0000,0000,,You have what other kind\Nof a velocity [INAUDIBLE]
Dialogue: 0,0:01:33.25,0:01:43.40,Default,,0000,0000,0000,,over the the main of the shape\Nx between a and be and y.
Dialogue: 0,0:01:43.40,0:01:49.07,Default,,0000,0000,0000,,You write wild and happy\Nfrom bottom to top.
Dialogue: 0,0:01:49.07,0:01:54.48,Default,,0000,0000,0000,,That's called the wild--\Nnot wild, the vertical strip
Dialogue: 0,0:01:54.48,0:01:58.99,Default,,0000,0000,0000,,method, where y will be\Nbetween the bottom function
Dialogue: 0,0:01:58.99,0:02:02.91,Default,,0000,0000,0000,,f of x and the top\Nfunction f of x.
Dialogue: 0,0:02:02.91,0:02:05.35,Default,,0000,0000,0000,,And last time I\Ntook examples where
Dialogue: 0,0:02:05.35,0:02:08.28,Default,,0000,0000,0000,,f and g were both positive, but\Nremember, you don't have to.
Dialogue: 0,0:02:08.28,0:02:12.53,Default,,0000,0000,0000,,All you have to have is that\Ng is always greater than f,
Dialogue: 0,0:02:12.53,0:02:14.10,Default,,0000,0000,0000,,or equal at some point.
Dialogue: 0,0:02:14.10,0:02:16.81,Default,,0000,0000,0000,,
Dialogue: 0,0:02:16.81,0:02:21.19,Default,,0000,0000,0000,,And then what else do\Nwe have for these cases?
Dialogue: 0,0:02:21.19,0:02:24.92,Default,,0000,0000,0000,,These are all\Ncontinuous functions.
Dialogue: 0,0:02:24.92,0:02:26.77,Default,,0000,0000,0000,,What else did we have?
Dialogue: 0,0:02:26.77,0:02:29.03,Default,,0000,0000,0000,,We had two domains.
Dialogue: 0,0:02:29.03,0:02:33.44,Default,,0000,0000,0000,,
Dialogue: 0,0:02:33.44,0:02:35.40,Default,,0000,0000,0000,,Had one and had two.
Dialogue: 0,0:02:35.40,0:02:38.34,Default,,0000,0000,0000,,
Dialogue: 0,0:02:38.34,0:02:42.98,Default,,0000,0000,0000,,Where what was going on,\Nwe have played a little bit
Dialogue: 0,0:02:42.98,0:02:50.12,Default,,0000,0000,0000,,around with y between c\Nand d limits with points.
Dialogue: 0,0:02:50.12,0:02:53.04,Default,,0000,0000,0000,,These are horizontal,\Nso we take the domain
Dialogue: 0,0:02:53.04,0:02:58.66,Default,,0000,0000,0000,,as being defined by these\Nhorizontal strips between let's
Dialogue: 0,0:02:58.66,0:03:00.04,Default,,0000,0000,0000,,say a function.
Dialogue: 0,0:03:00.04,0:03:03.84,Default,,0000,0000,0000,,Again, I need to rotate my head,\Nbut I didn't do my yoga today,
Dialogue: 0,0:03:03.84,0:03:07.14,Default,,0000,0000,0000,,so it's a little bit sticky.
Dialogue: 0,0:03:07.14,0:03:07.93,Default,,0000,0000,0000,,I'll try.
Dialogue: 0,0:03:07.93,0:03:19.35,Default,,0000,0000,0000,,x equals F of y, and x equals\NG of y, assuming, of course,
Dialogue: 0,0:03:19.35,0:03:24.07,Default,,0000,0000,0000,,that f of y is always greater\Nthan or equal to g of y,
Dialogue: 0,0:03:24.07,0:03:28.27,Default,,0000,0000,0000,,and the rest of the\Napparatus is in place.
Dialogue: 0,0:03:28.27,0:03:31.68,Default,,0000,0000,0000,,Those are not so\Nhard to understand.
Dialogue: 0,0:03:31.68,0:03:33.08,Default,,0000,0000,0000,,We played around.
Dialogue: 0,0:03:33.08,0:03:35.50,Default,,0000,0000,0000,,We switched the integrals.
Dialogue: 0,0:03:35.50,0:03:38.92,Default,,0000,0000,0000,,We changed the order of\Nintegration from dy dx
Dialogue: 0,0:03:38.92,0:03:43.19,Default,,0000,0000,0000,,to dx dy, so we have\Nto change the domain.
Dialogue: 0,0:03:43.19,0:03:45.78,Default,,0000,0000,0000,,We went from\Nvertical strip method
Dialogue: 0,0:03:45.78,0:03:52.05,Default,,0000,0000,0000,,to horizontal strip method\Nor the other way around.
Dialogue: 0,0:03:52.05,0:03:57.10,Default,,0000,0000,0000,,And for what kind of\Nexample, something
Dialogue: 0,0:03:57.10,0:03:59.94,Default,,0000,0000,0000,,like that-- I think it\Nwas a leaf like that,
Dialogue: 0,0:03:59.94,0:04:02.73,Default,,0000,0000,0000,,we said, let's compute\Nthe area or compute
Dialogue: 0,0:04:02.73,0:04:09.75,Default,,0000,0000,0000,,another kind of double\Nintegral over this leaf in two
Dialogue: 0,0:04:09.75,0:04:10.84,Default,,0000,0000,0000,,different ways.
Dialogue: 0,0:04:10.84,0:04:14.44,Default,,0000,0000,0000,,And we did it with\Nvertical strips,
Dialogue: 0,0:04:14.44,0:04:17.14,Default,,0000,0000,0000,,and we did the same\Nwith horizontal strips.
Dialogue: 0,0:04:17.14,0:04:20.33,Default,,0000,0000,0000,,
Dialogue: 0,0:04:20.33,0:04:22.65,Default,,0000,0000,0000,,So we reversed the\Norder of integration,
Dialogue: 0,0:04:22.65,0:04:27.10,Default,,0000,0000,0000,,and we said, I'm having the\Ndouble integral over domain
Dialogue: 0,0:04:27.10,0:04:31.56,Default,,0000,0000,0000,,of God knows what, f of\Nxy, continuous function,
Dialogue: 0,0:04:31.56,0:04:37.44,Default,,0000,0000,0000,,positive, continuous whenever\Nyou want, and we said da.
Dialogue: 0,0:04:37.44,0:04:40.16,Default,,0000,0000,0000,,We didn't quite specify\Nthe meaning of da.
Dialogue: 0,0:04:40.16,0:04:43.02,Default,,0000,0000,0000,,We said that da is\Nthe area element,
Dialogue: 0,0:04:43.02,0:04:47.27,Default,,0000,0000,0000,,but that sounds a little\Nbit weird, because it makes
Dialogue: 0,0:04:47.27,0:04:51.50,Default,,0000,0000,0000,,you think of surfaces,\Nand an area element
Dialogue: 0,0:04:51.50,0:04:53.88,Default,,0000,0000,0000,,doesn't have to be a\Nlittle square in general.
Dialogue: 0,0:04:53.88,0:04:59.29,Default,,0000,0000,0000,,It could be something like a\Npatch on a surface, bounded
Dialogue: 0,0:04:59.29,0:05:04.10,Default,,0000,0000,0000,,by two curves within your\Nsegments in each direction.
Dialogue: 0,0:05:04.10,0:05:06.47,Default,,0000,0000,0000,,So you think, well, I\Ndon't know what that is.
Dialogue: 0,0:05:06.47,0:05:07.84,Default,,0000,0000,0000,,I'll tell you\Ntoday what that is.
Dialogue: 0,0:05:07.84,0:05:11.13,Default,,0000,0000,0000,,It's a mysterious thing,\Nit's really beautiful,
Dialogue: 0,0:05:11.13,0:05:12.62,Default,,0000,0000,0000,,and we'll talk about it.
Dialogue: 0,0:05:12.62,0:05:15.44,Default,,0000,0000,0000,,Now, what did we do last time?
Dialogue: 0,0:05:15.44,0:05:19.36,Default,,0000,0000,0000,,We applied the two\Ntheorems that allowed
Dialogue: 0,0:05:19.36,0:05:23.78,Default,,0000,0000,0000,,us to do this both ways.
Dialogue: 0,0:05:23.78,0:05:29.36,Default,,0000,0000,0000,,Integral from a to b, what was\Nmy usual [? wrist ?] is down,
Dialogue: 0,0:05:29.36,0:05:32.43,Default,,0000,0000,0000,,f of x is in g of x, right?
Dialogue: 0,0:05:32.43,0:05:36.13,Default,,0000,0000,0000,,
Dialogue: 0,0:05:36.13,0:05:37.34,Default,,0000,0000,0000,,dy dx.
Dialogue: 0,0:05:37.34,0:05:39.75,Default,,0000,0000,0000,,So if you do it in\Nthis order, it's
Dialogue: 0,0:05:39.75,0:05:44.78,Default,,0000,0000,0000,,going to be the same as if\Nyou do it in the other order.
Dialogue: 0,0:05:44.78,0:05:53.46,Default,,0000,0000,0000,,ab are these guys, and then\Nthis was cd on the y-axis.
Dialogue: 0,0:05:53.46,0:05:56.81,Default,,0000,0000,0000,,This is the range between\Nc and d in altitudes.
Dialogue: 0,0:05:56.81,0:06:00.74,Default,,0000,0000,0000,,So we have integral from\Nc to d, integral from,
Dialogue: 0,0:06:00.74,0:06:02.91,Default,,0000,0000,0000,,I don't know what they will be.
Dialogue: 0,0:06:02.91,0:06:07.14,Default,,0000,0000,0000,,This big guy I'm talking--\Nwhich one is the one?
Dialogue: 0,0:06:07.14,0:06:11.34,Default,,0000,0000,0000,,This one, that's going to\Nbe called x equals f of y,
Dialogue: 0,0:06:11.34,0:06:17.60,Default,,0000,0000,0000,,or g of y, and let's put the\Nbig one G and the smaller one,
Dialogue: 0,0:06:17.60,0:06:19.56,Default,,0000,0000,0000,,x equals F of y.
Dialogue: 0,0:06:19.56,0:06:24.04,Default,,0000,0000,0000,,So you have to [? re-denote ?]\Nthese functions,
Dialogue: 0,0:06:24.04,0:06:31.43,Default,,0000,0000,0000,,these inverse functions, and\Nuse them as functions of y.
Dialogue: 0,0:06:31.43,0:06:34.64,Default,,0000,0000,0000,,So it makes sense to\Nsay-- what did we do?
Dialogue: 0,0:06:34.64,0:06:39.62,Default,,0000,0000,0000,,We first integrated respect to\Nx between two functions of y.
Dialogue: 0,0:06:39.62,0:06:44.17,Default,,0000,0000,0000,,That was the so-called\Nhorizontal strip method, dy.
Dialogue: 0,0:06:44.17,0:06:48.32,Default,,0000,0000,0000,,So I have summarized\Nthe ideas from last time
Dialogue: 0,0:06:48.32,0:06:53.35,Default,,0000,0000,0000,,that we worked with, generally\Nwith corners x and y.
Dialogue: 0,0:06:53.35,0:06:55.95,Default,,0000,0000,0000,,We were very happy about them.
Dialogue: 0,0:06:55.95,0:07:00.05,Default,,0000,0000,0000,,We had the rectangular\Ndomain, where x was between ab
Dialogue: 0,0:07:00.05,0:07:01.77,Default,,0000,0000,0000,,and y was between cd.
Dialogue: 0,0:07:01.77,0:07:05.64,Default,,0000,0000,0000,,Then we went to type 1, not\Ndiabetes, just type 1 region,
Dialogue: 0,0:07:05.64,0:07:09.07,Default,,0000,0000,0000,,type 2, and those\Nguys are related.
Dialogue: 0,0:07:09.07,0:07:12.32,Default,,0000,0000,0000,,So if you understood 1 and\Nunderstood the other one,
Dialogue: 0,0:07:12.32,0:07:15.30,Default,,0000,0000,0000,,and if you have a\Nnice domain like that,
Dialogue: 0,0:07:15.30,0:07:18.09,Default,,0000,0000,0000,,you can compute the\Narea or something.
Dialogue: 0,0:07:18.09,0:07:21.07,Default,,0000,0000,0000,,The area will correspond\Nto x equals 1.
Dialogue: 0,0:07:21.07,0:07:24.09,Default,,0000,0000,0000,,So if f is 1, then\Nthat's the area.
Dialogue: 0,0:07:24.09,0:07:28.85,Default,,0000,0000,0000,,That will also be a\Nvolume of a cylinder based
Dialogue: 0,0:07:28.85,0:07:33.13,Default,,0000,0000,0000,,on that region with height 1.
Dialogue: 0,0:07:33.13,0:07:36.97,Default,,0000,0000,0000,,Imagine a can of Coke\Nthat has height 1,
Dialogue: 0,0:07:36.97,0:07:40.92,Default,,0000,0000,0000,,and-- maybe better,\Nchocolate cake,
Dialogue: 0,0:07:40.92,0:07:43.82,Default,,0000,0000,0000,,that has the shape of\Nthis leaf on the bottom,
Dialogue: 0,0:07:43.82,0:07:47.71,Default,,0000,0000,0000,,and then its height\Nis 1 everywhere.
Dialogue: 0,0:07:47.71,0:07:51.79,Default,,0000,0000,0000,,So if you put 1 here, and\Nyou get the area element,
Dialogue: 0,0:07:51.79,0:07:54.82,Default,,0000,0000,0000,,and then everything\Nelse can be done
Dialogue: 0,0:07:54.82,0:07:59.96,Default,,0000,0000,0000,,by reversing the order of\Nintegration if f is continuous.
Dialogue: 0,0:07:59.96,0:08:02.86,Default,,0000,0000,0000,,But for polar\Ncoordinates, the situation
Dialogue: 0,0:08:02.86,0:08:08.19,Default,,0000,0000,0000,,has to be reconsidered almost\Nentirely, because the area
Dialogue: 0,0:08:08.19,0:08:17.74,Default,,0000,0000,0000,,element, da is called\Nthe area element for us,
Dialogue: 0,0:08:17.74,0:08:25.74,Default,,0000,0000,0000,,was equal to dx dy for the\Ncartesian coordinate case.
Dialogue: 0,0:08:25.74,0:08:32.16,Default,,0000,0000,0000,,
Dialogue: 0,0:08:32.16,0:08:36.62,Default,,0000,0000,0000,,And here I'm making a\Nweird face, I'm weird, no?
Dialogue: 0,0:08:36.62,0:08:39.95,Default,,0000,0000,0000,,Saying, what am I going\Nto do, what is this
Dialogue: 0,0:08:39.95,0:08:44.42,Default,,0000,0000,0000,,going to become for\Npolar coordinates?
Dialogue: 0,0:08:44.42,0:08:47.71,Default,,0000,0000,0000,,
Dialogue: 0,0:08:47.71,0:08:52.61,Default,,0000,0000,0000,,And now you go, oh my God,\Nnot polar coordinates.
Dialogue: 0,0:08:52.61,0:08:54.15,Default,,0000,0000,0000,,Those were my\Nenemies in Calc II.
Dialogue: 0,0:08:54.15,0:08:55.87,Default,,0000,0000,0000,,Many people told me that.
Dialogue: 0,0:08:55.87,0:09:01.87,Default,,0000,0000,0000,,And I tried to go\Ninto my time machine
Dialogue: 0,0:09:01.87,0:09:04.54,Default,,0000,0000,0000,,and go back something\Nlike 25 years ago
Dialogue: 0,0:09:04.54,0:09:07.98,Default,,0000,0000,0000,,and see how I felt about\Nthem, and I remember that.
Dialogue: 0,0:09:07.98,0:09:12.12,Default,,0000,0000,0000,,I didn't get them from\Nthe first 48 hours
Dialogue: 0,0:09:12.12,0:09:15.73,Default,,0000,0000,0000,,after I was exposed to them.
Dialogue: 0,0:09:15.73,0:09:18.04,Default,,0000,0000,0000,,Therefore, let's\Ndo some preview.
Dialogue: 0,0:09:18.04,0:09:21.49,Default,,0000,0000,0000,,What were those\Npolar coordinates?
Dialogue: 0,0:09:21.49,0:09:25.84,Default,,0000,0000,0000,,Polar coordinates were\Na beautiful thing,
Dialogue: 0,0:09:25.84,0:09:27.74,Default,,0000,0000,0000,,these guys from trig.
Dialogue: 0,0:09:27.74,0:09:31.50,Default,,0000,0000,0000,,Trig was your friend hopefully.
Dialogue: 0,0:09:31.50,0:09:34.66,Default,,0000,0000,0000,,And what did we have\Nin trigonometry?
Dialogue: 0,0:09:34.66,0:09:38.71,Default,,0000,0000,0000,,In trigonometry, we had\Na point on a circle.
Dialogue: 0,0:09:38.71,0:09:41.27,Default,,0000,0000,0000,,This is not the unit\Ntrigonometric circle,
Dialogue: 0,0:09:41.27,0:09:45.20,Default,,0000,0000,0000,,it's a circle of--\Nbless you-- radius r.
Dialogue: 0,0:09:45.20,0:09:49.76,Default,,0000,0000,0000,,I'm a little bit shifted\Nby a phase of phi 0.
Dialogue: 0,0:09:49.76,0:09:54.51,Default,,0000,0000,0000,,So you have a radius r.
Dialogue: 0,0:09:54.51,0:09:56.97,Default,,0000,0000,0000,,And let's call that little r.
Dialogue: 0,0:09:56.97,0:10:02.71,Default,,0000,0000,0000,,
Dialogue: 0,0:10:02.71,0:10:06.37,Default,,0000,0000,0000,,And then, we say, OK,\Nhow about the angle?
Dialogue: 0,0:10:06.37,0:10:08.80,Default,,0000,0000,0000,,That's the second\Npolar coordinate.
Dialogue: 0,0:10:08.80,0:10:16.40,Default,,0000,0000,0000,,The angle by measuring\Nfrom the, what
Dialogue: 0,0:10:16.40,0:10:17.84,Default,,0000,0000,0000,,is this called, the x-axis.
Dialogue: 0,0:10:17.84,0:10:21.24,Default,,0000,0000,0000,,
Dialogue: 0,0:10:21.24,0:10:25.71,Default,,0000,0000,0000,,Origin, x-axis, o, x,\Ngoing counterclockwise,
Dialogue: 0,0:10:25.71,0:10:28.37,Default,,0000,0000,0000,,because we are mathemeticians.
Dialogue: 0,0:10:28.37,0:10:30.90,Default,,0000,0000,0000,,Every normal person, when\Nthey mix into a bowl,
Dialogue: 0,0:10:30.90,0:10:32.93,Default,,0000,0000,0000,,they mix like that.
Dialogue: 0,0:10:32.93,0:10:35.48,Default,,0000,0000,0000,,Well, I've seen that\Nmost of my colleagues--
Dialogue: 0,0:10:35.48,0:10:37.67,Default,,0000,0000,0000,,this is just a\Npsychological test, OK?
Dialogue: 0,0:10:37.67,0:10:39.56,Default,,0000,0000,0000,,I wanted to see\Nhow they mix when
Dialogue: 0,0:10:39.56,0:10:41.73,Default,,0000,0000,0000,,they cook, or mix\Nup-- most of them
Dialogue: 0,0:10:41.73,0:10:44.02,Default,,0000,0000,0000,,mix in a trigonometric sense.
Dialogue: 0,0:10:44.02,0:10:47.61,Default,,0000,0000,0000,,I don't know if this has\Nanything to do with the brain
Dialogue: 0,0:10:47.61,0:10:51.37,Default,,0000,0000,0000,,connections, but I think\Nthat's [? kind of weird. ?]
Dialogue: 0,0:10:51.37,0:10:54.55,Default,,0000,0000,0000,,I don't have a statistical\Nresult, but most of the people
Dialogue: 0,0:10:54.55,0:10:58.59,Default,,0000,0000,0000,,I've seen that, and do\Nmathematics, mix like that.
Dialogue: 0,0:10:58.59,0:11:02.53,Default,,0000,0000,0000,,So trigonometric sense.
Dialogue: 0,0:11:02.53,0:11:09.01,Default,,0000,0000,0000,,What is the connection with the\Nactual Cartesian coordinates?
Dialogue: 0,0:11:09.01,0:11:13.65,Default,,0000,0000,0000,,D you know what Cartesian\Ncomes from as a word?
Dialogue: 0,0:11:13.65,0:11:15.77,Default,,0000,0000,0000,,Cartesian, that sounds weird.
Dialogue: 0,0:11:15.77,0:11:17.24,Default,,0000,0000,0000,,STUDENT: From Descartes.
Dialogue: 0,0:11:17.24,0:11:18.24,Default,,0000,0000,0000,,MAGDALENA TODA: Exactly.
Dialogue: 0,0:11:18.24,0:11:19.70,Default,,0000,0000,0000,,Who said that?
Dialogue: 0,0:11:19.70,0:11:21.18,Default,,0000,0000,0000,,Roberto, thank you so much.
Dialogue: 0,0:11:21.18,0:11:22.16,Default,,0000,0000,0000,,I'm impressed.
Dialogue: 0,0:11:22.16,0:11:22.99,Default,,0000,0000,0000,,Descartes was--
Dialogue: 0,0:11:22.99,0:11:23.66,Default,,0000,0000,0000,,STUDENT: French.
Dialogue: 0,0:11:23.66,0:11:26.01,Default,,0000,0000,0000,,MAGDALENA TODA: --a\NFrench mathematician.
Dialogue: 0,0:11:26.01,0:11:28.98,Default,,0000,0000,0000,,But actually, I mean,\Nhe was everything.
Dialogue: 0,0:11:28.98,0:11:30.91,Default,,0000,0000,0000,,He was a crazy lunatic.
Dialogue: 0,0:11:30.91,0:11:34.78,Default,,0000,0000,0000,,He was a philosopher,\Na mathematician,
Dialogue: 0,0:11:34.78,0:11:37.36,Default,,0000,0000,0000,,a scientist in general.
Dialogue: 0,0:11:37.36,0:11:40.85,Default,,0000,0000,0000,,He also knew a lot\Nabout life science.
Dialogue: 0,0:11:40.85,0:11:43.79,Default,,0000,0000,0000,,But at the time, I don't\Nknow if this is true.
Dialogue: 0,0:11:43.79,0:11:45.97,Default,,0000,0000,0000,,I should check with wiki,\Nor whoever can tell me.
Dialogue: 0,0:11:45.97,0:11:50.44,Default,,0000,0000,0000,,One of my professors in college\Ntold me that at that time,
Dialogue: 0,0:11:50.44,0:11:53.24,Default,,0000,0000,0000,,there was a fashion\Nthat people would
Dialogue: 0,0:11:53.24,0:11:57.06,Default,,0000,0000,0000,,change their names like they\Ndo on Facebook nowadays.
Dialogue: 0,0:11:57.06,0:11:59.98,Default,,0000,0000,0000,,So they and change their\Nname from Francesca
Dialogue: 0,0:11:59.98,0:12:04.78,Default,,0000,0000,0000,,to Frenchy, from Roberto\Nto Robby, from-- so
Dialogue: 0,0:12:04.78,0:12:08.51,Default,,0000,0000,0000,,if they would have to\Nclean up Facebook and see
Dialogue: 0,0:12:08.51,0:12:14.80,Default,,0000,0000,0000,,how many names correspond to\Nthe ID, I think less than 20%.
Dialogue: 0,0:12:14.80,0:12:16.86,Default,,0000,0000,0000,,At that time it was the same.
Dialogue: 0,0:12:16.86,0:12:22.78,Default,,0000,0000,0000,,All of the scientists loved\Nto romanize their names.
Dialogue: 0,0:12:22.78,0:12:26.15,Default,,0000,0000,0000,,And of course he was\Nof a romance language,
Dialogue: 0,0:12:26.15,0:12:30.37,Default,,0000,0000,0000,,but he said, what if I\Nmade my name a Latin name,
Dialogue: 0,0:12:30.37,0:12:32.25,Default,,0000,0000,0000,,I changed my name\Ninto a Latin name.
Dialogue: 0,0:12:32.25,0:12:36.64,Default,,0000,0000,0000,,So he himself, this is what\Nmy professor told me, he
Dialogue: 0,0:12:36.64,0:12:39.78,Default,,0000,0000,0000,,himself changed his\Nname to Cartesius.
Dialogue: 0,0:12:39.78,0:12:45.51,Default,,0000,0000,0000,,"Car-teh-see-yus" actually, in\NLatin, the way it should be.
Dialogue: 0,0:12:45.51,0:12:49.88,Default,,0000,0000,0000,,
Dialogue: 0,0:12:49.88,0:12:52.20,Default,,0000,0000,0000,,OK, very smart guy.
Dialogue: 0,0:12:52.20,0:12:56.80,Default,,0000,0000,0000,,Now, when we look\Na x and y, there
Dialogue: 0,0:12:56.80,0:13:04.40,Default,,0000,0000,0000,,has to be a connection between\Nx, y as the couple, and r theta
Dialogue: 0,0:13:04.40,0:13:09.23,Default,,0000,0000,0000,,as the same-- I mean a\Ncouple, not the couple,
Dialogue: 0,0:13:09.23,0:13:10.69,Default,,0000,0000,0000,,for the same point.
Dialogue: 0,0:13:10.69,0:13:11.19,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:13:11.19,0:13:11.98,Default,,0000,0000,0000,,STUDENT: Cartesius.
Dialogue: 0,0:13:11.98,0:13:14.75,Default,,0000,0000,0000,,Like meaning flat?
Dialogue: 0,0:13:14.75,0:13:15.25,Default,,0000,0000,0000,,The name?
Dialogue: 0,0:13:15.25,0:13:17.42,Default,,0000,0000,0000,,MAGDALENA TODA: These are\Nthe Cartesian coordinates,
Dialogue: 0,0:13:17.42,0:13:20.13,Default,,0000,0000,0000,,and it sounds like the word map.
Dialogue: 0,0:13:20.13,0:13:22.02,Default,,0000,0000,0000,,I think he had meant
Dialogue: 0,0:13:22.02,0:13:23.64,Default,,0000,0000,0000,,STUDENT: Because the\Nmeaning of carte--
Dialogue: 0,0:13:23.64,0:13:24.76,Default,,0000,0000,0000,,STUDENT: But look, look.
Dialogue: 0,0:13:24.76,0:13:27.95,Default,,0000,0000,0000,,Descartes means from the map.
Dialogue: 0,0:13:27.95,0:13:30.25,Default,,0000,0000,0000,,From the books, or from the map.
Dialogue: 0,0:13:30.25,0:13:33.35,Default,,0000,0000,0000,,So he thought what his\Nname would really mean,
Dialogue: 0,0:13:33.35,0:13:36.26,Default,,0000,0000,0000,,and so he recalled himself.
Dialogue: 0,0:13:36.26,0:13:39.00,Default,,0000,0000,0000,,There was no fun, no\NTwitter, no Facebook.
Dialogue: 0,0:13:39.00,0:13:43.55,Default,,0000,0000,0000,,So they had to do something\Nto enjoy themselves.
Dialogue: 0,0:13:43.55,0:13:46.18,Default,,0000,0000,0000,,Now, when it comes\Nto these triangles,
Dialogue: 0,0:13:46.18,0:13:49.78,Default,,0000,0000,0000,,we have to think of the\Nrelationship between x, y
Dialogue: 0,0:13:49.78,0:13:52.51,Default,,0000,0000,0000,,and r, theta.
Dialogue: 0,0:13:52.51,0:13:56.16,Default,,0000,0000,0000,,And could somebody tell me what\Nthe relationship between x, y
Dialogue: 0,0:13:56.16,0:13:59.24,Default,,0000,0000,0000,,and r, theta is?
Dialogue: 0,0:13:59.24,0:14:01.26,Default,,0000,0000,0000,,x represents
Dialogue: 0,0:14:01.26,0:14:02.61,Default,,0000,0000,0000,,STUDENT: R cosine theta.
Dialogue: 0,0:14:02.61,0:14:05.26,Default,,0000,0000,0000,,STUDENT: r cosine\Ntheta, who says that?
Dialogue: 0,0:14:05.26,0:14:07.90,Default,,0000,0000,0000,,Trigonometry taught us\Nthat, because that's
Dialogue: 0,0:14:07.90,0:14:14.39,Default,,0000,0000,0000,,the adjacent side over\Nthe hypotenuse for cosine.
Dialogue: 0,0:14:14.39,0:14:18.25,Default,,0000,0000,0000,,In terms of sine, you\Nknow what you have,
Dialogue: 0,0:14:18.25,0:14:22.59,Default,,0000,0000,0000,,so you're going to have\Ny equals r sine theta,
Dialogue: 0,0:14:22.59,0:14:26.74,Default,,0000,0000,0000,,and we have to decide\Nif x and y are allowed
Dialogue: 0,0:14:26.74,0:14:28.20,Default,,0000,0000,0000,,to be anywhere in plane.
Dialogue: 0,0:14:28.20,0:14:31.16,Default,,0000,0000,0000,,
Dialogue: 0,0:14:31.16,0:14:35.47,Default,,0000,0000,0000,,For the plane,\NI'll also write r2.
Dialogue: 0,0:14:35.47,0:14:40.83,Default,,0000,0000,0000,,R2, not R2 from the movie,\Njust r2 is the plane,
Dialogue: 0,0:14:40.83,0:14:44.10,Default,,0000,0000,0000,,and r3 is the space,\Nthe [? intriguing ?]
Dialogue: 0,0:14:44.10,0:14:46.85,Default,,0000,0000,0000,,space, three-dimensional one.
Dialogue: 0,0:14:46.85,0:14:50.87,Default,,0000,0000,0000,,r theta, is a couple where?
Dialogue: 0,0:14:50.87,0:14:52.44,Default,,0000,0000,0000,,That's a little bit tricky.
Dialogue: 0,0:14:52.44,0:14:54.12,Default,,0000,0000,0000,,We have to make a restriction.
Dialogue: 0,0:14:54.12,0:14:59.34,Default,,0000,0000,0000,,We allow r to be anywhere\Nbetween 0 and infinity.
Dialogue: 0,0:14:59.34,0:15:03.95,Default,,0000,0000,0000,,So it has to be a\Npositive number.
Dialogue: 0,0:15:03.95,0:15:13.05,Default,,0000,0000,0000,,And theta [INTERPOSING VOICES]\Nbetween 0 and 2 pi.
Dialogue: 0,0:15:13.05,0:15:14.90,Default,,0000,0000,0000,,STUDENT: I've been\Nsick since Tuesday.
Dialogue: 0,0:15:14.90,0:15:16.40,Default,,0000,0000,0000,,MAGDALENA TODA: I\Nbelieve you, Ryan.
Dialogue: 0,0:15:16.40,0:15:17.64,Default,,0000,0000,0000,,You sound sick to me.
Dialogue: 0,0:15:17.64,0:15:20.78,Default,,0000,0000,0000,,Take your viruses away from me.
Dialogue: 0,0:15:20.78,0:15:21.65,Default,,0000,0000,0000,,Take the germs away.
Dialogue: 0,0:15:21.65,0:15:25.32,Default,,0000,0000,0000,,I don't even have\Nthe-- I'm kidding,
Dialogue: 0,0:15:25.32,0:15:27.75,Default,,0000,0000,0000,,Alex, I hope you\Ndon't get offended.
Dialogue: 0,0:15:27.75,0:15:31.53,Default,,0000,0000,0000,,So, I hope this works this time.
Dialogue: 0,0:15:31.53,0:15:33.07,Default,,0000,0000,0000,,I'm making a\Nsarcastic-- it's really,
Dialogue: 0,0:15:33.07,0:15:34.52,Default,,0000,0000,0000,,I hope you're feeling better.
Dialogue: 0,0:15:34.52,0:15:35.97,Default,,0000,0000,0000,,I'm sorry about that.
Dialogue: 0,0:15:35.97,0:15:38.90,Default,,0000,0000,0000,,
Dialogue: 0,0:15:38.90,0:15:41.76,Default,,0000,0000,0000,,So you haven't missed much.
Dialogue: 0,0:15:41.76,0:15:42.63,Default,,0000,0000,0000,,Only the jokes.
Dialogue: 0,0:15:42.63,0:15:46.73,Default,,0000,0000,0000,,So x equals r cosine theta,\Ny equals r sine theta.
Dialogue: 0,0:15:46.73,0:15:49.64,Default,,0000,0000,0000,,Is that your\Nfavorite change that
Dialogue: 0,0:15:49.64,0:15:55.100,Default,,0000,0000,0000,,was a differential\Nmapping from the set x,
Dialogue: 0,0:15:55.100,0:15:58.81,Default,,0000,0000,0000,,y to the set r,\Ntheta back and forth.
Dialogue: 0,0:15:58.81,0:16:02.30,Default,,0000,0000,0000,,
Dialogue: 0,0:16:02.30,0:16:05.38,Default,,0000,0000,0000,,And you are going\Nto probably say, OK
Dialogue: 0,0:16:05.38,0:16:08.33,Default,,0000,0000,0000,,how do you denote such a map?
Dialogue: 0,0:16:08.33,0:16:11.46,Default,,0000,0000,0000,,I mean, going from x,\Ny to r, theta and back,
Dialogue: 0,0:16:11.46,0:16:14.69,Default,,0000,0000,0000,,let's suppose that we go\Nfrom r, theta to x, y,
Dialogue: 0,0:16:14.69,0:16:17.28,Default,,0000,0000,0000,,and that's going to be a big if.
Dialogue: 0,0:16:17.28,0:16:20.43,Default,,0000,0000,0000,,And going backwards is going\Nto be the inverse mapping.
Dialogue: 0,0:16:20.43,0:16:23.71,Default,,0000,0000,0000,,So I'm going to\Ncall it f inverse.
Dialogue: 0,0:16:23.71,0:16:30.77,Default,,0000,0000,0000,,So that's a map from a couple\Nto another couple of number.
Dialogue: 0,0:16:30.77,0:16:35.96,Default,,0000,0000,0000,,And you say, OK, but\Nwhy is that a map?
Dialogue: 0,0:16:35.96,0:16:38.26,Default,,0000,0000,0000,,All right, guys,\Nnow let me tell you.
Dialogue: 0,0:16:38.26,0:16:43.33,Default,,0000,0000,0000,,So x, you can do x as a\Nfunction of r, theta, right?
Dialogue: 0,0:16:43.33,0:16:45.75,Default,,0000,0000,0000,,It is a function of r and theta.
Dialogue: 0,0:16:45.75,0:16:48.42,Default,,0000,0000,0000,,It's a function\Nof two variables.
Dialogue: 0,0:16:48.42,0:16:51.96,Default,,0000,0000,0000,,And y is a function\Nof r and theta.
Dialogue: 0,0:16:51.96,0:16:53.74,Default,,0000,0000,0000,,It's another function\Nof two variables.
Dialogue: 0,0:16:53.74,0:16:58.17,Default,,0000,0000,0000,,They are both nice\Nand differentiable.
Dialogue: 0,0:16:58.17,0:17:02.71,Default,,0000,0000,0000,,We assume not only that\Nthey are differentiable,
Dialogue: 0,0:17:02.71,0:17:07.36,Default,,0000,0000,0000,,but the partial derivatives\Nwill be continuous.
Dialogue: 0,0:17:07.36,0:17:10.64,Default,,0000,0000,0000,,So it's really\Nnice as a mapping.
Dialogue: 0,0:17:10.64,0:17:14.66,Default,,0000,0000,0000,,And you think, could I\Nwrite the chain rule?
Dialogue: 0,0:17:14.66,0:17:16.23,Default,,0000,0000,0000,,That is the whole idea.
Dialogue: 0,0:17:16.23,0:17:18.04,Default,,0000,0000,0000,,What is the meaning\Nof differential?
Dialogue: 0,0:17:18.04,0:17:20.05,Default,,0000,0000,0000,,dx differential dy.
Dialogue: 0,0:17:20.05,0:17:23.39,Default,,0000,0000,0000,,Since I was chatting with\Nyou, once, [? Yuniel ?],
Dialogue: 0,0:17:23.39,0:17:28.60,Default,,0000,0000,0000,,and you asked me to\Nhelp you with homework,
Dialogue: 0,0:17:28.60,0:17:31.48,Default,,0000,0000,0000,,I had to go over\Ndifferential again.
Dialogue: 0,0:17:31.48,0:17:36.35,Default,,0000,0000,0000,,If you were to define,\Nlike Mr. Leibniz did,
Dialogue: 0,0:17:36.35,0:17:39.93,Default,,0000,0000,0000,,the differential of the\Nfunction x with respect
Dialogue: 0,0:17:39.93,0:17:44.06,Default,,0000,0000,0000,,to both variables, that\Nwas the sum, right?
Dialogue: 0,0:17:44.06,0:17:45.45,Default,,0000,0000,0000,,You've done that\Nin the homework,
Dialogue: 0,0:17:45.45,0:17:46.72,Default,,0000,0000,0000,,it's fresh in your mind.
Dialogue: 0,0:17:46.72,0:17:53.50,Default,,0000,0000,0000,,So you get x sub r,\Ndr, plus f x sub what?
Dialogue: 0,0:17:53.50,0:17:54.13,Default,,0000,0000,0000,,STUDENT: Theta.
Dialogue: 0,0:17:54.13,0:17:56.94,Default,,0000,0000,0000,,MAGDALENA TODA:\NSub theta d-theta.
Dialogue: 0,0:17:56.94,0:18:01.74,Default,,0000,0000,0000,,And somebody asked me,\Nwhat if I see skip the dr?
Dialogue: 0,0:18:01.74,0:18:02.49,Default,,0000,0000,0000,,No, don't do that.
Dialogue: 0,0:18:02.49,0:18:05.20,Default,,0000,0000,0000,,First of all, WeBWorK is not\Ngoing to take the answer.
Dialogue: 0,0:18:05.20,0:18:09.31,Default,,0000,0000,0000,,But second of all, the\Nmost important stuff
Dialogue: 0,0:18:09.31,0:18:13.19,Default,,0000,0000,0000,,here to remember is that these\Nare small, infinitesimally
Dialogue: 0,0:18:13.19,0:18:15.19,Default,,0000,0000,0000,,small, displacements.
Dialogue: 0,0:18:15.19,0:18:32.13,Default,,0000,0000,0000,,Infinitesimally small\Ndisplacements in the directions
Dialogue: 0,0:18:32.13,0:18:33.99,Default,,0000,0000,0000,,x and y, respectively.
Dialogue: 0,0:18:33.99,0:18:37.44,Default,,0000,0000,0000,,So you would say, what does\Nthat mean, infinitesimally?
Dialogue: 0,0:18:37.44,0:18:39.79,Default,,0000,0000,0000,,It doesn't mean delta-x small.
Dialogue: 0,0:18:39.79,0:18:43.95,Default,,0000,0000,0000,,Delta-x small would be like\Nme driving 7 feet, when
Dialogue: 0,0:18:43.95,0:18:48.92,Default,,0000,0000,0000,,I know I have to drive fast to\NAmarillo to be there in 1 hour.
Dialogue: 0,0:18:48.92,0:18:49.66,Default,,0000,0000,0000,,Well, OK.
Dialogue: 0,0:18:49.66,0:18:51.60,Default,,0000,0000,0000,,Don't tell anybody.
Dialogue: 0,0:18:51.60,0:18:55.16,Default,,0000,0000,0000,,But, it's about 2 hours, right?
Dialogue: 0,0:18:55.16,0:18:58.23,Default,,0000,0000,0000,,So I cannot be there in an hour.
Dialogue: 0,0:18:58.23,0:19:01.94,Default,,0000,0000,0000,,But driving those seven\Nfeet is like a delta x.
Dialogue: 0,0:19:01.94,0:19:07.10,Default,,0000,0000,0000,,Imagine, however, me\Nmeasuring that speed of mine
Dialogue: 0,0:19:07.10,0:19:10.13,Default,,0000,0000,0000,,in a much smaller\Nfraction of a second.
Dialogue: 0,0:19:10.13,0:19:15.100,Default,,0000,0000,0000,,So shrink that time to\Nsomething infinitesimally small,
Dialogue: 0,0:19:15.100,0:19:17.45,Default,,0000,0000,0000,,which is what you have here.
Dialogue: 0,0:19:17.45,0:19:19.21,Default,,0000,0000,0000,,That kind of quantity.
Dialogue: 0,0:19:19.21,0:19:25.44,Default,,0000,0000,0000,,And dy will be y sub r dr\Nplus y sub theta d-theta.
Dialogue: 0,0:19:25.44,0:19:28.75,Default,,0000,0000,0000,,
Dialogue: 0,0:19:28.75,0:19:32.72,Default,,0000,0000,0000,,And now, I'm not going\Nto go by the book.
Dialogue: 0,0:19:32.72,0:19:34.56,Default,,0000,0000,0000,,I'm going to go\Na little bit more
Dialogue: 0,0:19:34.56,0:19:39.63,Default,,0000,0000,0000,,in depth, because in the book--\NFirst of all, let me tell you,
Dialogue: 0,0:19:39.63,0:19:43.87,Default,,0000,0000,0000,,if I went by the book,\Nwhat I would come with.
Dialogue: 0,0:19:43.87,0:19:48.79,Default,,0000,0000,0000,,And of course the way\Nwe teach mathematics
Dialogue: 0,0:19:48.79,0:19:52.43,Default,,0000,0000,0000,,all through K-12 and through\Ncollege is swallow this theorem
Dialogue: 0,0:19:52.43,0:19:53.81,Default,,0000,0000,0000,,and believe it.
Dialogue: 0,0:19:53.81,0:19:58.91,Default,,0000,0000,0000,,So practically you accept\Nwhatever we give you
Dialogue: 0,0:19:58.91,0:20:02.12,Default,,0000,0000,0000,,without controlling it, without\Nchecking if we're right,
Dialogue: 0,0:20:02.12,0:20:05.15,Default,,0000,0000,0000,,without trying to prove it.
Dialogue: 0,0:20:05.15,0:20:06.65,Default,,0000,0000,0000,,Practically, the\Ntheorem in the book
Dialogue: 0,0:20:06.65,0:20:09.30,Default,,0000,0000,0000,,says that if you\Nhave a bunch of x,
Dialogue: 0,0:20:09.30,0:20:14.40,Default,,0000,0000,0000,,y that is continuous\Nover a domain, D,
Dialogue: 0,0:20:14.40,0:20:21.32,Default,,0000,0000,0000,,and you do change\Nthe variables over--
Dialogue: 0,0:20:21.32,0:20:22.71,Default,,0000,0000,0000,,STUDENT: I forgot my glasses.
Dialogue: 0,0:20:22.71,0:20:25.04,Default,,0000,0000,0000,,So I'm going to sit very close.
Dialogue: 0,0:20:25.04,0:20:28.57,Default,,0000,0000,0000,,MAGDALENA TODA:\NWhat do you wear?
Dialogue: 0,0:20:28.57,0:20:30.63,Default,,0000,0000,0000,,What [INAUDIBLE]?
Dialogue: 0,0:20:30.63,0:20:31.84,Default,,0000,0000,0000,,STUDENT: I couldn't tell you.
Dialogue: 0,0:20:31.84,0:20:33.10,Default,,0000,0000,0000,,I can see from here.
Dialogue: 0,0:20:33.10,0:20:33.57,Default,,0000,0000,0000,,MAGDALENA TODA: You can?
Dialogue: 0,0:20:33.57,0:20:34.03,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:20:34.03,0:20:35.07,Default,,0000,0000,0000,,My vision's not terrible.
Dialogue: 0,0:20:35.07,0:20:41.60,Default,,0000,0000,0000,,MAGDALENA TODA: All\Nright. f of x, y da.
Dialogue: 0,0:20:41.60,0:20:47.12,Default,,0000,0000,0000,,If I change this da\Nas dx dy, let's say,
Dialogue: 0,0:20:47.12,0:20:49.82,Default,,0000,0000,0000,,to a perspective\Nof something else
Dialogue: 0,0:20:49.82,0:20:52.47,Default,,0000,0000,0000,,in terms of polar\Ncoordinates, then
Dialogue: 0,0:20:52.47,0:20:57.38,Default,,0000,0000,0000,,the integral I'm going to get is\Nover the corresponding domain D
Dialogue: 0,0:20:57.38,0:21:00.53,Default,,0000,0000,0000,,star, whatever that would be.
Dialogue: 0,0:21:00.53,0:21:06.48,Default,,0000,0000,0000,,Then I'm going to have f of\Nx of r theta, y of r theta,
Dialogue: 0,0:21:06.48,0:21:09.87,Default,,0000,0000,0000,,everything expressed\Nin terms of r theta.
Dialogue: 0,0:21:09.87,0:21:13.99,Default,,0000,0000,0000,,And instead of\Nthe a-- so we just
Dialogue: 0,0:21:13.99,0:21:18.50,Default,,0000,0000,0000,,feed you this piece of\Ncake and say, believe it,
Dialogue: 0,0:21:18.50,0:21:21.93,Default,,0000,0000,0000,,believe it and leave us alone.
Dialogue: 0,0:21:21.93,0:21:22.43,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:21:22.43,0:21:26.78,Default,,0000,0000,0000,,That's what it does in\Nthe book in section 11.3.
Dialogue: 0,0:21:26.78,0:21:33.45,Default,,0000,0000,0000,,So without understanding why\Nyou have to-- instead of the r
Dialogue: 0,0:21:33.45,0:21:35.79,Default,,0000,0000,0000,,d theta and multiply it by an r.
Dialogue: 0,0:21:35.79,0:21:36.48,Default,,0000,0000,0000,,What is that?
Dialogue: 0,0:21:36.48,0:21:38.11,Default,,0000,0000,0000,,You don't know why you do that.
Dialogue: 0,0:21:38.11,0:21:40.52,Default,,0000,0000,0000,,And I thought, that's\Nthe way we thought it
Dialogue: 0,0:21:40.52,0:21:42.58,Default,,0000,0000,0000,,for way too many years.
Dialogue: 0,0:21:42.58,0:21:45.92,Default,,0000,0000,0000,,I'm sick and tired\Nof not explaining why
Dialogue: 0,0:21:45.92,0:21:50.74,Default,,0000,0000,0000,,you multiply that with an r.
Dialogue: 0,0:21:50.74,0:21:55.10,Default,,0000,0000,0000,,So I will tell you something\Nthat's quite interesting,
Dialogue: 0,0:21:55.10,0:21:58.29,Default,,0000,0000,0000,,and something that I learned\Nlate in graduate school.
Dialogue: 0,0:21:58.29,0:22:00.63,Default,,0000,0000,0000,,I was late already.
Dialogue: 0,0:22:00.63,0:22:05.79,Default,,0000,0000,0000,,I was in my 20s when I\Nstudied differential forms
Dialogue: 0,0:22:05.79,0:22:07.59,Default,,0000,0000,0000,,for the first time.
Dialogue: 0,0:22:07.59,0:22:12.88,Default,,0000,0000,0000,,And differential\Nforms have some sort
Dialogue: 0,0:22:12.88,0:22:25.06,Default,,0000,0000,0000,,of special wedge product, which\Nis very physical in nature.
Dialogue: 0,0:22:25.06,0:22:30.26,Default,,0000,0000,0000,,So if you love physics, you\Nwill understand more or less
Dialogue: 0,0:22:30.26,0:22:34.01,Default,,0000,0000,0000,,what I'm talking about.
Dialogue: 0,0:22:34.01,0:22:40.66,Default,,0000,0000,0000,,Imagine that you have two\Nvectors, vector a and vector b.
Dialogue: 0,0:22:40.66,0:22:44.17,Default,,0000,0000,0000,,
Dialogue: 0,0:22:44.17,0:22:48.25,Default,,0000,0000,0000,,For these vectors,\Nyou go, oh my God.
Dialogue: 0,0:22:48.25,0:22:54.09,Default,,0000,0000,0000,,If these would be vectors in\Na tangent plane to a surface,
Dialogue: 0,0:22:54.09,0:22:56.40,Default,,0000,0000,0000,,you think, some\Nof these would be
Dialogue: 0,0:22:56.40,0:22:59.98,Default,,0000,0000,0000,,tangent vectors to a surface.
Dialogue: 0,0:22:59.98,0:23:02.38,Default,,0000,0000,0000,,This is the tangent\Nplane and everything.
Dialogue: 0,0:23:02.38,0:23:07.03,Default,,0000,0000,0000,,You go, OK, if these\Nwere infinitesimally
Dialogue: 0,0:23:07.03,0:23:11.63,Default,,0000,0000,0000,,small displacements-- which they\Nare not, but assume they would
Dialogue: 0,0:23:11.63,0:23:19.49,Default,,0000,0000,0000,,be-- how would you do the area\Nof the infinitesimally small
Dialogue: 0,0:23:19.49,0:23:22.37,Default,,0000,0000,0000,,parallelogram that\Nthey have between them.
Dialogue: 0,0:23:22.37,0:23:31.17,Default,,0000,0000,0000,,This is actually the area\Nelement right here, ea.
Dialogue: 0,0:23:31.17,0:23:35.15,Default,,0000,0000,0000,,So instead of dx dy, you're\Nnot going to have dx dy,
Dialogue: 0,0:23:35.15,0:23:39.100,Default,,0000,0000,0000,,you're going to have some\Nsort of, I don't know,
Dialogue: 0,0:23:39.100,0:23:48.17,Default,,0000,0000,0000,,this is like a\Nd-something, d u, and this
Dialogue: 0,0:23:48.17,0:23:55.18,Default,,0000,0000,0000,,is a d v. And when I compute\Nthe area of the parallelogram,
Dialogue: 0,0:23:55.18,0:23:58.12,Default,,0000,0000,0000,,I consider these to\Nbe vectors, and I
Dialogue: 0,0:23:58.12,0:24:02.18,Default,,0000,0000,0000,,say, how did we get\Nit from the vectors
Dialogue: 0,0:24:02.18,0:24:06.26,Default,,0000,0000,0000,,to the area of\Nthe parallelogram?
Dialogue: 0,0:24:06.26,0:24:10.06,Default,,0000,0000,0000,,We took the vectors,\Nwe shook them off.
Dialogue: 0,0:24:10.06,0:24:19.32,Default,,0000,0000,0000,,We made a cross product\Nof them, and then we
Dialogue: 0,0:24:19.32,0:24:23.37,Default,,0000,0000,0000,,took the norm, the\Nmagnitude of that.
Dialogue: 0,0:24:23.37,0:24:26.81,Default,,0000,0000,0000,,Does this makes sense,\Ncompared to this parallelogram?
Dialogue: 0,0:24:26.81,0:24:27.31,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:24:27.31,0:24:30.55,Default,,0000,0000,0000,,Remember, guys, this\Nwas like, how big
Dialogue: 0,0:24:30.55,0:24:33.17,Default,,0000,0000,0000,,is du, a small\Ninfinitesimal displacement,
Dialogue: 0,0:24:33.17,0:24:36.34,Default,,0000,0000,0000,,but that would be like the\Nwidth, one of the dimensions.
Dialogue: 0,0:24:36.34,0:24:39.99,Default,,0000,0000,0000,,There's the other of the\Ndimension of the area element
Dialogue: 0,0:24:39.99,0:24:44.05,Default,,0000,0000,0000,,times-- this area element\Nis that tiny pixel that
Dialogue: 0,0:24:44.05,0:24:49.01,Default,,0000,0000,0000,,is sitting on the surface\Nin the tangent plane, yeah?
Dialogue: 0,0:24:49.01,0:24:54.22,Default,,0000,0000,0000,,Sine of the angle\Nbetween the guys.
Dialogue: 0,0:24:54.22,0:24:54.76,Default,,0000,0000,0000,,Oh, OK.
Dialogue: 0,0:24:54.76,0:25:00.76,Default,,0000,0000,0000,,So if the guys are not\Nperpendicular to one another,
Dialogue: 0,0:25:00.76,0:25:03.73,Default,,0000,0000,0000,,if the two displacements are not\Nperpendicular to one another,
Dialogue: 0,0:25:03.73,0:25:07.34,Default,,0000,0000,0000,,you still have to multiply\Nthe sine of theta.
Dialogue: 0,0:25:07.34,0:25:09.19,Default,,0000,0000,0000,,Otherwise you don't\Nget the element
Dialogue: 0,0:25:09.19,0:25:12.32,Default,,0000,0000,0000,,of the area of\Nthis parallelogram.
Dialogue: 0,0:25:12.32,0:25:17.53,Default,,0000,0000,0000,,So why did the Cartesian\Ncoordinates not pose a problem?
Dialogue: 0,0:25:17.53,0:25:19.56,Default,,0000,0000,0000,,For Cartesian\Ncoordinates, it's easy.
Dialogue: 0,0:25:19.56,0:25:22.62,Default,,0000,0000,0000,,
Dialogue: 0,0:25:22.62,0:25:23.49,Default,,0000,0000,0000,,It's a piece of cake.
Dialogue: 0,0:25:23.49,0:25:24.36,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:25:24.36,0:25:32.13,Default,,0000,0000,0000,,Because this is the x, this is\Nthe y, as little tiny measures
Dialogue: 0,0:25:32.13,0:25:33.33,Default,,0000,0000,0000,,multiplied.
Dialogue: 0,0:25:33.33,0:25:37.38,Default,,0000,0000,0000,,How much is sine of theta\Nbetween Cartesian coordinates?
Dialogue: 0,0:25:37.38,0:25:37.88,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,0:25:37.88,0:25:40.97,Default,,0000,0000,0000,,MAGDALENA TODA: It's 1,\Nbecause its 90 degrees.
Dialogue: 0,0:25:40.97,0:25:43.16,Default,,0000,0000,0000,,When they are\Northogonal coordinates,
Dialogue: 0,0:25:43.16,0:25:46.88,Default,,0000,0000,0000,,it's a piece of cake,\Nbecause you have 1 there,
Dialogue: 0,0:25:46.88,0:25:48.30,Default,,0000,0000,0000,,and then your life\Nbecomes easier.
Dialogue: 0,0:25:48.30,0:25:50.94,Default,,0000,0000,0000,,
Dialogue: 0,0:25:50.94,0:25:57.03,Default,,0000,0000,0000,,So in general, what\Nis the area limit?
Dialogue: 0,0:25:57.03,0:26:02.03,Default,,0000,0000,0000,,The area limit for\Narbitrary coordinates--
Dialogue: 0,0:26:02.03,0:26:17.02,Default,,0000,0000,0000,,So area limit for some\Narbitrary coordinates
Dialogue: 0,0:26:17.02,0:26:20.31,Default,,0000,0000,0000,,should be defined\Nas the sined area.
Dialogue: 0,0:26:20.31,0:26:29.32,Default,,0000,0000,0000,,
Dialogue: 0,0:26:29.32,0:26:32.19,Default,,0000,0000,0000,,And you say, what do you\Nmean that's a sined area,
Dialogue: 0,0:26:32.19,0:26:34.58,Default,,0000,0000,0000,,and why would you do that.?
Dialogue: 0,0:26:34.58,0:26:38.28,Default,,0000,0000,0000,,Well, it's not so\Nhard to understand.
Dialogue: 0,0:26:38.28,0:26:41.74,Default,,0000,0000,0000,,Imagine that you have a\Nconvention, and you say,
Dialogue: 0,0:26:41.74,0:26:54.81,Default,,0000,0000,0000,,OK, dx times dy equals\Nnegative dy times dx.
Dialogue: 0,0:26:54.81,0:26:56.92,Default,,0000,0000,0000,,And you say, what, what?
Dialogue: 0,0:26:56.92,0:27:00.52,Default,,0000,0000,0000,,If you change the\Norder of dx dy,
Dialogue: 0,0:27:00.52,0:27:06.60,Default,,0000,0000,0000,,this wedge stuff works exactly\Nlike the-- what is that called?
Dialogue: 0,0:27:06.60,0:27:07.79,Default,,0000,0000,0000,,Cross product.
Dialogue: 0,0:27:07.79,0:27:13.15,Default,,0000,0000,0000,,So the wedge works just\Nlike the cross product.
Dialogue: 0,0:27:13.15,0:27:17.51,Default,,0000,0000,0000,,Just like the cross product.
Dialogue: 0,0:27:17.51,0:27:23.32,Default,,0000,0000,0000,,In some other ways, suppose\Nthat I am here, right?
Dialogue: 0,0:27:23.32,0:27:27.72,Default,,0000,0000,0000,,And this is a vector, like an\Ninfinitesimal displacement,
Dialogue: 0,0:27:27.72,0:27:29.37,Default,,0000,0000,0000,,and that's the other one.
Dialogue: 0,0:27:29.37,0:27:33.80,Default,,0000,0000,0000,,If I multiply them\None after the other,
Dialogue: 0,0:27:33.80,0:27:38.06,Default,,0000,0000,0000,,and I use this strange wedge\N[INTERPOSING VOICES] the area,
Dialogue: 0,0:27:38.06,0:27:40.97,Default,,0000,0000,0000,,I'm going to have an orientation\Nfor that tangent line,
Dialogue: 0,0:27:40.97,0:27:46.39,Default,,0000,0000,0000,,and it's going to go\Nup, the orientation.
Dialogue: 0,0:27:46.39,0:27:48.33,Default,,0000,0000,0000,,The orientation is important.
Dialogue: 0,0:27:48.33,0:27:50.99,Default,,0000,0000,0000,,But if dx dy and\NI switched them,
Dialogue: 0,0:27:50.99,0:27:56.05,Default,,0000,0000,0000,,I said, dy, swap with dx,\Nwhat's going to happen?
Dialogue: 0,0:27:56.05,0:28:01.53,Default,,0000,0000,0000,,I have to change to\Nchange to clockwise.
Dialogue: 0,0:28:01.53,0:28:03.61,Default,,0000,0000,0000,,And then the\Norientation goes down.
Dialogue: 0,0:28:03.61,0:28:06.72,Default,,0000,0000,0000,,And that's what they use\Nin mechanics when it comes
Dialogue: 0,0:28:06.72,0:28:09.13,Default,,0000,0000,0000,,to the normal to the surface.
Dialogue: 0,0:28:09.13,0:28:12.77,Default,,0000,0000,0000,,So again, you guys remember,\Nwe had 2 vector products,
Dialogue: 0,0:28:12.77,0:28:16.37,Default,,0000,0000,0000,,and we did the cross product,\Nand we got the normal.
Dialogue: 0,0:28:16.37,0:28:18.80,Default,,0000,0000,0000,,If it's from this\None to this one,
Dialogue: 0,0:28:18.80,0:28:20.74,Default,,0000,0000,0000,,it's counterclockwise\Nand goes up,
Dialogue: 0,0:28:20.74,0:28:23.86,Default,,0000,0000,0000,,but if it's from this\Nvector to this other vector,
Dialogue: 0,0:28:23.86,0:28:26.99,Default,,0000,0000,0000,,it's clockwise and goes down.
Dialogue: 0,0:28:26.99,0:28:29.82,Default,,0000,0000,0000,,This is how a\Nmechanical engineer
Dialogue: 0,0:28:29.82,0:28:32.82,Default,,0000,0000,0000,,will know how the\Nsurface is oriented
Dialogue: 0,0:28:32.82,0:28:35.73,Default,,0000,0000,0000,,based on the partial\Nvelocities, for example
Dialogue: 0,0:28:35.73,0:28:39.37,Default,,0000,0000,0000,,He has the partial\Nvelocities along a surface,
Dialogue: 0,0:28:39.37,0:28:42.75,Default,,0000,0000,0000,,and somebody says, take the\Nnormal, take the unit normal.
Dialogue: 0,0:28:42.75,0:28:44.76,Default,,0000,0000,0000,,He goes, like, are\Nyou a physicist?
Dialogue: 0,0:28:44.76,0:28:46.44,Default,,0000,0000,0000,,No, I'm an engineer.
Dialogue: 0,0:28:46.44,0:28:48.79,Default,,0000,0000,0000,,You don't know how\Nto take the normal.
Dialogue: 0,0:28:48.79,0:28:50.11,Default,,0000,0000,0000,,And of course, he knows.
Dialogue: 0,0:28:50.11,0:28:53.50,Default,,0000,0000,0000,,He knows the convention\Nby this right-hand rule,
Dialogue: 0,0:28:53.50,0:28:55.19,Default,,0000,0000,0000,,whatever you guys call it.
Dialogue: 0,0:28:55.19,0:28:57.26,Default,,0000,0000,0000,,I call it the faucet rule.
Dialogue: 0,0:28:57.26,0:29:01.40,Default,,0000,0000,0000,,It goes like this,\Nor it goes like that.
Dialogue: 0,0:29:01.40,0:29:04.27,Default,,0000,0000,0000,,It's the same for a faucet,\Nfor any type of screw,
Dialogue: 0,0:29:04.27,0:29:08.35,Default,,0000,0000,0000,,for the right-hand\Nrule, whatever.
Dialogue: 0,0:29:08.35,0:29:11.36,Default,,0000,0000,0000,,What else do you have\Nto believe me are true?
Dialogue: 0,0:29:11.36,0:29:14.56,Default,,0000,0000,0000,,dx wedge dx is 0.
Dialogue: 0,0:29:14.56,0:29:17.74,Default,,0000,0000,0000,,Can somebody tell me why\Nthat is natural to introduce
Dialogue: 0,0:29:17.74,0:29:19.49,Default,,0000,0000,0000,,such a wedge product?
Dialogue: 0,0:29:19.49,0:29:22.36,Default,,0000,0000,0000,,STUDENT: Because the sine of\Nthe angle between those is 0.
Dialogue: 0,0:29:22.36,0:29:23.28,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,0:29:23.28,0:29:28.66,Default,,0000,0000,0000,,Once you flatten this, once\Nyou flatten the parallelogram,
Dialogue: 0,0:29:28.66,0:29:29.83,Default,,0000,0000,0000,,there is no area.
Dialogue: 0,0:29:29.83,0:29:31.47,Default,,0000,0000,0000,,So the area is 0.
Dialogue: 0,0:29:31.47,0:29:34.96,Default,,0000,0000,0000,,How about dy dy sined area?
Dialogue: 0,0:29:34.96,0:29:35.93,Default,,0000,0000,0000,,0.
Dialogue: 0,0:29:35.93,0:29:37.81,Default,,0000,0000,0000,,So these are all\Nthe properties you
Dialogue: 0,0:29:37.81,0:29:41.44,Default,,0000,0000,0000,,need to know of the\Nsine area, sined areas.
Dialogue: 0,0:29:41.44,0:29:44.27,Default,,0000,0000,0000,,
Dialogue: 0,0:29:44.27,0:29:46.53,Default,,0000,0000,0000,,OK, so now let's\Nsee what happens
Dialogue: 0,0:29:46.53,0:29:51.15,Default,,0000,0000,0000,,if we take this element,\Nwhich is a differential,
Dialogue: 0,0:29:51.15,0:29:55.35,Default,,0000,0000,0000,,and wedge it with this element,\Nwhich is also a differential.
Dialogue: 0,0:29:55.35,0:29:56.37,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:29:56.37,0:29:59.92,Default,,0000,0000,0000,,Oh my God, I'm shaking\Nonly thinking about it.
Dialogue: 0,0:29:59.92,0:30:01.86,Default,,0000,0000,0000,,I'm going to get\Nsomething weird.
Dialogue: 0,0:30:01.86,0:30:04.07,Default,,0000,0000,0000,,But I mean, mad weird.
Dialogue: 0,0:30:04.07,0:30:06.66,Default,,0000,0000,0000,,Let's see what happens.
Dialogue: 0,0:30:06.66,0:30:13.96,Default,,0000,0000,0000,,dx wedge dy equals-- do\Nyou guys have questions?
Dialogue: 0,0:30:13.96,0:30:18.33,Default,,0000,0000,0000,,Let's see what the mechanics are\Nfor this type of computation.
Dialogue: 0,0:30:18.33,0:30:21.25,Default,,0000,0000,0000,,
Dialogue: 0,0:30:21.25,0:30:27.69,Default,,0000,0000,0000,,I go-- this is like\Na-- displacement wedge
Dialogue: 0,0:30:27.69,0:30:29.59,Default,,0000,0000,0000,,this other displacement.
Dialogue: 0,0:30:29.59,0:30:32.74,Default,,0000,0000,0000,,
Dialogue: 0,0:30:32.74,0:30:36.11,Default,,0000,0000,0000,,Think of them as true\Nvector displacements,
Dialogue: 0,0:30:36.11,0:30:41.15,Default,,0000,0000,0000,,and as if you had a cross\Nproduct, or something.
Dialogue: 0,0:30:41.15,0:30:42.08,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:30:42.08,0:30:43.66,Default,,0000,0000,0000,,How does this go?
Dialogue: 0,0:30:43.66,0:30:44.82,Default,,0000,0000,0000,,It's distributed.
Dialogue: 0,0:30:44.82,0:30:47.77,Default,,0000,0000,0000,,It's linear functions,\Nbecause we've
Dialogue: 0,0:30:47.77,0:30:51.14,Default,,0000,0000,0000,,studied the\Nproperties of vectors,
Dialogue: 0,0:30:51.14,0:30:52.74,Default,,0000,0000,0000,,this acts by linearity.
Dialogue: 0,0:30:52.74,0:30:58.18,Default,,0000,0000,0000,,So you go and say, first\Nfirst, times plus first times
Dialogue: 0,0:30:58.18,0:31:02.64,Default,,0000,0000,0000,,second-- and times is\Nthis guy, this weirdo--
Dialogue: 0,0:31:02.64,0:31:06.94,Default,,0000,0000,0000,,plus second times first,\Nplus second times second,
Dialogue: 0,0:31:06.94,0:31:09.20,Default,,0000,0000,0000,,where the wedge is\Nthe operator that
Dialogue: 0,0:31:09.20,0:31:11.28,Default,,0000,0000,0000,,has to satisfy these functions.
Dialogue: 0,0:31:11.28,0:31:14.06,Default,,0000,0000,0000,,It's similar to\Nthe cross product.
Dialogue: 0,0:31:14.06,0:31:15.19,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:31:15.19,0:31:21.37,Default,,0000,0000,0000,,Then let's go x\Nsub r, y sub r, dr
Dialogue: 0,0:31:21.37,0:31:26.88,Default,,0000,0000,0000,,wedge dr. Oh, let's 0 go away.
Dialogue: 0,0:31:26.88,0:31:30.34,Default,,0000,0000,0000,,I say, leave me alone,\Nyou're making my life hard.
Dialogue: 0,0:31:30.34,0:31:37.69,Default,,0000,0000,0000,,Then I go plus x sub r--\Nthis is a small function.
Dialogue: 0,0:31:37.69,0:31:40.52,Default,,0000,0000,0000,,y sub theta, another\Nsmall function.
Dialogue: 0,0:31:40.52,0:31:44.05,Default,,0000,0000,0000,,What of this\Ndisplacement, dr d theta.
Dialogue: 0,0:31:44.05,0:31:46.65,Default,,0000,0000,0000,,I'm like those d\Nsomething, d something,
Dialogue: 0,0:31:46.65,0:31:49.35,Default,,0000,0000,0000,,two small displacements\Nin the cross product.
Dialogue: 0,0:31:49.35,0:31:52.62,Default,,0000,0000,0000,,OK, plus.
Dialogue: 0,0:31:52.62,0:31:55.27,Default,,0000,0000,0000,,Who is telling me what next?
Dialogue: 0,0:31:55.27,0:31:56.02,Default,,0000,0000,0000,,STUDENT: x theta--
Dialogue: 0,0:31:56.02,0:32:05.55,Default,,0000,0000,0000,,MAGDALENA TODA: x theta\Nyr, d theta dr. Is it fair?
Dialogue: 0,0:32:05.55,0:32:10.20,Default,,0000,0000,0000,,I did the second guy from the\Nfirst one with the first guy
Dialogue: 0,0:32:10.20,0:32:11.79,Default,,0000,0000,0000,,from the second one.
Dialogue: 0,0:32:11.79,0:32:14.72,Default,,0000,0000,0000,,And finally, I'm too\Nlazy to write it down.
Dialogue: 0,0:32:14.72,0:32:15.96,Default,,0000,0000,0000,,What do I get?
Dialogue: 0,0:32:15.96,0:32:16.86,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:32:16.86,0:32:17.02,Default,,0000,0000,0000,,MAGDALENA TODA: 0.
Dialogue: 0,0:32:17.02,0:32:17.68,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:32:17.68,0:32:20.07,Default,,0000,0000,0000,,Because d theta,\Nalways d theta is 0.
Dialogue: 0,0:32:20.07,0:32:27.16,Default,,0000,0000,0000,,It's like you are flattening--\Nthere is no more parallelogram.
Dialogue: 0,0:32:27.16,0:32:27.94,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:32:27.94,0:32:32.33,Default,,0000,0000,0000,,So the two dimensions of\Nthe parallelogram become 0.
Dialogue: 0,0:32:32.33,0:32:37.07,Default,,0000,0000,0000,,The parallelogram would\Nbecome [? a secant. ?]
Dialogue: 0,0:32:37.07,0:32:39.93,Default,,0000,0000,0000,,What you get is\Nsomething really weak.
Dialogue: 0,0:32:39.93,0:32:42.21,Default,,0000,0000,0000,,And we don't talk\Nabout it in the book,
Dialogue: 0,0:32:42.21,0:32:45.02,Default,,0000,0000,0000,,but that's called the Jacobian.
Dialogue: 0,0:32:45.02,0:32:51.15,Default,,0000,0000,0000,,dr d theta and d theta dr, once\Nyou introduce the sine area,
Dialogue: 0,0:32:51.15,0:32:55.92,Default,,0000,0000,0000,,you finally understand\Nwhy you get this r here,
Dialogue: 0,0:32:55.92,0:32:57.74,Default,,0000,0000,0000,,what the Jacobian is.
Dialogue: 0,0:32:57.74,0:32:59.37,Default,,0000,0000,0000,,If you don't introduce\Nthe sine area,
Dialogue: 0,0:32:59.37,0:33:02.34,Default,,0000,0000,0000,,you will never understand,\Nand you cannot explain it
Dialogue: 0,0:33:02.34,0:33:06.14,Default,,0000,0000,0000,,to anybody, any student have.
Dialogue: 0,0:33:06.14,0:33:11.53,Default,,0000,0000,0000,,OK, so this guy, d theta,\Nwhich the r is just
Dialogue: 0,0:33:11.53,0:33:13.69,Default,,0000,0000,0000,,swapping the two displacements.
Dialogue: 0,0:33:13.69,0:33:16.99,Default,,0000,0000,0000,,So it's going to be\Nminus dr d theta.
Dialogue: 0,0:33:16.99,0:33:18.67,Default,,0000,0000,0000,,Why is that, guys?
Dialogue: 0,0:33:18.67,0:33:23.03,Default,,0000,0000,0000,,Because that's how I said, every\Ntime I swap two displacements,
Dialogue: 0,0:33:23.03,0:33:25.44,Default,,0000,0000,0000,,I'm changing the orientation.
Dialogue: 0,0:33:25.44,0:33:28.08,Default,,0000,0000,0000,,It's like the cross\Nproduct between a and b,
Dialogue: 0,0:33:28.08,0:33:30.08,Default,,0000,0000,0000,,and the cross product\Nbetween b and a.
Dialogue: 0,0:33:30.08,0:33:34.52,Default,,0000,0000,0000,,So I'm going up or I'm going\Ndown, I'm changing orientation.
Dialogue: 0,0:33:34.52,0:33:35.90,Default,,0000,0000,0000,,What's left in the end?
Dialogue: 0,0:33:35.90,0:33:39.00,Default,,0000,0000,0000,,It's really just this\Nguy that's really weird.
Dialogue: 0,0:33:39.00,0:33:41.29,Default,,0000,0000,0000,,I'm going to collect the terms.
Dialogue: 0,0:33:41.29,0:33:44.93,Default,,0000,0000,0000,,One from here, one\Nfrom here, and a minus.
Dialogue: 0,0:33:44.93,0:33:45.43,Default,,0000,0000,0000,,Go ahead.
Dialogue: 0,0:33:45.43,0:33:49.03,Default,,0000,0000,0000,,STUDENT: Do the wedges\Njust cancel out?
Dialogue: 0,0:33:49.03,0:33:50.35,Default,,0000,0000,0000,,MAGDALENA TODA: This was 0.
Dialogue: 0,0:33:50.35,0:33:52.35,Default,,0000,0000,0000,,This was 0.
Dialogue: 0,0:33:52.35,0:33:57.58,Default,,0000,0000,0000,,And this dr d theta is nonzero,\Nbut is the common factor.
Dialogue: 0,0:33:57.58,0:34:00.02,Default,,0000,0000,0000,,So I pull him out from here.
Dialogue: 0,0:34:00.02,0:34:01.89,Default,,0000,0000,0000,,I pull him out from here.
Dialogue: 0,0:34:01.89,0:34:02.39,Default,,0000,0000,0000,,Out.
Dialogue: 0,0:34:02.39,0:34:08.47,Default,,0000,0000,0000,,Factor out, and what's\Nleft is this guy over here
Dialogue: 0,0:34:08.47,0:34:10.56,Default,,0000,0000,0000,,who is this guy over here.
Dialogue: 0,0:34:10.56,0:34:14.58,Default,,0000,0000,0000,,And this guy over\Nhere with a minus
Dialogue: 0,0:34:14.58,0:34:20.32,Default,,0000,0000,0000,,who gives me minus d theta yr.
Dialogue: 0,0:34:20.32,0:34:21.00,Default,,0000,0000,0000,,That's all.
Dialogue: 0,0:34:21.00,0:34:25.27,Default,,0000,0000,0000,,So now you will understand\Nwhy I am going to get r.
Dialogue: 0,0:34:25.27,0:34:30.44,Default,,0000,0000,0000,,So the general rule will\Nbe that the area element dx
Dialogue: 0,0:34:30.44,0:34:35.86,Default,,0000,0000,0000,,dy, the wedge sined\Narea, will be--
Dialogue: 0,0:34:35.86,0:34:39.21,Default,,0000,0000,0000,,you have to help me\Nwith this individual,
Dialogue: 0,0:34:39.21,0:34:42.99,Default,,0000,0000,0000,,because he really looks weird.
Dialogue: 0,0:34:42.99,0:34:46.48,Default,,0000,0000,0000,,Do you know of a name for it?
Dialogue: 0,0:34:46.48,0:34:49.91,Default,,0000,0000,0000,,Do you know what\Nthis is going to be?
Dialogue: 0,0:34:49.91,0:34:52.40,Default,,0000,0000,0000,,Linear algebra people,\Nonly two of you.
Dialogue: 0,0:34:52.40,0:34:56.65,Default,,0000,0000,0000,,Maybe you have an idea.
Dialogue: 0,0:34:56.65,0:34:59.95,Default,,0000,0000,0000,,So it's like, I\Ntake this fellow,
Dialogue: 0,0:34:59.95,0:35:01.82,Default,,0000,0000,0000,,and I multiply by that fellow.
Dialogue: 0,0:35:01.82,0:35:04.50,Default,,0000,0000,0000,,
Dialogue: 0,0:35:04.50,0:35:06.55,Default,,0000,0000,0000,,Multiply these two.
Dialogue: 0,0:35:06.55,0:35:12.97,Default,,0000,0000,0000,,And I go minus this\Nfellow times that fellow.
Dialogue: 0,0:35:12.97,0:35:14.82,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:35:14.82,0:35:17.59,Default,,0000,0000,0000,,MAGDALENA TODA: It's like\Na determinant of something.
Dialogue: 0,0:35:17.59,0:35:23.38,Default,,0000,0000,0000,,So when people write\Nthe differential system,
Dialogue: 0,0:35:23.38,0:35:26.46,Default,,0000,0000,0000,,[INTERPOSING VOICES]\N51, you will understand
Dialogue: 0,0:35:26.46,0:35:27.94,Default,,0000,0000,0000,,that this is a system.
Dialogue: 0,0:35:27.94,0:35:28.44,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:35:28.44,0:35:29.94,Default,,0000,0000,0000,,It's a system of two equations.
Dialogue: 0,0:35:29.94,0:35:32.16,Default,,0000,0000,0000,,
Dialogue: 0,0:35:32.16,0:35:34.03,Default,,0000,0000,0000,,The other little, like,\Nvector displacements,
Dialogue: 0,0:35:34.03,0:35:36.37,Default,,0000,0000,0000,,you are going to\Nwrite it like that.
Dialogue: 0,0:35:36.37,0:35:45.95,Default,,0000,0000,0000,,dx dy will be matrix\Nmultiplication dr d theta.
Dialogue: 0,0:35:45.95,0:35:50.21,Default,,0000,0000,0000,,And how do you multiply\Nx sub r x sub theta?
Dialogue: 0,0:35:50.21,0:35:55.19,Default,,0000,0000,0000,,So you go first row times\Nfirst column give you that.
Dialogue: 0,0:35:55.19,0:35:59.51,Default,,0000,0000,0000,,And second row times the\Ncolumn gives you this.
Dialogue: 0,0:35:59.51,0:36:02.15,Default,,0000,0000,0000,,y sub r, y sub theta.
Dialogue: 0,0:36:02.15,0:36:06.34,Default,,0000,0000,0000,,This is a magic guy\Ncalled Jacobian.
Dialogue: 0,0:36:06.34,0:36:09.88,Default,,0000,0000,0000,,We keep this a secret, and\Nmost Professors don't even
Dialogue: 0,0:36:09.88,0:36:13.05,Default,,0000,0000,0000,,cover 12.8, because\Nthey don't want to tell
Dialogue: 0,0:36:13.05,0:36:15.06,Default,,0000,0000,0000,,people what a Jacobian is.
Dialogue: 0,0:36:15.06,0:36:16.89,Default,,0000,0000,0000,,This is little r.
Dialogue: 0,0:36:16.89,0:36:20.86,Default,,0000,0000,0000,,I know you don't believe me, but\Nthe determinant of this matrix
Dialogue: 0,0:36:20.86,0:36:22.52,Default,,0000,0000,0000,,must be little r.
Dialogue: 0,0:36:22.52,0:36:24.91,Default,,0000,0000,0000,,You have to help me prove that.
Dialogue: 0,0:36:24.91,0:36:27.34,Default,,0000,0000,0000,,And this is the Jacobian.
Dialogue: 0,0:36:27.34,0:36:30.38,Default,,0000,0000,0000,,Do you guys know why\Nit's called Jacobian?
Dialogue: 0,0:36:30.38,0:36:33.36,Default,,0000,0000,0000,,It's the determinant\Nof this matrix.
Dialogue: 0,0:36:33.36,0:36:43.26,Default,,0000,0000,0000,,Let's call this\Nmatrix J. And this
Dialogue: 0,0:36:43.26,0:36:49.21,Default,,0000,0000,0000,,is J, determinant\Nof [? scripture. ?]
Dialogue: 0,0:36:49.21,0:36:50.48,Default,,0000,0000,0000,,This is called Jacobian.
Dialogue: 0,0:36:50.48,0:36:54.16,Default,,0000,0000,0000,,
Dialogue: 0,0:36:54.16,0:36:55.08,Default,,0000,0000,0000,,Why is it r?
Dialogue: 0,0:36:55.08,0:36:57.85,Default,,0000,0000,0000,,Let's-- I don't know.
Dialogue: 0,0:36:57.85,0:36:59.71,Default,,0000,0000,0000,,Let's see how we do it.
Dialogue: 0,0:36:59.71,0:37:03.54,Default,,0000,0000,0000,,
Dialogue: 0,0:37:03.54,0:37:06.90,Default,,0000,0000,0000,,This is r cosine theta, right?
Dialogue: 0,0:37:06.90,0:37:09.79,Default,,0000,0000,0000,,This is r sine theta.
Dialogue: 0,0:37:09.79,0:37:14.79,Default,,0000,0000,0000,,So dx must be what x sub r?
Dialogue: 0,0:37:14.79,0:37:19.73,Default,,0000,0000,0000,,X sub r, x sub r, cosine theta.
Dialogue: 0,0:37:19.73,0:37:21.75,Default,,0000,0000,0000,,d plus.
Dialogue: 0,0:37:21.75,0:37:23.57,Default,,0000,0000,0000,,What is x sub t?
Dialogue: 0,0:37:23.57,0:37:26.38,Default,,0000,0000,0000,,
Dialogue: 0,0:37:26.38,0:37:28.55,Default,,0000,0000,0000,,x sub theta.
Dialogue: 0,0:37:28.55,0:37:31.60,Default,,0000,0000,0000,,I need to differentiate\Nthis with respect to theta.
Dialogue: 0,0:37:31.60,0:37:33.60,Default,,0000,0000,0000,,STUDENT: It's going to\Nbe negative r sine theta.
Dialogue: 0,0:37:33.60,0:37:36.39,Default,,0000,0000,0000,,MAGDALENA TODA: Minus r\Nsine theta, very good.
Dialogue: 0,0:37:36.39,0:37:38.09,Default,,0000,0000,0000,,And d theta.
Dialogue: 0,0:37:38.09,0:37:44.20,Default,,0000,0000,0000,,Then I go dy was\Nsine theta-- dr,
Dialogue: 0,0:37:44.20,0:37:46.45,Default,,0000,0000,0000,,I'm looking at these\Nequations, and I'm
Dialogue: 0,0:37:46.45,0:37:49.02,Default,,0000,0000,0000,,repeating them for my case.
Dialogue: 0,0:37:49.02,0:37:52.89,Default,,0000,0000,0000,,This is true in general for\Nany kind of coordinates.
Dialogue: 0,0:37:52.89,0:37:56.64,Default,,0000,0000,0000,,So it's a general equation\Nfor any kind of coordinate,
Dialogue: 0,0:37:56.64,0:37:58.83,Default,,0000,0000,0000,,two coordinates,\Ntwo coordinates,
Dialogue: 0,0:37:58.83,0:38:00.63,Default,,0000,0000,0000,,any kind of\Ncoordinates in plane,
Dialogue: 0,0:38:00.63,0:38:04.94,Default,,0000,0000,0000,,you can choose any\Nfunctions, f of uv, g of uv,
Dialogue: 0,0:38:04.94,0:38:06.60,Default,,0000,0000,0000,,whatever you want.
Dialogue: 0,0:38:06.60,0:38:09.46,Default,,0000,0000,0000,,But for this particular\Ncase of polar coordinates
Dialogue: 0,0:38:09.46,0:38:12.27,Default,,0000,0000,0000,,is going to look really\Npretty in the end.
Dialogue: 0,0:38:12.27,0:38:15.61,Default,,0000,0000,0000,,What do I get when I do y theta?
Dialogue: 0,0:38:15.61,0:38:17.48,Default,,0000,0000,0000,,r cosine theta.
Dialogue: 0,0:38:17.48,0:38:18.83,Default,,0000,0000,0000,,Am I right, guys?
Dialogue: 0,0:38:18.83,0:38:20.73,Default,,0000,0000,0000,,Keen an eye on it.
Dialogue: 0,0:38:20.73,0:38:27.28,Default,,0000,0000,0000,,So this will become-- the\Narea element will become what?
Dialogue: 0,0:38:27.28,0:38:31.31,Default,,0000,0000,0000,,The determinant of this matrix.
Dialogue: 0,0:38:31.31,0:38:34.57,Default,,0000,0000,0000,,Red, red, red, red.
Dialogue: 0,0:38:34.57,0:38:35.89,Default,,0000,0000,0000,,How do I compute a term?
Dialogue: 0,0:38:35.89,0:38:39.41,Default,,0000,0000,0000,,Not everybody knows,\Nand it's this times
Dialogue: 0,0:38:39.41,0:38:45.15,Default,,0000,0000,0000,,that minus this times that.
Dialogue: 0,0:38:45.15,0:38:46.37,Default,,0000,0000,0000,,OK, let's do that.
Dialogue: 0,0:38:46.37,0:38:53.44,Default,,0000,0000,0000,,So I get r cosine squared\Ntheta minus minus plus r sine
Dialogue: 0,0:38:53.44,0:38:56.49,Default,,0000,0000,0000,,squared theta.
Dialogue: 0,0:38:56.49,0:38:58.87,Default,,0000,0000,0000,,dr, d theta, and our wedge.
Dialogue: 0,0:38:58.87,0:39:00.00,Default,,0000,0000,0000,,What is this?
Dialogue: 0,0:39:00.00,0:39:00.60,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,0:39:00.60,0:39:04.32,Default,,0000,0000,0000,,MAGDALENA TODA:\NJacobian is r times 1,
Dialogue: 0,0:39:04.32,0:39:07.43,Default,,0000,0000,0000,,because that's the\NPythagorean theorem, right?
Dialogue: 0,0:39:07.43,0:39:12.33,Default,,0000,0000,0000,,So we have r, and this is\Nthe meaning of r, here.
Dialogue: 0,0:39:12.33,0:39:16.83,Default,,0000,0000,0000,,So when I moved from dx dy,\NI originally had the wedge
Dialogue: 0,0:39:16.83,0:39:19.47,Default,,0000,0000,0000,,that I didn't tell you about.
Dialogue: 0,0:39:19.47,0:39:23.09,Default,,0000,0000,0000,,And this wedge\Nbecomes r dr d theta,
Dialogue: 0,0:39:23.09,0:39:27.29,Default,,0000,0000,0000,,and that's the\Ncorrect way to explain
Dialogue: 0,0:39:27.29,0:39:29.89,Default,,0000,0000,0000,,why you get the Jacobian there.
Dialogue: 0,0:39:29.89,0:39:31.50,Default,,0000,0000,0000,,We don't do that in the book.
Dialogue: 0,0:39:31.50,0:39:34.86,Default,,0000,0000,0000,,We do it later, and we\Nsort of smuggle through.
Dialogue: 0,0:39:34.86,0:39:37.10,Default,,0000,0000,0000,,We don't do a very thorough job.
Dialogue: 0,0:39:37.10,0:39:39.98,Default,,0000,0000,0000,,When you go into\Nadvanced calculus,
Dialogue: 0,0:39:39.98,0:39:43.24,Default,,0000,0000,0000,,you would see that again the\Nway I explained it to you.
Dialogue: 0,0:39:43.24,0:39:47.37,Default,,0000,0000,0000,,If you ever want to\Ngo to graduate school,
Dialogue: 0,0:39:47.37,0:39:52.44,Default,,0000,0000,0000,,then you need to take the\NAdvanced Calculus I, 4350
Dialogue: 0,0:39:52.44,0:39:57.53,Default,,0000,0000,0000,,and 4351 where you are\Ngoing to learn about this.
Dialogue: 0,0:39:57.53,0:40:01.21,Default,,0000,0000,0000,,If you take those as a math\Nmajor or engineering major,
Dialogue: 0,0:40:01.21,0:40:01.96,Default,,0000,0000,0000,,it doesn't matter.
Dialogue: 0,0:40:01.96,0:40:03.92,Default,,0000,0000,0000,,When you go to\Ngraduate school, they
Dialogue: 0,0:40:03.92,0:40:07.47,Default,,0000,0000,0000,,don't make you take\Nadvanced calculus again
Dialogue: 0,0:40:07.47,0:40:09.38,Default,,0000,0000,0000,,at graduate school.
Dialogue: 0,0:40:09.38,0:40:12.74,Default,,0000,0000,0000,,So it's somewhere borderline\Nbetween senior year
Dialogue: 0,0:40:12.74,0:40:19.01,Default,,0000,0000,0000,,and graduate school, it's like\Nthe first course you would take
Dialogue: 0,0:40:19.01,0:40:22.02,Default,,0000,0000,0000,,in graduate school, for many.
Dialogue: 0,0:40:22.02,0:40:22.67,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:40:22.67,0:40:29.89,Default,,0000,0000,0000,,So an example of\Nthis transformation
Dialogue: 0,0:40:29.89,0:40:33.27,Default,,0000,0000,0000,,where we know what\Nwe are talking about.
Dialogue: 0,0:40:33.27,0:40:39.13,Default,,0000,0000,0000,,Let's say I have\Na picture, and I
Dialogue: 0,0:40:39.13,0:40:42.73,Default,,0000,0000,0000,,have a domain D, which\Nis-- this is x squared
Dialogue: 0,0:40:42.73,0:40:44.95,Default,,0000,0000,0000,,plus y squared equals 1.
Dialogue: 0,0:40:44.95,0:40:48.37,Default,,0000,0000,0000,,I have the domain as being\N[INTERPOSING VOICES].
Dialogue: 0,0:40:48.37,0:40:51.80,Default,,0000,0000,0000,,
Dialogue: 0,0:40:51.80,0:40:58.12,Default,,0000,0000,0000,,And then I say, I would\Nlike-- what would I like?
Dialogue: 0,0:40:58.12,0:41:04.29,Default,,0000,0000,0000,,I would like the\Nvolume of the-- this
Dialogue: 0,0:41:04.29,0:41:10.22,Default,,0000,0000,0000,,is a paraboloid, z equals\Nx squared plus y squared.
Dialogue: 0,0:41:10.22,0:41:12.62,Default,,0000,0000,0000,,I would like the\Nvolume of this object.
Dialogue: 0,0:41:12.62,0:41:13.82,Default,,0000,0000,0000,,This is my obsession.
Dialogue: 0,0:41:13.82,0:41:17.58,Default,,0000,0000,0000,,I'm going to create a\Nvase some day like that.
Dialogue: 0,0:41:17.58,0:41:22.56,Default,,0000,0000,0000,,So you want this\Npiece to be a solid.
Dialogue: 0,0:41:22.56,0:41:25.42,Default,,0000,0000,0000,,In cross section,\Nit will just this.
Dialogue: 0,0:41:25.42,0:41:26.25,Default,,0000,0000,0000,,In cross section.
Dialogue: 0,0:41:26.25,0:41:27.83,Default,,0000,0000,0000,,And it's a solid of revolution.
Dialogue: 0,0:41:27.83,0:41:30.30,Default,,0000,0000,0000,,In this cross section,\Nyou have to imagine
Dialogue: 0,0:41:30.30,0:41:36.10,Default,,0000,0000,0000,,revolving it around the z-axis,\Nthen creating a heavy object.
Dialogue: 0,0:41:36.10,0:41:38.44,Default,,0000,0000,0000,,From the outside, don't\Nsee what's inside.
Dialogue: 0,0:41:38.44,0:41:39.53,Default,,0000,0000,0000,,It looks like a cylinder.
Dialogue: 0,0:41:39.53,0:41:42.46,Default,,0000,0000,0000,,But you go inside and\Nyou see the valley.
Dialogue: 0,0:41:42.46,0:41:46.26,Default,,0000,0000,0000,,So it's between a\Nparaboloid and a disc,
Dialogue: 0,0:41:46.26,0:41:48.46,Default,,0000,0000,0000,,a unit disc on the floor.
Dialogue: 0,0:41:48.46,0:41:51.40,Default,,0000,0000,0000,,How are we going\Nto try and do that?
Dialogue: 0,0:41:51.40,0:41:53.96,Default,,0000,0000,0000,,And what did I\Nteach you last time?
Dialogue: 0,0:41:53.96,0:42:02.02,Default,,0000,0000,0000,,Last time, I taught you that--\Nwe have to go over a domain D.
Dialogue: 0,0:42:02.02,0:42:04.19,Default,,0000,0000,0000,,But that domain\ND, unfortunately,
Dialogue: 0,0:42:04.19,0:42:05.78,Default,,0000,0000,0000,,is hard to express.
Dialogue: 0,0:42:05.78,0:42:09.22,Default,,0000,0000,0000,,How would you express D\Nin Cartesian coordinates?
Dialogue: 0,0:42:09.22,0:42:14.63,Default,,0000,0000,0000,,
Dialogue: 0,0:42:14.63,0:42:15.84,Default,,0000,0000,0000,,You can do it.
Dialogue: 0,0:42:15.84,0:42:18.77,Default,,0000,0000,0000,,It's going to be a headache.
Dialogue: 0,0:42:18.77,0:42:22.27,Default,,0000,0000,0000,,x is between minus 1 and 1.
Dialogue: 0,0:42:22.27,0:42:23.77,Default,,0000,0000,0000,,Am I right, guys?
Dialogue: 0,0:42:23.77,0:42:28.27,Default,,0000,0000,0000,,And y will be between--\Nnow I have two branches.
Dialogue: 0,0:42:28.27,0:42:30.23,Default,,0000,0000,0000,,One, and the other one.
Dialogue: 0,0:42:30.23,0:42:33.10,Default,,0000,0000,0000,,One branch would be square--\NI hate square roots.
Dialogue: 0,0:42:33.10,0:42:36.25,Default,,0000,0000,0000,,I absolutely hate them.
Dialogue: 0,0:42:36.25,0:42:40.33,Default,,0000,0000,0000,,y is between 1 minus\Nsquare root x squared,
Dialogue: 0,0:42:40.33,0:42:43.30,Default,,0000,0000,0000,,minus square root\N1 minus x squared.
Dialogue: 0,0:42:43.30,0:42:47.65,Default,,0000,0000,0000,,So if I were to ask you to do\Nthe integral like last time,
Dialogue: 0,0:42:47.65,0:42:50.79,Default,,0000,0000,0000,,how would you set\Nup the integral?
Dialogue: 0,0:42:50.79,0:42:53.38,Default,,0000,0000,0000,,You go, OK, I know what this is.
Dialogue: 0,0:42:53.38,0:43:01.38,Default,,0000,0000,0000,,Integral over D of\Nf of x, y, dx dy.
Dialogue: 0,0:43:01.38,0:43:02.90,Default,,0000,0000,0000,,This is actually a wedge.
Dialogue: 0,0:43:02.90,0:43:06.06,Default,,0000,0000,0000,,In my case, we avoided that.
Dialogue: 0,0:43:06.06,0:43:07.54,Default,,0000,0000,0000,,We said dh.
Dialogue: 0,0:43:07.54,0:43:09.91,Default,,0000,0000,0000,,And we said, what is f of x, y?
Dialogue: 0,0:43:09.91,0:43:11.77,Default,,0000,0000,0000,,x squared plus y\Nsquared, because I
Dialogue: 0,0:43:11.77,0:43:16.04,Default,,0000,0000,0000,,want everything that's under\Nthe graph, not above the graph.
Dialogue: 0,0:43:16.04,0:43:18.100,Default,,0000,0000,0000,,So everything that's\Nunder the graph.
Dialogue: 0,0:43:18.100,0:43:26.60,Default,,0000,0000,0000,,F of x, y is this guy.
Dialogue: 0,0:43:26.60,0:43:28.43,Default,,0000,0000,0000,,And the I have to\Nstart thinking,
Dialogue: 0,0:43:28.43,0:43:31.54,Default,,0000,0000,0000,,because it's a type 1 or type 2?
Dialogue: 0,0:43:31.54,0:43:35.70,Default,,0000,0000,0000,,It's a type 1 the\Nway I set it up,
Dialogue: 0,0:43:35.70,0:43:39.06,Default,,0000,0000,0000,,but I can make it\Ntype 2 by reversing
Dialogue: 0,0:43:39.06,0:43:41.52,Default,,0000,0000,0000,,the order of integration\Nlike I did last time.
Dialogue: 0,0:43:41.52,0:43:44.04,Default,,0000,0000,0000,,If I treat it like\Nthat, it's going
Dialogue: 0,0:43:44.04,0:43:46.42,Default,,0000,0000,0000,,to be type 1, though, right?
Dialogue: 0,0:43:46.42,0:43:50.64,Default,,0000,0000,0000,,So I have to put\Ndy first, and then
Dialogue: 0,0:43:50.64,0:43:54.57,Default,,0000,0000,0000,,change the color of the dx.
Dialogue: 0,0:43:54.57,0:43:58.28,Default,,0000,0000,0000,,And since mister y\Nis the purple guy,
Dialogue: 0,0:43:58.28,0:44:03.00,Default,,0000,0000,0000,,y would be going between\Nthese ugly square roots that
Dialogue: 0,0:44:03.00,0:44:04.22,Default,,0000,0000,0000,,to go on my nerves.
Dialogue: 0,0:44:04.22,0:44:10.36,Default,,0000,0000,0000,,
Dialogue: 0,0:44:10.36,0:44:17.48,Default,,0000,0000,0000,,And then x goes\Nbetween minus 1 and 1.
Dialogue: 0,0:44:17.48,0:44:20.87,Default,,0000,0000,0000,,It's a little bit of a headache.
Dialogue: 0,0:44:20.87,0:44:22.98,Default,,0000,0000,0000,,Why is it a headache, guys?
Dialogue: 0,0:44:22.98,0:44:27.47,Default,,0000,0000,0000,,Let's anticipate what we need to\Ndo if we do it like last time.
Dialogue: 0,0:44:27.47,0:44:32.11,Default,,0000,0000,0000,,We need to integrate this\Nugly fellow in terms of y,
Dialogue: 0,0:44:32.11,0:44:35.51,Default,,0000,0000,0000,,and when we integrate this in\Nterms of y, what do we get?
Dialogue: 0,0:44:35.51,0:44:38.45,Default,,0000,0000,0000,,Don't write it, because\Nit's going to be a mess.
Dialogue: 0,0:44:38.45,0:44:44.87,Default,,0000,0000,0000,,We get x squared times\Ny plus y cubed over 3.
Dialogue: 0,0:44:44.87,0:44:47.48,Default,,0000,0000,0000,,And then, instead of y, I have\Nto replace those square roots,
Dialogue: 0,0:44:47.48,0:44:49.60,Default,,0000,0000,0000,,and I'll never get rid\Nof the square roots.
Dialogue: 0,0:44:49.60,0:44:52.76,Default,,0000,0000,0000,,It's going to be a mess, indeed.
Dialogue: 0,0:44:52.76,0:44:56.25,Default,,0000,0000,0000,,And I may even-- in\Ngeneral, I may not even
Dialogue: 0,0:44:56.25,0:44:58.86,Default,,0000,0000,0000,,be able to solve the\Nintegral, and that's
Dialogue: 0,0:44:58.86,0:45:00.78,Default,,0000,0000,0000,,a bit headache,\Nbecause I'll start
Dialogue: 0,0:45:00.78,0:45:03.44,Default,,0000,0000,0000,,crying, I'll get depressed,\NI'll take Prozac, whatever
Dialogue: 0,0:45:03.44,0:45:04.82,Default,,0000,0000,0000,,you take for depression.
Dialogue: 0,0:45:04.82,0:45:07.56,Default,,0000,0000,0000,,I don't know, I never took it,\Nbecause I'm never depressed.
Dialogue: 0,0:45:07.56,0:45:10.96,Default,,0000,0000,0000,,So what do you do in that case?
Dialogue: 0,0:45:10.96,0:45:12.22,Default,,0000,0000,0000,,STUDENT: Switch to polar.
Dialogue: 0,0:45:12.22,0:45:13.72,Default,,0000,0000,0000,,MAGDALENA TODA: You\Nswitch to polar.
Dialogue: 0,0:45:13.72,0:45:18.61,Default,,0000,0000,0000,,So you use this big polar-switch\Ntheorem, the theorem that
Dialogue: 0,0:45:18.61,0:45:23.94,Default,,0000,0000,0000,,tells you, be smart,\Napply this theorem,
Dialogue: 0,0:45:23.94,0:45:30.70,Default,,0000,0000,0000,,and have to understand that\Nthe D, which was this expressed
Dialogue: 0,0:45:30.70,0:45:32.97,Default,,0000,0000,0000,,in [INTERPOSING VOICES]\NCartesian coordinates
Dialogue: 0,0:45:32.97,0:45:37.48,Default,,0000,0000,0000,,is D. If you want express\Nthe same thing as D star,
Dialogue: 0,0:45:37.48,0:45:39.60,Default,,0000,0000,0000,,D star will be in\Npolar coordinates.
Dialogue: 0,0:45:39.60,0:45:44.01,Default,,0000,0000,0000,,You have to be a little bit\Nsmarter, and say r theta,
Dialogue: 0,0:45:44.01,0:45:48.98,Default,,0000,0000,0000,,where now you have to put\Nthe bounds that limit--
Dialogue: 0,0:45:48.98,0:45:49.59,Default,,0000,0000,0000,,STUDENT: r.
Dialogue: 0,0:45:49.59,0:45:50.69,Default,,0000,0000,0000,,MAGDALENA TODA: r from?
Dialogue: 0,0:45:50.69,0:45:51.36,Default,,0000,0000,0000,,STUDENT: 0 to 1.
Dialogue: 0,0:45:51.36,0:45:52.78,Default,,0000,0000,0000,,MAGDALENA TODA: 0\Nto 1, excellent.
Dialogue: 0,0:45:52.78,0:45:56.90,Default,,0000,0000,0000,,You cannot let r go to\Ninfinity, because the vase is
Dialogue: 0,0:45:56.90,0:45:57.44,Default,,0000,0000,0000,,increasingly.
Dialogue: 0,0:45:57.44,0:46:01.31,Default,,0000,0000,0000,,You only needs the vase that\Nhas the radius 1 on the bottom.
Dialogue: 0,0:46:01.31,0:46:08.72,Default,,0000,0000,0000,,So r is 0 to 1, and\Ntheta is 0 to 1 pi.
Dialogue: 0,0:46:08.72,0:46:10.64,Default,,0000,0000,0000,,And there you have\Nyour domain this time.
Dialogue: 0,0:46:10.64,0:46:15.75,Default,,0000,0000,0000,,So I need to be smart\Nand say integral.
Dialogue: 0,0:46:15.75,0:46:18.00,Default,,0000,0000,0000,,Integral, what do\Nyou want to do first?
Dialogue: 0,0:46:18.00,0:46:21.85,Default,,0000,0000,0000,,Well, it doesn't matter, dr,\Nd theta, whatever you want.
Dialogue: 0,0:46:21.85,0:46:26.31,Default,,0000,0000,0000,,So mister theta will\Nbe the last of the two.
Dialogue: 0,0:46:26.31,0:46:32.27,Default,,0000,0000,0000,,So theta will be between 0\Nand 2 pi, a complete rotation.
Dialogue: 0,0:46:32.27,0:46:35.86,Default,,0000,0000,0000,,r between 0 and 1.
Dialogue: 0,0:46:35.86,0:46:37.97,Default,,0000,0000,0000,,And inside here I\Nhave to be smart
Dialogue: 0,0:46:37.97,0:46:41.71,Default,,0000,0000,0000,,and see that life\Ncan be fun when
Dialogue: 0,0:46:41.71,0:46:44.32,Default,,0000,0000,0000,,I work with polar coordinates.
Dialogue: 0,0:46:44.32,0:46:45.64,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:46:45.64,0:46:47.06,Default,,0000,0000,0000,,What is the integral?
Dialogue: 0,0:46:47.06,0:46:48.11,Default,,0000,0000,0000,,x squared plus y squared.
Dialogue: 0,0:46:48.11,0:46:50.68,Default,,0000,0000,0000,,I've seen him\Nsomewhere before when
Dialogue: 0,0:46:50.68,0:46:54.99,Default,,0000,0000,0000,,it came to polar coordinates.
Dialogue: 0,0:46:54.99,0:46:55.78,Default,,0000,0000,0000,,STUDENT: R squared.
Dialogue: 0,0:46:55.78,0:46:57.11,Default,,0000,0000,0000,,STUDENT: That will be r squared.
Dialogue: 0,0:46:57.11,0:46:59.60,Default,,0000,0000,0000,,MAGDALENA TODA: That\Nwill be r squared.
Dialogue: 0,0:46:59.60,0:47:04.48,Default,,0000,0000,0000,,r squared times-- never\Nforget the Jacobian,
Dialogue: 0,0:47:04.48,0:47:07.91,Default,,0000,0000,0000,,and the Jacobian is mister r.
Dialogue: 0,0:47:07.91,0:47:13.03,Default,,0000,0000,0000,,And now I'm going to\Ntake all this integral.
Dialogue: 0,0:47:13.03,0:47:16.49,Default,,0000,0000,0000,,I'll finally compute\Nthe volume of my vase.
Dialogue: 0,0:47:16.49,0:47:19.96,Default,,0000,0000,0000,,Imagine if this vase\Nwould be made of gold.
Dialogue: 0,0:47:19.96,0:47:21.69,Default,,0000,0000,0000,,This is my dream.
Dialogue: 0,0:47:21.69,0:47:24.97,Default,,0000,0000,0000,,So imagine that this\Nvase would have,
Dialogue: 0,0:47:24.97,0:47:26.79,Default,,0000,0000,0000,,I don't know what dimensions.
Dialogue: 0,0:47:26.79,0:47:29.39,Default,,0000,0000,0000,,I need to find the\Nvolume, and multiply it
Dialogue: 0,0:47:29.39,0:47:32.40,Default,,0000,0000,0000,,by the density of gold\Nand find out-- yes, sir?
Dialogue: 0,0:47:32.40,0:47:35.66,Default,,0000,0000,0000,,STUDENT: Professor, like in this\Nquestion, b time is dt by dr,
Dialogue: 0,0:47:35.66,0:47:38.06,Default,,0000,0000,0000,,but you can't switch it--
Dialogue: 0,0:47:38.06,0:47:39.27,Default,,0000,0000,0000,,MAGDALENA TODA: Yes, you can.
Dialogue: 0,0:47:39.27,0:47:41.32,Default,,0000,0000,0000,,That's exactly my point.
Dialogue: 0,0:47:41.32,0:47:42.69,Default,,0000,0000,0000,,I'll tell you in a second.
Dialogue: 0,0:47:42.69,0:47:47.98,Default,,0000,0000,0000,,When can you replace d theta dr?
Dialogue: 0,0:47:47.98,0:47:52.45,Default,,0000,0000,0000,,You can always do that when\Nyou have something under here,
Dialogue: 0,0:47:52.45,0:47:55.69,Default,,0000,0000,0000,,which is a big\Nfunction of theta times
Dialogue: 0,0:47:55.69,0:48:01.63,Default,,0000,0000,0000,,a bit function of r, because\Nyou can treat them differently.
Dialogue: 0,0:48:01.63,0:48:05.05,Default,,0000,0000,0000,,We will work about this later.
Dialogue: 0,0:48:05.05,0:48:08.64,Default,,0000,0000,0000,,Now, this has no theta.
Dialogue: 0,0:48:08.64,0:48:13.72,Default,,0000,0000,0000,,So actually, the\Ntheta is not going
Dialogue: 0,0:48:13.72,0:48:18.70,Default,,0000,0000,0000,,to affect your computation.
Dialogue: 0,0:48:18.70,0:48:22.41,Default,,0000,0000,0000,,Let's not even think about\Ntheta for the time being.
Dialogue: 0,0:48:22.41,0:48:29.90,Default,,0000,0000,0000,,What you have inside is Calculus\NI. When you have a product,
Dialogue: 0,0:48:29.90,0:48:31.40,Default,,0000,0000,0000,,you can always switch.
Dialogue: 0,0:48:31.40,0:48:33.88,Default,,0000,0000,0000,,And I'll give you\Na theorem later.
Dialogue: 0,0:48:33.88,0:48:39.15,Default,,0000,0000,0000,,0 over 1, r cubed,\Nthank God, this
Dialogue: 0,0:48:39.15,0:48:42.50,Default,,0000,0000,0000,,is Calc I. Integral\Nfrom 0 to 1, r
Dialogue: 0,0:48:42.50,0:48:47.00,Default,,0000,0000,0000,,cubed dr. That's Calc\NI. How much is that?
Dialogue: 0,0:48:47.00,0:48:47.62,Default,,0000,0000,0000,,I'm lazy.
Dialogue: 0,0:48:47.62,0:48:50.11,Default,,0000,0000,0000,,I don't want to do it.
Dialogue: 0,0:48:50.11,0:48:51.18,Default,,0000,0000,0000,,STUDENT: 1/4.
Dialogue: 0,0:48:51.18,0:48:52.22,Default,,0000,0000,0000,,MAGDALENA TODA: It's 1/4.
Dialogue: 0,0:48:52.22,0:48:52.72,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:48:52.72,0:48:53.91,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:48:53.91,0:48:58.46,Default,,0000,0000,0000,,And if I get further, and I'm a\Nlittle bi lazy, what do I get?
Dialogue: 0,0:48:58.46,0:49:01.50,Default,,0000,0000,0000,,1/4 is the constant,\Nit pulls out.
Dialogue: 0,0:49:01.50,0:49:03.14,Default,,0000,0000,0000,,STUDENT: So, they don't--
Dialogue: 0,0:49:03.14,0:49:09.78,Default,,0000,0000,0000,,MAGDALENA TODA: So I get 2 pi\Nover 4, which is pi over 2.
Dialogue: 0,0:49:09.78,0:49:10.54,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,0:49:10.54,0:49:11.12,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:49:11.12,0:49:12.87,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nthis constant gets out,
Dialogue: 0,0:49:12.87,0:49:14.20,Default,,0000,0000,0000,,integral comes in through 2 pi.
Dialogue: 0,0:49:14.20,0:49:16.22,Default,,0000,0000,0000,,It will be 2 pi, and\Nthis is my answer.
Dialogue: 0,0:49:16.22,0:49:19.52,Default,,0000,0000,0000,,So pi over 2 is the volume.
Dialogue: 0,0:49:19.52,0:49:22.57,Default,,0000,0000,0000,,If I have a 1-inch\Ndiameter, and I
Dialogue: 0,0:49:22.57,0:49:26.54,Default,,0000,0000,0000,,have this vase made of gold,\Nwhich is a piece of jewelry,
Dialogue: 0,0:49:26.54,0:49:34.16,Default,,0000,0000,0000,,really beautiful, then I'm going\Nto have pi over 2 the volume.
Dialogue: 0,0:49:34.16,0:49:36.33,Default,,0000,0000,0000,,That will be a little\Nbit hard to see
Dialogue: 0,0:49:36.33,0:49:38.93,Default,,0000,0000,0000,,what we have in square inches.
Dialogue: 0,0:49:38.93,0:49:43.92,Default,,0000,0000,0000,,We have 1.5-something\Nsquare inches, and then--
Dialogue: 0,0:49:43.92,0:49:45.10,Default,,0000,0000,0000,,STUDENT: More.
Dialogue: 0,0:49:45.10,0:49:46.48,Default,,0000,0000,0000,,MAGDALENA TODA:\NAnd then multiply
Dialogue: 0,0:49:46.48,0:49:50.35,Default,,0000,0000,0000,,by the density of\Ngold, and estimate,
Dialogue: 0,0:49:50.35,0:49:57.73,Default,,0000,0000,0000,,based on the mass, how much\Nmoney that's going to be.
Dialogue: 0,0:49:57.73,0:49:59.88,Default,,0000,0000,0000,,What did I want to\Ntell [? Miteish? ?]
Dialogue: 0,0:49:59.88,0:50:02.63,Default,,0000,0000,0000,,I don't want to forget what\Nhe asked me, because that
Dialogue: 0,0:50:02.63,0:50:04.24,Default,,0000,0000,0000,,was a smart question.
Dialogue: 0,0:50:04.24,0:50:08.62,Default,,0000,0000,0000,,When can we reverse the\Norder of integration?
Dialogue: 0,0:50:08.62,0:50:11.100,Default,,0000,0000,0000,,In general, it's\Nhard to compute.
Dialogue: 0,0:50:11.100,0:50:14.54,Default,,0000,0000,0000,,But in this case, I'm you\Nare the luckiest person
Dialogue: 0,0:50:14.54,0:50:16.79,Default,,0000,0000,0000,,in the world, because\Njust take a look at me.
Dialogue: 0,0:50:16.79,0:50:22.18,Default,,0000,0000,0000,,I have, let's see, my\Nr between 0 and 2 pi,
Dialogue: 0,0:50:22.18,0:50:29.47,Default,,0000,0000,0000,,and my theta between 0 and 2\Npi, and my r between 0 and 1.
Dialogue: 0,0:50:29.47,0:50:31.97,Default,,0000,0000,0000,,Whatever, it doesn't matter,\Nit could be anything.
Dialogue: 0,0:50:31.97,0:50:36.39,Default,,0000,0000,0000,,And here I have a function of r\Nand a function g of theta only.
Dialogue: 0,0:50:36.39,0:50:38.06,Default,,0000,0000,0000,,And it's a product.
Dialogue: 0,0:50:38.06,0:50:40.79,Default,,0000,0000,0000,,The variables are separate.
Dialogue: 0,0:50:40.79,0:50:45.80,Default,,0000,0000,0000,,When I do-- what do I\Ndo for dr or d theta?
Dialogue: 0,0:50:45.80,0:50:49.24,Default,,0000,0000,0000,,dr. When I do dr--\Nwith respect to dr,
Dialogue: 0,0:50:49.24,0:50:52.70,Default,,0000,0000,0000,,this fellow goes, I\Ndon't belong in here.
Dialogue: 0,0:50:52.70,0:50:55.65,Default,,0000,0000,0000,,I'm mister theta that\Ndoesn't belong in here.
Dialogue: 0,0:50:55.65,0:50:56.93,Default,,0000,0000,0000,,I'm independent.
Dialogue: 0,0:50:56.93,0:50:59.16,Default,,0000,0000,0000,,I want to go out.
Dialogue: 0,0:50:59.16,0:51:01.60,Default,,0000,0000,0000,,And he wants out.
Dialogue: 0,0:51:01.60,0:51:10.48,Default,,0000,0000,0000,,So you have some integrals\Nthat you got out a g of theta,
Dialogue: 0,0:51:10.48,0:51:16.44,Default,,0000,0000,0000,,and another integral, and you\Nhave f of r dr in a bracket,
Dialogue: 0,0:51:16.44,0:51:20.88,Default,,0000,0000,0000,,and then you go d theta.
Dialogue: 0,0:51:20.88,0:51:23.08,Default,,0000,0000,0000,,What is going to happen next?
Dialogue: 0,0:51:23.08,0:51:26.79,Default,,0000,0000,0000,,You solve this integral, and\Nit's going to be a number.
Dialogue: 0,0:51:26.79,0:51:30.40,Default,,0000,0000,0000,,This number could be 8,\N7, 9.2, God knows what.
Dialogue: 0,0:51:30.40,0:51:33.23,Default,,0000,0000,0000,,Why don't you pull that\Nconstant out right now?
Dialogue: 0,0:51:33.23,0:51:35.48,Default,,0000,0000,0000,,So you say, OK, I can do that.
Dialogue: 0,0:51:35.48,0:51:37.13,Default,,0000,0000,0000,,It's just a number.
Dialogue: 0,0:51:37.13,0:51:37.63,Default,,0000,0000,0000,,Whatever.
Dialogue: 0,0:51:37.63,0:51:41.61,Default,,0000,0000,0000,,That's going to be\Nintegral f dr, times
Dialogue: 0,0:51:41.61,0:51:44.32,Default,,0000,0000,0000,,what do you have left\Nwhen you pull that out?
Dialogue: 0,0:51:44.32,0:51:44.82,Default,,0000,0000,0000,,A what?
Dialogue: 0,0:51:44.82,0:51:45.62,Default,,0000,0000,0000,,STUDENT: Integral.
Dialogue: 0,0:51:45.62,0:51:49.46,Default,,0000,0000,0000,,MAGDALENA TODA: Integral of\NG, the integral of g of theta,
Dialogue: 0,0:51:49.46,0:51:51.00,Default,,0000,0000,0000,,d theta.
Dialogue: 0,0:51:51.00,0:51:54.44,Default,,0000,0000,0000,,So we just proved a theorem\Nthat is really pretty.
Dialogue: 0,0:51:54.44,0:51:59.24,Default,,0000,0000,0000,,If you have to integrate,\Nand I will try to do it here.
Dialogue: 0,0:51:59.24,0:52:03.20,Default,,0000,0000,0000,,
Dialogue: 0,0:52:03.20,0:52:03.70,Default,,0000,0000,0000,,No--
Dialogue: 0,0:52:03.70,0:52:06.24,Default,,0000,0000,0000,,STUDENT: So essentially, when\Nyou're integrating with respect
Dialogue: 0,0:52:06.24,0:52:11.24,Default,,0000,0000,0000,,to r, you can treat any function\Nof only theta as a constant?
Dialogue: 0,0:52:11.24,0:52:12.23,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,0:52:12.23,0:52:15.05,Default,,0000,0000,0000,,I'll tell you in a second\Nwhat it means, because--
Dialogue: 0,0:52:15.05,0:52:15.81,Default,,0000,0000,0000,,STUDENT: Sorry.
Dialogue: 0,0:52:15.81,0:52:16.98,Default,,0000,0000,0000,,MAGDALENA TODA: You're fine.
Dialogue: 0,0:52:16.98,0:52:21.60,Default,,0000,0000,0000,,Integrate for domain,\Nrectangular domains,
Dialogue: 0,0:52:21.60,0:52:25.77,Default,,0000,0000,0000,,let's say u between alpha,\Nbeta, u between gamma,
Dialogue: 0,0:52:25.77,0:52:29.71,Default,,0000,0000,0000,,delta, then what's\Ngoing to happen?
Dialogue: 0,0:52:29.71,0:52:35.38,Default,,0000,0000,0000,,As you said very well,\Nintegral from-- what
Dialogue: 0,0:52:35.38,0:52:38.33,Default,,0000,0000,0000,,do you want first, dv or du?
Dialogue: 0,0:52:38.33,0:52:41.13,Default,,0000,0000,0000,,dv, du, it doesn't matter.
Dialogue: 0,0:52:41.13,0:52:44.11,Default,,0000,0000,0000,,v is between gamma, delta.
Dialogue: 0,0:52:44.11,0:52:47.08,Default,,0000,0000,0000,,v is the first guy inside, OK.
Dialogue: 0,0:52:47.08,0:52:48.57,Default,,0000,0000,0000,,Gamma, delta.
Dialogue: 0,0:52:48.57,0:52:50.06,Default,,0000,0000,0000,,I should have cd.
Dialogue: 0,0:52:50.06,0:52:51.08,Default,,0000,0000,0000,,It's all Greek to me.
Dialogue: 0,0:52:51.08,0:52:55.06,Default,,0000,0000,0000,,Why did I pick\Nthat three people?
Dialogue: 0,0:52:55.06,0:52:59.60,Default,,0000,0000,0000,,If this is going to be a product\Nof two functions, one is in u
Dialogue: 0,0:52:59.60,0:53:06.21,Default,,0000,0000,0000,,and one is in v. Let's\Nsay A of u and B of v,
Dialogue: 0,0:53:06.21,0:53:11.10,Default,,0000,0000,0000,,I can go ahead and say\Nproduct of two constants.
Dialogue: 0,0:53:11.10,0:53:14.04,Default,,0000,0000,0000,,And who are those two\Nconstants I was referring to?
Dialogue: 0,0:53:14.04,0:53:16.00,Default,,0000,0000,0000,,You can do that directly.
Dialogue: 0,0:53:16.00,0:53:18.94,Default,,0000,0000,0000,,If the two variables are\Nseparated through a product,
Dialogue: 0,0:53:18.94,0:53:22.73,Default,,0000,0000,0000,,you have a product of\Ntwo separate variables.
Dialogue: 0,0:53:22.73,0:53:26.32,Default,,0000,0000,0000,,A is only in u, it\Ndepends only on u.
Dialogue: 0,0:53:26.32,0:53:30.82,Default,,0000,0000,0000,,And B is only on v. They have\Nnothing to do with one another.
Dialogue: 0,0:53:30.82,0:53:35.15,Default,,0000,0000,0000,,Then you can go ahead and do\Nthe first integral with respect
Dialogue: 0,0:53:35.15,0:53:43.31,Default,,0000,0000,0000,,to u only of a of u, du,\Nu between alpha, beta.
Dialogue: 0,0:53:43.31,0:53:45.94,Default,,0000,0000,0000,,That was your first variable.
Dialogue: 0,0:53:45.94,0:53:48.62,Default,,0000,0000,0000,,Times this other constant.
Dialogue: 0,0:53:48.62,0:53:54.49,Default,,0000,0000,0000,,Integral of B of v,\Nwhere v is moving,
Dialogue: 0,0:53:54.49,0:53:59.07,Default,,0000,0000,0000,,v is moving between\Ngamma, delta.
Dialogue: 0,0:53:59.07,0:54:00.98,Default,,0000,0000,0000,,Instead of alpha,\Nbeta, gamma, delta,
Dialogue: 0,0:54:00.98,0:54:03.97,Default,,0000,0000,0000,,put any numbers you want.
Dialogue: 0,0:54:03.97,0:54:04.85,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:54:04.85,0:54:06.18,Default,,0000,0000,0000,,This is the lucky case.
Dialogue: 0,0:54:06.18,0:54:09.20,Default,,0000,0000,0000,,So you're always hoping\Nthat on the final,
Dialogue: 0,0:54:09.20,0:54:12.84,Default,,0000,0000,0000,,you can get something\Nwhere you can separate.
Dialogue: 0,0:54:12.84,0:54:13.95,Default,,0000,0000,0000,,Here you have no theta.
Dialogue: 0,0:54:13.95,0:54:16.33,Default,,0000,0000,0000,,This is the luckiest\Ncase in the world.
Dialogue: 0,0:54:16.33,0:54:18.55,Default,,0000,0000,0000,,So it's just r\Ncubed times theta.
Dialogue: 0,0:54:18.55,0:54:21.44,Default,,0000,0000,0000,,But you can still\Nhave a lucky case
Dialogue: 0,0:54:21.44,0:54:24.53,Default,,0000,0000,0000,,when you have something\Nlike a function of r
Dialogue: 0,0:54:24.53,0:54:25.94,Default,,0000,0000,0000,,times a function of theta.
Dialogue: 0,0:54:25.94,0:54:28.55,Default,,0000,0000,0000,,And then you have\Nanother beautiful polar
Dialogue: 0,0:54:28.55,0:54:31.60,Default,,0000,0000,0000,,coordinate integral\Nthat you're not going
Dialogue: 0,0:54:31.60,0:54:35.00,Default,,0000,0000,0000,,to struggle with for very long.
Dialogue: 0,0:54:35.00,0:54:37.45,Default,,0000,0000,0000,,OK, I'm going to erase here.
Dialogue: 0,0:54:37.45,0:54:56.10,Default,,0000,0000,0000,,
Dialogue: 0,0:54:56.10,0:55:01.56,Default,,0000,0000,0000,,For example, let me\Ngive you another one.
Dialogue: 0,0:55:01.56,0:55:04.11,Default,,0000,0000,0000,,Suppose that somebody\Nwas really mean to you,
Dialogue: 0,0:55:04.11,0:55:08.40,Default,,0000,0000,0000,,and wanted to kill\Nyou in the final,
Dialogue: 0,0:55:08.40,0:55:10.06,Default,,0000,0000,0000,,and they gave you the\Nfollowing problem.
Dialogue: 0,0:55:10.06,0:55:12.59,Default,,0000,0000,0000,,
Dialogue: 0,0:55:12.59,0:55:17.37,Default,,0000,0000,0000,,Assume the domain D-- they\Ndon't even tell you what it is.
Dialogue: 0,0:55:17.37,0:55:19.35,Default,,0000,0000,0000,,They just want to\Nchallenge you--
Dialogue: 0,0:55:19.35,0:55:25.47,Default,,0000,0000,0000,,will be x, y with the\Nproperty that x squared plus y
Dialogue: 0,0:55:25.47,0:55:32.08,Default,,0000,0000,0000,,squared is between a 1 and a 4.
Dialogue: 0,0:55:32.08,0:55:36.37,Default,,0000,0000,0000,,
Dialogue: 0,0:55:36.37,0:55:52.92,Default,,0000,0000,0000,,Compute the integral over D of\Nr [? pan ?] of y over x and da,
Dialogue: 0,0:55:52.92,0:55:57.33,Default,,0000,0000,0000,,where bi would be ds dy.
Dialogue: 0,0:55:57.33,0:56:00.71,Default,,0000,0000,0000,,So you look at this\Ncross-eyed and say, gosh,
Dialogue: 0,0:56:00.71,0:56:04.22,Default,,0000,0000,0000,,whoever-- we don't do that.
Dialogue: 0,0:56:04.22,0:56:05.31,Default,,0000,0000,0000,,But I've seen schools.
Dialogue: 0,0:56:05.31,0:56:08.60,Default,,0000,0000,0000,,I've seen this given at a\Nschool, when they covered
Dialogue: 0,0:56:08.60,0:56:11.62,Default,,0000,0000,0000,,this particular\Nexample, they've covered
Dialogue: 0,0:56:11.62,0:56:14.71,Default,,0000,0000,0000,,something like the previous\None that I showed you.
Dialogue: 0,0:56:14.71,0:56:16.20,Default,,0000,0000,0000,,But they never covered this.
Dialogue: 0,0:56:16.20,0:56:18.46,Default,,0000,0000,0000,,And they said,\NOK, they're smart,
Dialogue: 0,0:56:18.46,0:56:19.99,Default,,0000,0000,0000,,let them figure this out.
Dialogue: 0,0:56:19.99,0:56:23.36,Default,,0000,0000,0000,,And I think it was Texas A&M.\NThey gave something like that
Dialogue: 0,0:56:23.36,0:56:26.35,Default,,0000,0000,0000,,without working this in class.
Dialogue: 0,0:56:26.35,0:56:28.58,Default,,0000,0000,0000,,They assumed that\Nthe students should
Dialogue: 0,0:56:28.58,0:56:31.12,Default,,0000,0000,0000,,be good enough to\Nfigure out what
Dialogue: 0,0:56:31.12,0:56:35.36,Default,,0000,0000,0000,,this is in polar coordinates.
Dialogue: 0,0:56:35.36,0:56:39.79,Default,,0000,0000,0000,,So in polar coordinates,\Nwhat does the theorem say?
Dialogue: 0,0:56:39.79,0:56:44.36,Default,,0000,0000,0000,,We should switch to a domain\ND star that corresponds to D.
Dialogue: 0,0:56:44.36,0:56:48.22,Default,,0000,0000,0000,,Now, D was given like that.
Dialogue: 0,0:56:48.22,0:56:50.66,Default,,0000,0000,0000,,But we have to say\Nthe corresponding D
Dialogue: 0,0:56:50.66,0:56:55.09,Default,,0000,0000,0000,,star, reinterpreted\Nin polar coordinates,
Dialogue: 0,0:56:55.09,0:56:59.71,Default,,0000,0000,0000,,r theta has to be also\Nwritten beautifully out.
Dialogue: 0,0:56:59.71,0:57:03.91,Default,,0000,0000,0000,,Unless you draw the picture,\Nfirst of all, you cannot do it.
Dialogue: 0,0:57:03.91,0:57:07.79,Default,,0000,0000,0000,,So the prof at Texas A&M didn't\Neven say, draw the picture,
Dialogue: 0,0:57:07.79,0:57:10.70,Default,,0000,0000,0000,,and think of the\Nmeaning of that.
Dialogue: 0,0:57:10.70,0:57:14.95,Default,,0000,0000,0000,,What is the meaning of\Nthis set, geometric set,
Dialogue: 0,0:57:14.95,0:57:17.07,Default,,0000,0000,0000,,geometric locus of points.
Dialogue: 0,0:57:17.07,0:57:18.88,Default,,0000,0000,0000,,STUDENT: You've\Ngot a circle sub-
Dialogue: 0,0:57:18.88,0:57:21.58,Default,,0000,0000,0000,,MAGDALENA TODA: You\Nhave concentric circles,
Dialogue: 0,0:57:21.58,0:57:26.95,Default,,0000,0000,0000,,sub-radius 1 and 2, and it's\Nlike a ring, it's an annulus.
Dialogue: 0,0:57:26.95,0:57:30.02,Default,,0000,0000,0000,,And he said, well,\NI didn't do it.
Dialogue: 0,0:57:30.02,0:57:33.02,Default,,0000,0000,0000,,I mean they were smart.
Dialogue: 0,0:57:33.02,0:57:35.45,Default,,0000,0000,0000,,I gave it to them to do.
Dialogue: 0,0:57:35.45,0:57:40.67,Default,,0000,0000,0000,,So if the students don't see\Nat least an example like that,
Dialogue: 0,0:57:40.67,0:57:44.55,Default,,0000,0000,0000,,they have difficulty,\Nin my experience.
Dialogue: 0,0:57:44.55,0:57:47.30,Default,,0000,0000,0000,,OK, for this kind\Nof annulus, you
Dialogue: 0,0:57:47.30,0:57:50.81,Default,,0000,0000,0000,,see the radius would start\Nhere, but the dotted part
Dialogue: 0,0:57:50.81,0:57:53.49,Default,,0000,0000,0000,,is not included in your domain.
Dialogue: 0,0:57:53.49,0:57:57.11,Default,,0000,0000,0000,,So you have to be smart and\Nsay, wait a minute, my radius
Dialogue: 0,0:57:57.11,0:57:58.55,Default,,0000,0000,0000,,is not starting at 0.
Dialogue: 0,0:57:58.55,0:58:01.53,Default,,0000,0000,0000,,It's starting at 1\Nand it's ending at 2.
Dialogue: 0,0:58:01.53,0:58:05.98,Default,,0000,0000,0000,,And I put that here.
Dialogue: 0,0:58:05.98,0:58:11.02,Default,,0000,0000,0000,,And theta is the whole\Nring, so from 0 to 2 pi.
Dialogue: 0,0:58:11.02,0:58:14.49,Default,,0000,0000,0000,,
Dialogue: 0,0:58:14.49,0:58:18.25,Default,,0000,0000,0000,,Whether you do that\Nover the open set,
Dialogue: 0,0:58:18.25,0:58:21.36,Default,,0000,0000,0000,,that's called annulus\Nwithout the boundaries,
Dialogue: 0,0:58:21.36,0:58:25.26,Default,,0000,0000,0000,,or you do it about the\None with the boundaries,
Dialogue: 0,0:58:25.26,0:58:28.24,Default,,0000,0000,0000,,it doesn't matter, the integral\Nis not going to change.
Dialogue: 0,0:58:28.24,0:58:33.18,Default,,0000,0000,0000,,And you are going to learn\Nthat in Advanced Calculus, why
Dialogue: 0,0:58:33.18,0:58:36.65,Default,,0000,0000,0000,,it doesn't matter that if\Nyou remove the boundary,
Dialogue: 0,0:58:36.65,0:58:38.63,Default,,0000,0000,0000,,you put back the boundary.
Dialogue: 0,0:58:38.63,0:58:42.97,Default,,0000,0000,0000,,That is a certain set of a\Nmeasure 0 for your integration.
Dialogue: 0,0:58:42.97,0:58:46.00,Default,,0000,0000,0000,,It's not going to\Nchange your results.
Dialogue: 0,0:58:46.00,0:58:48.74,Default,,0000,0000,0000,,So no matter how you\Nexpress it-- maybe
Dialogue: 0,0:58:48.74,0:58:51.59,Default,,0000,0000,0000,,you want to express\Nit like an open set.
Dialogue: 0,0:58:51.59,0:58:55.36,Default,,0000,0000,0000,,You still have\Nthe same integral.
Dialogue: 0,0:58:55.36,0:58:57.87,Default,,0000,0000,0000,,Double integral\Nof D star, this is
Dialogue: 0,0:58:57.87,0:59:01.68,Default,,0000,0000,0000,,going to give me a headache,\Nunless you help me.
Dialogue: 0,0:59:01.68,0:59:05.66,Default,,0000,0000,0000,,What is this in\Npolar coordinates?
Dialogue: 0,0:59:05.66,0:59:06.49,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:59:06.49,0:59:09.78,Default,,0000,0000,0000,,
Dialogue: 0,0:59:09.78,0:59:11.41,Default,,0000,0000,0000,,MAGDALENA TODA: I\Nknow when-- once I've
Dialogue: 0,0:59:11.41,0:59:13.24,Default,,0000,0000,0000,,figured out the\Nintegrand, I'm going
Dialogue: 0,0:59:13.24,0:59:16.73,Default,,0000,0000,0000,,to remember to always\Nmultiply by an r,
Dialogue: 0,0:59:16.73,0:59:18.62,Default,,0000,0000,0000,,because if I don't,\NI'm in big trouble.
Dialogue: 0,0:59:18.62,0:59:23.54,Default,,0000,0000,0000,,And then I go dr d theta.
Dialogue: 0,0:59:23.54,0:59:26.31,Default,,0000,0000,0000,,But I don't know what this is.
Dialogue: 0,0:59:26.31,0:59:28.28,Default,,0000,0000,0000,,STUDENT: r.
Dialogue: 0,0:59:28.28,0:59:34.31,Default,,0000,0000,0000,,MAGDALENA TODA: Nope, but\Nyou're-- so r cosine theta is
Dialogue: 0,0:59:34.31,0:59:37.68,Default,,0000,0000,0000,,x, r sine theta is y.
Dialogue: 0,0:59:37.68,0:59:41.26,Default,,0000,0000,0000,,When you do y over\Nx, what do you get?
Dialogue: 0,0:59:41.26,0:59:43.71,Default,,0000,0000,0000,,Always tangent of theta.
Dialogue: 0,0:59:43.71,0:59:47.68,Default,,0000,0000,0000,,And if you do arctangent\Nof tangent, you get theta.
Dialogue: 0,0:59:47.68,0:59:50.52,Default,,0000,0000,0000,,So that was not hard,\Nbut the students did
Dialogue: 0,0:59:50.52,0:59:53.13,Default,,0000,0000,0000,,not-- in that\Nclass, I was talking
Dialogue: 0,0:59:53.13,0:59:56.60,Default,,0000,0000,0000,,to whoever gave the exam,\N70-something percent
Dialogue: 0,0:59:56.60,0:59:58.92,Default,,0000,0000,0000,,of the students did\Nnot know how to do it,
Dialogue: 0,0:59:58.92,1:00:01.49,Default,,0000,0000,0000,,because they had never\Nseen something similar,
Dialogue: 0,1:00:01.49,1:00:07.23,Default,,0000,0000,0000,,and they didn't think how\Nto express this theta in r.
Dialogue: 0,1:00:07.23,1:00:08.86,Default,,0000,0000,0000,,So what do we mean to do?
Dialogue: 0,1:00:08.86,1:00:11.63,Default,,0000,0000,0000,,We mean, is this a product?
Dialogue: 0,1:00:11.63,1:00:13.17,Default,,0000,0000,0000,,It's a beautiful product.
Dialogue: 0,1:00:13.17,1:00:17.62,Default,,0000,0000,0000,,They are separate variables like\N[INAUDIBLE] [? shafts. ?] Now,
Dialogue: 0,1:00:17.62,1:00:19.83,Default,,0000,0000,0000,,you see, you can separate them.
Dialogue: 0,1:00:19.83,1:00:26.73,Default,,0000,0000,0000,,The r is between 1 and 2,\Nso I can do-- eventually I
Dialogue: 0,1:00:26.73,1:00:27.98,Default,,0000,0000,0000,,can do the r first.
Dialogue: 0,1:00:27.98,1:00:33.32,Default,,0000,0000,0000,,And theta is between 0 and\N2 pi, and as I taught you
Dialogue: 0,1:00:33.32,1:00:37.65,Default,,0000,0000,0000,,by the previous theorem, you\Ncan separate the two integrals,
Dialogue: 0,1:00:37.65,1:00:39.97,Default,,0000,0000,0000,,because this one gets out.
Dialogue: 0,1:00:39.97,1:00:41.28,Default,,0000,0000,0000,,It's a constant.
Dialogue: 0,1:00:41.28,1:00:46.58,Default,,0000,0000,0000,,So you're left with integral\Nfrom 0 to 2 pi theta d
Dialogue: 0,1:00:46.58,1:01:04.93,Default,,0000,0000,0000,,theta, and the integral from 1\Nto 2 r dr. r dr theta d theta.
Dialogue: 0,1:01:04.93,1:01:06.40,Default,,0000,0000,0000,,This should be a piece of cake.
Dialogue: 0,1:01:06.40,1:01:13.94,Default,,0000,0000,0000,,The only thing we have to\Ndo is some easy Calculus I.
Dialogue: 0,1:01:13.94,1:01:18.44,Default,,0000,0000,0000,,So what is integral\Nof theta d theta?
Dialogue: 0,1:01:18.44,1:01:20.48,Default,,0000,0000,0000,,I'm not going to rush anywhere.
Dialogue: 0,1:01:20.48,1:01:27.16,Default,,0000,0000,0000,,Theta squared over 2\Nbetween theta equals 0 down
Dialogue: 0,1:01:27.16,1:01:30.94,Default,,0000,0000,0000,,and theta equals 2 pi up.
Dialogue: 0,1:01:30.94,1:01:32.35,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:01:32.35,1:01:33.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:01:33.74,1:01:34.61,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:01:34.61,1:01:35.56,Default,,0000,0000,0000,,I'll do that later.
Dialogue: 0,1:01:35.56,1:01:36.52,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,1:01:36.52,1:01:41.14,Default,,0000,0000,0000,,This is going to be r squared\Nover 2 between 1 and 2.
Dialogue: 0,1:01:41.14,1:01:44.40,Default,,0000,0000,0000,,So the numerical\Nanswer, if I know
Dialogue: 0,1:01:44.40,1:01:51.06,Default,,0000,0000,0000,,how to do any math like\Nthat, is going to be--
Dialogue: 0,1:01:51.06,1:01:52.12,Default,,0000,0000,0000,,STUDENT: 2 pi squared.
Dialogue: 0,1:01:52.12,1:01:53.75,Default,,0000,0000,0000,,MAGDALENA TODA: 2 pi\Nsquared, because I
Dialogue: 0,1:01:53.75,1:01:57.77,Default,,0000,0000,0000,,have 4 pi squared over\N2, so the first guy
Dialogue: 0,1:01:57.77,1:02:07.89,Default,,0000,0000,0000,,is 2 pi squared, times-- I\Nget a 4 and 4 minus 1-- are
Dialogue: 0,1:02:07.89,1:02:09.25,Default,,0000,0000,0000,,you guys with me?
Dialogue: 0,1:02:09.25,1:02:12.62,Default,,0000,0000,0000,,So I get a-- let me\Nwrite it like that.
Dialogue: 0,1:02:12.62,1:02:16.53,Default,,0000,0000,0000,,4 over 2 minus 1 over 2.
Dialogue: 0,1:02:16.53,1:02:18.78,Default,,0000,0000,0000,,What's going to\Nhappen to the over 2?
Dialogue: 0,1:02:18.78,1:02:20.20,Default,,0000,0000,0000,,We'll simplify.
Dialogue: 0,1:02:20.20,1:02:23.54,Default,,0000,0000,0000,,So this is going\Nto be 3 pi squared.
Dialogue: 0,1:02:23.54,1:02:24.87,Default,,0000,0000,0000,,Okey Dokey?
Dialogue: 0,1:02:24.87,1:02:25.38,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:02:25.38,1:02:28.38,Default,,0000,0000,0000,,STUDENT: How did you split it\Ninto two integrals, right here?
Dialogue: 0,1:02:28.38,1:02:31.10,Default,,0000,0000,0000,,MAGDALENA TODA: That's exactly\Nwhat I taught you before.
Dialogue: 0,1:02:31.10,1:02:34.04,Default,,0000,0000,0000,,So if I had not\Ntaught you before,
Dialogue: 0,1:02:34.04,1:02:36.83,Default,,0000,0000,0000,,how did I prove that theorem?
Dialogue: 0,1:02:36.83,1:02:41.43,Default,,0000,0000,0000,,The theorem that was\Nbefore was like that.
Dialogue: 0,1:02:41.43,1:02:44.38,Default,,0000,0000,0000,,What was it?
Dialogue: 0,1:02:44.38,1:02:48.70,Default,,0000,0000,0000,,Suppose I have a function of\Ntheta, and a function of r,
Dialogue: 0,1:02:48.70,1:02:52.54,Default,,0000,0000,0000,,and I have d theta\Ndr. And I think
Dialogue: 0,1:02:52.54,1:02:55.78,Default,,0000,0000,0000,,this weather got to us,\Nbecause several people have
Dialogue: 0,1:02:55.78,1:02:57.77,Default,,0000,0000,0000,,the cold and the flu.
Dialogue: 0,1:02:57.77,1:02:59.27,Default,,0000,0000,0000,,Wash your hands a lot.
Dialogue: 0,1:02:59.27,1:03:03.62,Default,,0000,0000,0000,,It's full of--\Nmathematicians full of germs.
Dialogue: 0,1:03:03.62,1:03:08.56,Default,,0000,0000,0000,,So theta, you want theta to\Nbe between whatever you want.
Dialogue: 0,1:03:08.56,1:03:11.03,Default,,0000,0000,0000,,Any two numbers.
Dialogue: 0,1:03:11.03,1:03:12.29,Default,,0000,0000,0000,,Alpha and beta.
Dialogue: 0,1:03:12.29,1:03:14.84,Default,,0000,0000,0000,,And r between gamma, delta.
Dialogue: 0,1:03:14.84,1:03:17.50,Default,,0000,0000,0000,,This is what I\Nexplained last time.
Dialogue: 0,1:03:17.50,1:03:22.45,Default,,0000,0000,0000,,So when you integrate with\Nrespect to theta first inside,
Dialogue: 0,1:03:22.45,1:03:26.42,Default,,0000,0000,0000,,g of r says I have nothing\Nto do with these guys.
Dialogue: 0,1:03:26.42,1:03:28.39,Default,,0000,0000,0000,,They're not my type,\Nthey're not my gang.
Dialogue: 0,1:03:28.39,1:03:31.36,Default,,0000,0000,0000,,I'm going out, have\Na beer by myself.
Dialogue: 0,1:03:31.36,1:03:39.22,Default,,0000,0000,0000,,So he goes out and\Njoins the r group,
Dialogue: 0,1:03:39.22,1:03:41.42,Default,,0000,0000,0000,,because theta and r\Nhave nothing in common.
Dialogue: 0,1:03:41.42,1:03:44.56,Default,,0000,0000,0000,,They are separate variables.
Dialogue: 0,1:03:44.56,1:03:46.34,Default,,0000,0000,0000,,This is a function\Nof r only, and that's
Dialogue: 0,1:03:46.34,1:03:48.10,Default,,0000,0000,0000,,a function of theta only.
Dialogue: 0,1:03:48.10,1:03:50.41,Default,,0000,0000,0000,,This is what I'm talking about.
Dialogue: 0,1:03:50.41,1:03:52.42,Default,,0000,0000,0000,,OK, so that's a constant.
Dialogue: 0,1:03:52.42,1:03:55.62,Default,,0000,0000,0000,,That constant pulls out.
Dialogue: 0,1:03:55.62,1:03:59.52,Default,,0000,0000,0000,,So in the end, what you have is\Nthat constant that pulled out
Dialogue: 0,1:03:59.52,1:04:06.27,Default,,0000,0000,0000,,is going to be alpha, beta, f of\Nbeta d theta as a number, times
Dialogue: 0,1:04:06.27,1:04:07.66,Default,,0000,0000,0000,,what's left inside?
Dialogue: 0,1:04:07.66,1:04:11.25,Default,,0000,0000,0000,,Integral from gamma\Nto delta g of r
Dialogue: 0,1:04:11.25,1:04:17.78,Default,,0000,0000,0000,,dr. So when the two functions\NF and G are functions of theta,
Dialogue: 0,1:04:17.78,1:04:22.17,Default,,0000,0000,0000,,respectively, r only, they have\Nnothing to do with one another,
Dialogue: 0,1:04:22.17,1:04:24.74,Default,,0000,0000,0000,,and you can write\Nthe original integral
Dialogue: 0,1:04:24.74,1:04:28.57,Default,,0000,0000,0000,,as the product of integrals,\Nand it's really a lucky case.
Dialogue: 0,1:04:28.57,1:04:33.26,Default,,0000,0000,0000,,But you are going to encounter\Nthis lucky case many times
Dialogue: 0,1:04:33.26,1:04:38.90,Default,,0000,0000,0000,,in your final, in the midterm,\Nin-- OK, now thinking of what
Dialogue: 0,1:04:38.90,1:04:41.37,Default,,0000,0000,0000,,I wanted to put on the midterm.
Dialogue: 0,1:04:41.37,1:04:45.31,Default,,0000,0000,0000,,
Dialogue: 0,1:04:45.31,1:04:47.88,Default,,0000,0000,0000,,Somebody asked me if I'm going\Nto put-- they looked already
Dialogue: 0,1:04:47.88,1:04:52.18,Default,,0000,0000,0000,,at the homework and at the\Nbook, and they asked me,
Dialogue: 0,1:04:52.18,1:04:57.57,Default,,0000,0000,0000,,are we going to have something\Nlike the area of the cardioid?
Dialogue: 0,1:04:57.57,1:05:01.13,Default,,0000,0000,0000,,Maybe not necessarily\Nthat-- or area
Dialogue: 0,1:05:01.13,1:05:05.43,Default,,0000,0000,0000,,between a cardioid and a circle\Nthat intersect each other.
Dialogue: 0,1:05:05.43,1:05:10.13,Default,,0000,0000,0000,,Those were doable\Neven with Calc II.
Dialogue: 0,1:05:10.13,1:05:12.71,Default,,0000,0000,0000,,Something like that, that\Nwas doable with Calc II,
Dialogue: 0,1:05:12.71,1:05:16.37,Default,,0000,0000,0000,,I don't want to do it with a\Ndouble integral in Calc III,
Dialogue: 0,1:05:16.37,1:05:22.58,Default,,0000,0000,0000,,and I want to give some problems\Nthat are relevant to you guys.
Dialogue: 0,1:05:22.58,1:05:26.59,Default,,0000,0000,0000,,
Dialogue: 0,1:05:26.59,1:05:29.22,Default,,0000,0000,0000,,The question, what's going\Nto be on the midterm?
Dialogue: 0,1:05:29.22,1:05:32.82,Default,,0000,0000,0000,,is not-- OK, what's going\Nto be on the midterm?
Dialogue: 0,1:05:32.82,1:05:36.06,Default,,0000,0000,0000,,It's going to be something\Nvery similar to the sample
Dialogue: 0,1:05:36.06,1:05:37.82,Default,,0000,0000,0000,,that I'm going to write.
Dialogue: 0,1:05:37.82,1:05:40.69,Default,,0000,0000,0000,,And I have already\Nincluded in that sample
Dialogue: 0,1:05:40.69,1:05:44.73,Default,,0000,0000,0000,,the volume of a\Nsphere of radius r.
Dialogue: 0,1:05:44.73,1:05:50.39,Default,,0000,0000,0000,,So how do you compute out\Nthe weight-- exercise 3 or 4,
Dialogue: 0,1:05:50.39,1:06:07.41,Default,,0000,0000,0000,,whatever that is-- we compute\Nthe volume of a sphere using
Dialogue: 0,1:06:07.41,1:06:08.36,Default,,0000,0000,0000,,double integrals.
Dialogue: 0,1:06:08.36,1:06:16.64,Default,,0000,0000,0000,,
Dialogue: 0,1:06:16.64,1:06:20.21,Default,,0000,0000,0000,,I don't know if we have time to\Ndo this problem, but if we do,
Dialogue: 0,1:06:20.21,1:06:25.39,Default,,0000,0000,0000,,that will be the last problem--\Nwhen you ask you teacher,
Dialogue: 0,1:06:25.39,1:06:28.100,Default,,0000,0000,0000,,why is the volume inside the\Nsphere, volume of a ball,
Dialogue: 0,1:06:28.100,1:06:29.89,Default,,0000,0000,0000,,actually.
Dialogue: 0,1:06:29.89,1:06:33.21,Default,,0000,0000,0000,,Well, the size-- the solid ball.
Dialogue: 0,1:06:33.21,1:06:35.83,Default,,0000,0000,0000,,Why is it 4 pi r cubed over 2?
Dialogue: 0,1:06:35.83,1:06:38.44,Default,,0000,0000,0000,,Your, did she tell\Nyou, or she told
Dialogue: 0,1:06:38.44,1:06:42.84,Default,,0000,0000,0000,,you something that you asked,\NMr. [? Jaime ?], for example?
Dialogue: 0,1:06:42.84,1:06:47.51,Default,,0000,0000,0000,,They were supposed to tell\Nyou that you can prove that
Dialogue: 0,1:06:47.51,1:06:49.02,Default,,0000,0000,0000,,with Calc II or Calc III.
Dialogue: 0,1:06:49.02,1:06:51.06,Default,,0000,0000,0000,,It's not easy.
Dialogue: 0,1:06:51.06,1:06:52.88,Default,,0000,0000,0000,,It's not an elementary formula.
Dialogue: 0,1:06:52.88,1:06:54.26,Default,,0000,0000,0000,,In the ancient\Ntimes, they didn't
Dialogue: 0,1:06:54.26,1:06:57.03,Default,,0000,0000,0000,,know how to do it, because\Nthey didn't know calculus.
Dialogue: 0,1:06:57.03,1:07:00.50,Default,,0000,0000,0000,,So what they tried to is\Nto approximate it and see
Dialogue: 0,1:07:00.50,1:07:02.77,Default,,0000,0000,0000,,how it goes.
Dialogue: 0,1:07:02.77,1:07:07.30,Default,,0000,0000,0000,,Assume you have the\Nsphere of radius r,
Dialogue: 0,1:07:07.30,1:07:09.49,Default,,0000,0000,0000,,and r is from here\Nto here, and I'm
Dialogue: 0,1:07:09.49,1:07:12.93,Default,,0000,0000,0000,,going to go ahead and draw the\Nequator, the upper hemisphere,
Dialogue: 0,1:07:12.93,1:07:18.51,Default,,0000,0000,0000,,the lower hemisphere, and\Nyou shouldn't help me,
Dialogue: 0,1:07:18.51,1:07:25.42,Default,,0000,0000,0000,,because isn't enough to say\Nit's twice the upper hemisphere
Dialogue: 0,1:07:25.42,1:07:28.64,Default,,0000,0000,0000,,volume, right?
Dialogue: 0,1:07:28.64,1:07:34.28,Default,,0000,0000,0000,,So if I knew the--\Nwhat is this called?
Dialogue: 0,1:07:34.28,1:07:36.56,Default,,0000,0000,0000,,If I knew the\Nexpression z equals
Dialogue: 0,1:07:36.56,1:07:41.14,Default,,0000,0000,0000,,f of x, y of the spherical\Ncap of the hemisphere,
Dialogue: 0,1:07:41.14,1:07:45.39,Default,,0000,0000,0000,,of the northern hemisphere,\NI would be in business.
Dialogue: 0,1:07:45.39,1:07:49.75,Default,,0000,0000,0000,,So if somebody even\Ntries-- one of my students,
Dialogue: 0,1:07:49.75,1:07:53.22,Default,,0000,0000,0000,,I gave him that, he didn't know\Npolar coordinates very well,
Dialogue: 0,1:07:53.22,1:07:57.62,Default,,0000,0000,0000,,so what he tried to do,\Nhe was trying to do,
Dialogue: 0,1:07:57.62,1:08:03.87,Default,,0000,0000,0000,,let's say z is going\Nto be square root of r
Dialogue: 0,1:08:03.87,1:08:09.77,Default,,0000,0000,0000,,squared minus z squared minus\Ny squared over the domain.
Dialogue: 0,1:08:09.77,1:08:13.30,Default,,0000,0000,0000,,So D will be what\Ndomain? x squared
Dialogue: 0,1:08:13.30,1:08:21.69,Default,,0000,0000,0000,,plus y squared between 0 and\Nr squared, am I right guys?
Dialogue: 0,1:08:21.69,1:08:25.89,Default,,0000,0000,0000,,So the D is on\Nthe floor, means x
Dialogue: 0,1:08:25.89,1:08:28.62,Default,,0000,0000,0000,,squared plus y squared\Nbetween 0 and r squared.
Dialogue: 0,1:08:28.62,1:08:32.34,Default,,0000,0000,0000,,This is the D that we have.
Dialogue: 0,1:08:32.34,1:08:35.89,Default,,0000,0000,0000,,This is D So twice what?
Dialogue: 0,1:08:35.89,1:08:37.11,Default,,0000,0000,0000,,f of x, y.
Dialogue: 0,1:08:37.11,1:08:40.42,Default,,0000,0000,0000,,
Dialogue: 0,1:08:40.42,1:08:42.01,Default,,0000,0000,0000,,The volume of the\Nupper hemisphere
Dialogue: 0,1:08:42.01,1:08:44.96,Default,,0000,0000,0000,,is the volume of everything\Nunder this graph, which
Dialogue: 0,1:08:44.96,1:08:46.38,Default,,0000,0000,0000,,is like a half.
Dialogue: 0,1:08:46.38,1:08:49.91,Default,,0000,0000,0000,,It's the northern hemisphere.
Dialogue: 0,1:08:49.91,1:08:52.82,Default,,0000,0000,0000,,dx dy, whatever is dx.
Dialogue: 0,1:08:52.82,1:08:55.15,Default,,0000,0000,0000,,So he tried to do\Nit, and he came up
Dialogue: 0,1:08:55.15,1:08:58.46,Default,,0000,0000,0000,,with something very ugly.
Dialogue: 0,1:08:58.46,1:09:02.08,Default,,0000,0000,0000,,Of course you can imagine\Nwhat he came up with.
Dialogue: 0,1:09:02.08,1:09:03.26,Default,,0000,0000,0000,,What would it be?
Dialogue: 0,1:09:03.26,1:09:04.18,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:09:04.18,1:09:05.86,Default,,0000,0000,0000,,Oh, God.
Dialogue: 0,1:09:05.86,1:09:10.34,Default,,0000,0000,0000,,x between minus r to r.
Dialogue: 0,1:09:10.34,1:09:30.68,Default,,0000,0000,0000,,y would be between 0\Nand-- you have to draw it.
Dialogue: 0,1:09:30.68,1:09:32.06,Default,,0000,0000,0000,,STUDENT: It's\Ngoing to be 0 or r.
Dialogue: 0,1:09:32.06,1:09:32.32,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,1:09:32.32,1:09:33.08,Default,,0000,0000,0000,,STUDENT: Oh, no.
Dialogue: 0,1:09:33.08,1:09:35.34,Default,,0000,0000,0000,,MAGDALENA TODA: So x\Nis between minus r--
Dialogue: 0,1:09:35.34,1:09:36.32,Default,,0000,0000,0000,,STUDENT: It's going to\Nbe as a function of x.
Dialogue: 0,1:09:36.32,1:09:37.80,Default,,0000,0000,0000,,MAGDALENA TODA: And this is x.
Dialogue: 0,1:09:37.80,1:09:39.38,Default,,0000,0000,0000,,And it's a function of x.
Dialogue: 0,1:09:39.38,1:09:44.56,Default,,0000,0000,0000,,And then you go square root\Nr squared minus x squared.
Dialogue: 0,1:09:44.56,1:09:47.05,Default,,0000,0000,0000,,It looks awful in\NCartesian coordinates.
Dialogue: 0,1:09:47.05,1:09:53.61,Default,,0000,0000,0000,,And then for f, he just\Nplugged in that thingy,
Dialogue: 0,1:09:53.61,1:09:55.57,Default,,0000,0000,0000,,and he said dy dx.
Dialogue: 0,1:09:55.57,1:09:58.06,Default,,0000,0000,0000,,And he would be\Nright, except that I
Dialogue: 0,1:09:58.06,1:09:59.53,Default,,0000,0000,0000,,would get a headache\Njust looking
Dialogue: 0,1:09:59.53,1:10:03.59,Default,,0000,0000,0000,,at it, because it's a mess.
Dialogue: 0,1:10:03.59,1:10:05.93,Default,,0000,0000,0000,,It's a horrible, horrible mess.
Dialogue: 0,1:10:05.93,1:10:09.10,Default,,0000,0000,0000,,I don't like it.
Dialogue: 0,1:10:09.10,1:10:13.86,Default,,0000,0000,0000,,So how am I going to solve\Nthis in polar coordinates?
Dialogue: 0,1:10:13.86,1:10:15.54,Default,,0000,0000,0000,,I still have the 2.
Dialogue: 0,1:10:15.54,1:10:16.81,Default,,0000,0000,0000,,I cannot get rid of the 2.
Dialogue: 0,1:10:16.81,1:10:21.35,Default,,0000,0000,0000,,How do I express--\Nin polar coordinates,
Dialogue: 0,1:10:21.35,1:10:25.77,Default,,0000,0000,0000,,the 2 would be one for the upper\Npart, one for the lower part--
Dialogue: 0,1:10:25.77,1:10:29.34,Default,,0000,0000,0000,,How do I express in polar\Ncoordinates the disc?
Dialogue: 0,1:10:29.34,1:10:31.21,Default,,0000,0000,0000,,Rho or r.
Dialogue: 0,1:10:31.21,1:10:37.97,Default,,0000,0000,0000,,r between 0 to R, and theta,\Nall the way from 0 to 2 pi.
Dialogue: 0,1:10:37.97,1:10:41.14,Default,,0000,0000,0000,,So I'm still sort of lucky\Nthat I'm in business.
Dialogue: 0,1:10:41.14,1:10:46.62,Default,,0000,0000,0000,,I go 0 to 2 pi\Nintegral from 0 to r,
Dialogue: 0,1:10:46.62,1:10:51.03,Default,,0000,0000,0000,,and for that guy, that\Nis in the integrand,
Dialogue: 0,1:10:51.03,1:10:54.26,Default,,0000,0000,0000,,I'm going to say squared of z.
Dialogue: 0,1:10:54.26,1:11:03.60,Default,,0000,0000,0000,,z is r squared minus-- who\Nis z squared plus y squared
Dialogue: 0,1:11:03.60,1:11:06.68,Default,,0000,0000,0000,,in polar coordinates?
Dialogue: 0,1:11:06.68,1:11:10.27,Default,,0000,0000,0000,,r squared. very good. r squared.
Dialogue: 0,1:11:10.27,1:11:13.64,Default,,0000,0000,0000,,Don't forget that\Ninstead of dy dx,
Dialogue: 0,1:11:13.64,1:11:19.58,Default,,0000,0000,0000,,you have to say times r,\Nthe Jacobian, dr d theta.
Dialogue: 0,1:11:19.58,1:11:23.56,Default,,0000,0000,0000,,Can we solve this, and\Nfind the correct formula?
Dialogue: 0,1:11:23.56,1:11:25.84,Default,,0000,0000,0000,,That's what I'm talking about.
Dialogue: 0,1:11:25.84,1:11:27.41,Default,,0000,0000,0000,,We need the u substitution.
Dialogue: 0,1:11:27.41,1:11:30.80,Default,,0000,0000,0000,,Without the u substitution,\Nwe will be dead meat.
Dialogue: 0,1:11:30.80,1:11:33.06,Default,,0000,0000,0000,,But I don't know how\Nto do u substitution,
Dialogue: 0,1:11:33.06,1:11:35.38,Default,,0000,0000,0000,,so I need your help.
Dialogue: 0,1:11:35.38,1:11:37.77,Default,,0000,0000,0000,,Of course you can help me.
Dialogue: 0,1:11:37.77,1:11:39.20,Default,,0000,0000,0000,,Who is the constant?
Dialogue: 0,1:11:39.20,1:11:41.11,Default,,0000,0000,0000,,R is the constant.
Dialogue: 0,1:11:41.11,1:11:43.03,Default,,0000,0000,0000,,It's a number.
Dialogue: 0,1:11:43.03,1:11:46.31,Default,,0000,0000,0000,,Little r is a variable.
Dialogue: 0,1:11:46.31,1:11:48.26,Default,,0000,0000,0000,,Little r is a variable.
Dialogue: 0,1:11:48.26,1:11:53.62,Default,,0000,0000,0000,,
Dialogue: 0,1:11:53.62,1:11:55.57,Default,,0000,0000,0000,,STUDENT: r squared\Nis going to be the u.
Dialogue: 0,1:11:55.57,1:11:56.78,Default,,0000,0000,0000,,MAGDALENA TODA: u, very good.
Dialogue: 0,1:11:56.78,1:11:58.88,Default,,0000,0000,0000,,r squared minus r squared.
Dialogue: 0,1:11:58.88,1:12:01.74,Default,,0000,0000,0000,,How come this is\Nworking so well?
Dialogue: 0,1:12:01.74,1:12:07.34,Default,,0000,0000,0000,,Look why du will be\Nconstant prime 0 minus 2rdr.
Dialogue: 0,1:12:07.34,1:12:10.01,Default,,0000,0000,0000,,
Dialogue: 0,1:12:10.01,1:12:18.43,Default,,0000,0000,0000,,So I take this couple\Ncalled rdr, this block,
Dialogue: 0,1:12:18.43,1:12:21.78,Default,,0000,0000,0000,,and I identify the\Nblock over here.
Dialogue: 0,1:12:21.78,1:12:31.11,Default,,0000,0000,0000,,And rdr represents du\Nover minus 2, right?
Dialogue: 0,1:12:31.11,1:12:32.82,Default,,0000,0000,0000,,So I have to be\Nsmart and attentive,
Dialogue: 0,1:12:32.82,1:12:36.62,Default,,0000,0000,0000,,because if I make a mistake\Nat the end, it's all over.
Dialogue: 0,1:12:36.62,1:12:41.33,Default,,0000,0000,0000,,So 2 tiomes integral\Nfrom 0 to 2 pi.
Dialogue: 0,1:12:41.33,1:12:44.62,Default,,0000,0000,0000,,I could get rid of\Nthat and say just 2 pi.
Dialogue: 0,1:12:44.62,1:12:46.28,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:12:46.28,1:12:53.38,Default,,0000,0000,0000,,I could say 1 is theta-- as\Nthe product, go out-- times--
Dialogue: 0,1:12:53.38,1:12:57.38,Default,,0000,0000,0000,,and this is my integral that\NI'm worried about, the one only
Dialogue: 0,1:12:57.38,1:13:00.33,Default,,0000,0000,0000,,in r.
Dialogue: 0,1:13:00.33,1:13:01.80,Default,,0000,0000,0000,,Let me review it.
Dialogue: 0,1:13:01.80,1:13:06.72,Default,,0000,0000,0000,,
Dialogue: 0,1:13:06.72,1:13:09.14,Default,,0000,0000,0000,,This is the only one\NI'm worried about.
Dialogue: 0,1:13:09.14,1:13:10.84,Default,,0000,0000,0000,,This is a piece of cake.
Dialogue: 0,1:13:10.84,1:13:12.61,Default,,0000,0000,0000,,This is 2, this is 2 pi.
Dialogue: 0,1:13:12.61,1:13:14.01,Default,,0000,0000,0000,,This whole thing is 4 pi a.
Dialogue: 0,1:13:14.01,1:13:18.36,Default,,0000,0000,0000,,At least I got some 4 pi out.
Dialogue: 0,1:13:18.36,1:13:19.97,Default,,0000,0000,0000,,What have I done in here?
Dialogue: 0,1:13:19.97,1:13:23.30,Default,,0000,0000,0000,,I've applied the u\Nsubstitution, and I
Dialogue: 0,1:13:23.30,1:13:25.10,Default,,0000,0000,0000,,have to be doing a better job.
Dialogue: 0,1:13:25.10,1:13:30.69,Default,,0000,0000,0000,,I get 4 pi times what is\Nit after u substitution.
Dialogue: 0,1:13:30.69,1:13:37.08,Default,,0000,0000,0000,,This guy was minus\N1/2 du, right?
Dialogue: 0,1:13:37.08,1:13:40.30,Default,,0000,0000,0000,,This fellow is squared\Nu, [? squared ?]
Dialogue: 0,1:13:40.30,1:13:42.11,Default,,0000,0000,0000,,squared u as a power.
Dialogue: 0,1:13:42.11,1:13:43.11,Default,,0000,0000,0000,,STUDENT: u to the 1/2.
Dialogue: 0,1:13:43.11,1:13:44.82,Default,,0000,0000,0000,,MAGDALENA TODA: u\Nto the one half.
Dialogue: 0,1:13:44.82,1:13:51.88,Default,,0000,0000,0000,,And for the integral, what\Nin the world do I write?
Dialogue: 0,1:13:51.88,1:13:52.71,Default,,0000,0000,0000,,STUDENT: r squared--
Dialogue: 0,1:13:52.71,1:13:54.46,Default,,0000,0000,0000,,MAGDALENA TODA: OK.
Dialogue: 0,1:13:54.46,1:14:03.23,Default,,0000,0000,0000,,So when little r is 0, u\Nis going to be r squared.
Dialogue: 0,1:14:03.23,1:14:08.79,Default,,0000,0000,0000,,When little r is\Nbig R, you get 0.
Dialogue: 0,1:14:08.79,1:14:11.02,Default,,0000,0000,0000,,Now you have to\Nhelp me finish this.
Dialogue: 0,1:14:11.02,1:14:12.71,Default,,0000,0000,0000,,It should be a piece of cake.
Dialogue: 0,1:14:12.71,1:14:15.65,Default,,0000,0000,0000,,I cannot believe it's hard.
Dialogue: 0,1:14:15.65,1:14:19.44,Default,,0000,0000,0000,,What is the integral of 4 pi?
Dialogue: 0,1:14:19.44,1:14:20.86,Default,,0000,0000,0000,,Copy and paste.
Dialogue: 0,1:14:20.86,1:14:25.19,Default,,0000,0000,0000,,Minus 1/2, integrate\Ny to the 1/2.
Dialogue: 0,1:14:25.19,1:14:27.17,Default,,0000,0000,0000,,STUDENT: 2/3u to the 3/2.
Dialogue: 0,1:14:27.17,1:14:34.32,Default,,0000,0000,0000,,MAGDALENA TODA: 2/3 u to the\N3/2, between u equals 0 up,
Dialogue: 0,1:14:34.32,1:14:37.57,Default,,0000,0000,0000,,and u equals r squared down.
Dialogue: 0,1:14:37.57,1:14:38.61,Default,,0000,0000,0000,,It still looks bad, but--
Dialogue: 0,1:14:38.61,1:14:40.11,Default,,0000,0000,0000,,STUDENT: You've got\Na negative sign.
Dialogue: 0,1:14:40.11,1:14:41.96,Default,,0000,0000,0000,,MAGDALENA TODA: I've\Ngot a negative sign.
Dialogue: 0,1:14:41.96,1:14:42.94,Default,,0000,0000,0000,,STUDENT: Where is it--
Dialogue: 0,1:14:42.94,1:14:46.09,Default,,0000,0000,0000,,MAGDALENA TODA: So when\NI go this minus that,
Dialogue: 0,1:14:46.09,1:14:47.78,Default,,0000,0000,0000,,it's going to be very nice.
Dialogue: 0,1:14:47.78,1:14:48.47,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:14:48.47,1:14:56.39,Default,,0000,0000,0000,,I'm going to say minus 4\Npi over 2 times 2 over 3.
Dialogue: 0,1:14:56.39,1:14:59.19,Default,,0000,0000,0000,,I should have simplified\Nthem from the beginning.
Dialogue: 0,1:14:59.19,1:15:05.22,Default,,0000,0000,0000,,I have minus 5 pi over\N3 times at 0 I have 0.
Dialogue: 0,1:15:05.22,1:15:09.26,Default,,0000,0000,0000,,At r squared, I have r\Nsquared, and the square root
Dialogue: 0,1:15:09.26,1:15:11.74,Default,,0000,0000,0000,,is r, r cubed.
Dialogue: 0,1:15:11.74,1:15:12.73,Default,,0000,0000,0000,,r cubed.
Dialogue: 0,1:15:12.73,1:15:19.69,Default,,0000,0000,0000,,
Dialogue: 0,1:15:19.69,1:15:22.06,Default,,0000,0000,0000,,Oh my God, look how\Nbeautiful it is.
Dialogue: 0,1:15:22.06,1:15:24.00,Default,,0000,0000,0000,,Two minuses in a row.
Dialogue: 0,1:15:24.00,1:15:27.15,Default,,0000,0000,0000,,Multiply, give me a plus.
Dialogue: 0,1:15:27.15,1:15:28.32,Default,,0000,0000,0000,,STUDENT: This is the answer.
Dialogue: 0,1:15:28.32,1:15:29.77,Default,,0000,0000,0000,,MAGDALENA TODA: Plus.
Dialogue: 0,1:15:29.77,1:15:37.15,Default,,0000,0000,0000,,4 pi up over 3 down, r cubed.
Dialogue: 0,1:15:37.15,1:15:40.67,Default,,0000,0000,0000,,So we proved something\Nthat is essential,
Dialogue: 0,1:15:40.67,1:15:42.90,Default,,0000,0000,0000,,and we knew it from\Nwhen we were in school,
Dialogue: 0,1:15:42.90,1:15:46.14,Default,,0000,0000,0000,,but they told us that\Nwe cannot prove it,
Dialogue: 0,1:15:46.14,1:15:50.56,Default,,0000,0000,0000,,because we couldn't prove that\Nthe volume of a ball was 4 pi r
Dialogue: 0,1:15:50.56,1:15:51.70,Default,,0000,0000,0000,,cubed over 3.
Dialogue: 0,1:15:51.70,1:15:52.76,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:15:52.76,1:15:55.72,Default,,0000,0000,0000,,STUDENT: Why are the limits\Nof integration reversed?
Dialogue: 0,1:15:55.72,1:15:57.01,Default,,0000,0000,0000,,Why is r squared on the bottom?
Dialogue: 0,1:15:57.01,1:16:02.35,Default,,0000,0000,0000,,MAGDALENA TODA: Because\Nfirst comes little r, 0,
Dialogue: 0,1:16:02.35,1:16:06.31,Default,,0000,0000,0000,,and then comes little r to\Nbe big R. When I plug them
Dialogue: 0,1:16:06.31,1:16:09.64,Default,,0000,0000,0000,,in in this order-- so\Nlet's plug them in first,
Dialogue: 0,1:16:09.64,1:16:11.05,Default,,0000,0000,0000,,little r equals 0.
Dialogue: 0,1:16:11.05,1:16:15.57,Default,,0000,0000,0000,,I get, for the bottom part,\NI get u equals r squared,
Dialogue: 0,1:16:15.57,1:16:18.93,Default,,0000,0000,0000,,and when little\Nr equals big R, I
Dialogue: 0,1:16:18.93,1:16:21.81,Default,,0000,0000,0000,,get big R squared minus\Nbig R squared equals 0.
Dialogue: 0,1:16:21.81,1:16:24.06,Default,,0000,0000,0000,,And that's the good\Nthing, because when
Dialogue: 0,1:16:24.06,1:16:28.75,Default,,0000,0000,0000,,I do that, I get a minus, and\Nwith the minus I already had,
Dialogue: 0,1:16:28.75,1:16:29.80,Default,,0000,0000,0000,,I get a plus.
Dialogue: 0,1:16:29.80,1:16:33.47,Default,,0000,0000,0000,,And the volume is a positive\Nvolume, like every volume.
Dialogue: 0,1:16:33.47,1:16:36.11,Default,,0000,0000,0000,,4 pi [INAUDIBLE].
Dialogue: 0,1:16:36.11,1:16:39.38,Default,,0000,0000,0000,,So that's it for today.
Dialogue: 0,1:16:39.38,1:16:42.05,Default,,0000,0000,0000,,We finished 12-- what is that?
Dialogue: 0,1:16:42.05,1:16:44.07,Default,,0000,0000,0000,,12.3, polar coordinates.
Dialogue: 0,1:16:44.07,1:16:49.54,Default,,0000,0000,0000,,And we will next time\Ndo some homework.
Dialogue: 0,1:16:49.54,1:16:52.03,Default,,0000,0000,0000,,Ah, I opened the\Nhomework for you.
Dialogue: 0,1:16:52.03,1:16:55.01,Default,,0000,0000,0000,,So go ahead and do at least\Nthe first 10 problems.
Dialogue: 0,1:16:55.01,1:16:57.99,Default,,0000,0000,0000,,If you have difficulties,\Nlet me know on Tuesday,
Dialogue: 0,1:16:57.99,1:17:02.46,Default,,0000,0000,0000,,so we can work some in class.
Dialogue: 0,1:17:02.46,1:17:04.95,Default,,0000,0000,0000,,STUDENT: [? You do ?] so much.
Dialogue: 0,1:17:04.95,1:17:09.58,Default,,0000,0000,0000,,STUDENT: So, I went to the\N[INAUDIBLE], and I asked them,
Dialogue: 0,1:17:09.58,1:17:10.42,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:17:10.42,1:17:13.89,Default,,0000,0000,0000,,
Dialogue: 0,1:17:13.89,1:17:16.38,Default,,0000,0000,0000,,[SIDE CONVERSATION]
Dialogue: 0,1:17:16.38,1:18:34.30,Default,,0000,0000,0000,,
Dialogue: 0,1:18:34.30,1:18:35.80,Default,,0000,0000,0000,,STUDENT: Can you\Nimagine two years
Dialogue: 0,1:18:35.80,1:18:38.29,Default,,0000,0000,0000,,of a calculus that's the\Nequivalent to [? American ?]
Dialogue: 0,1:18:38.29,1:18:39.66,Default,,0000,0000,0000,,and only two credits?
Dialogue: 0,1:18:39.66,1:18:41.29,Default,,0000,0000,0000,,MAGDALENA TODA:\NBecause in your system,
Dialogue: 0,1:18:41.29,1:18:44.28,Default,,0000,0000,0000,,everything was pretty\Nmuch accelerated.
Dialogue: 0,1:18:44.28,1:18:46.78,Default,,0000,0000,0000,,STUDENT: Yeah, and\Nthey say, no, no, no--
Dialogue: 0,1:18:46.78,1:18:48.27,Default,,0000,0000,0000,,I had to ask him again.
Dialogue: 0,1:18:48.27,1:18:52.76,Default,,0000,0000,0000,,[INAUDIBLE] calculus,\Nin two years,
Dialogue: 0,1:18:52.76,1:18:56.26,Default,,0000,0000,0000,,that is only equivalent\Nto two credits.
Dialogue: 0,1:18:56.26,1:18:58.25,Default,,0000,0000,0000,,I was like--
Dialogue: 0,1:18:58.25,1:18:59.75,Default,,0000,0000,0000,,MAGDALENA TODA:\NAnyway, what happens
Dialogue: 0,1:18:59.75,1:19:03.24,Default,,0000,0000,0000,,is that we used to have\Nvery good evaluators
Dialogue: 0,1:19:03.24,1:19:06.24,Default,,0000,0000,0000,,in the registrar's office, and\Nmost of those people retired
Dialogue: 0,1:19:06.24,1:19:09.23,Default,,0000,0000,0000,,or they got promoted in other\Nadministrative positions.
Dialogue: 0,1:19:09.23,1:19:11.73,Default,,0000,0000,0000,,So they have three new hires.
Dialogue: 0,1:19:11.73,1:19:14.60,Default,,0000,0000,0000,,Those guys, they don't\Nknow what they are doing.
Dialogue: 0,1:19:14.60,1:19:17.06,Default,,0000,0000,0000,,Imagine, you would\Nfinish, graduate, today,
Dialogue: 0,1:19:17.06,1:19:19.53,Default,,0000,0000,0000,,next week, you go\Nfor the registrar.
Dialogue: 0,1:19:19.53,1:19:21.50,Default,,0000,0000,0000,,You don't know\Nwhat you're doing.
Dialogue: 0,1:19:21.50,1:19:22.48,Default,,0000,0000,0000,,You need time.
Dialogue: 0,1:19:22.48,1:19:22.98,Default,,0000,0000,0000,,Yes?
Dialogue: 0,1:19:22.98,1:19:25.44,Default,,0000,0000,0000,,STUDENT: I had a question\Nabout the homework.
Dialogue: 0,1:19:25.44,1:19:27.40,Default,,0000,0000,0000,,I'll wait for [INAUDIBLE].
Dialogue: 0,1:19:27.40,1:19:28.40,Default,,0000,0000,0000,,MAGDALENA TODA: It's OK.
Dialogue: 0,1:19:28.40,1:19:30.38,Default,,0000,0000,0000,,Do you have secrets?
Dialogue: 0,1:19:30.38,1:19:31.85,Default,,0000,0000,0000,,STUDENT: No, I don't.
Dialogue: 0,1:19:31.85,1:19:33.83,Default,,0000,0000,0000,,MAGDALENA TODA: Homework\Nis due the 32st.
Dialogue: 0,1:19:33.83,1:19:34.81,Default,,0000,0000,0000,,STUDENT: No, I had a\Nquestion from the homework.
Dialogue: 0,1:19:34.81,1:19:35.30,Default,,0000,0000,0000,,Like I had a problem that I\Nwas working on, and I was like
Dialogue: 0,1:19:35.30,1:19:36.72,Default,,0000,0000,0000,,MAGDALENA TODA:\NFrom the homework.
Dialogue: 0,1:19:36.72,1:19:39.25,Default,,0000,0000,0000,,OK You can wait.
Dialogue: 0,1:19:39.25,1:19:42.21,Default,,0000,0000,0000,,You guys have other,\Nmore basic questions?
Dialogue: 0,1:19:42.21,1:19:43.04,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:19:43.04,1:19:49.21,Default,,0000,0000,0000,,
Dialogue: 0,1:19:49.21,1:19:50.96,Default,,0000,0000,0000,,MAGDALENA TODA: There\Nis only one meeting.
Dialogue: 0,1:19:50.96,1:19:53.93,Default,,0000,0000,0000,,Oh, you mean-- Ah.
Dialogue: 0,1:19:53.93,1:19:55.42,Default,,0000,0000,0000,,Yes, I do.
Dialogue: 0,1:19:55.42,1:19:59.87,Default,,0000,0000,0000,,I have the following\Nthree-- Tuesday,
Dialogue: 0,1:19:59.87,1:20:04.99,Default,,0000,0000,0000,,Wednesday, and Friday-\Nno, Tuesday, Wednesday,
Dialogue: 0,1:20:04.99,1:20:05.62,Default,,0000,0000,0000,,and Thursday.
Dialogue: 0,1:20:05.62,1:20:09.87,Default,,0000,0000,0000,,On Friday we can have something,\Nsome special arrangement.
Dialogue: 0,1:20:09.87,1:20:12.41,Default,,0000,0000,0000,,This Friday?
Dialogue: 0,1:20:12.41,1:20:16.75,Default,,0000,0000,0000,,OK, how about like 11:15.
Dialogue: 0,1:20:16.75,1:20:22.57,Default,,0000,0000,0000,,Today, I have--\NI have right now.
Dialogue: 0,1:20:22.57,1:20:23.52,Default,,0000,0000,0000,,2:00.
Dialogue: 0,1:20:23.52,1:20:26.78,Default,,0000,0000,0000,,And I think the grad\Nstudents will come later.
Dialogue: 0,1:20:26.78,1:20:28.77,Default,,0000,0000,0000,,So you can just right now.
Dialogue: 0,1:20:28.77,1:20:32.24,Default,,0000,0000,0000,,And tomorrow around 11:15,\Nbecause I have meetings
Dialogue: 0,1:20:32.24,1:20:34.73,Default,,0000,0000,0000,,before 11 at the college.
Dialogue: 0,1:20:34.73,1:20:37.26,Default,,0000,0000,0000,,STUDENT: Do you mind if I go\Nget something to eat first?
Dialogue: 0,1:20:37.26,1:20:39.13,Default,,0000,0000,0000,,Or how long do you think\Nthey'll be in your office?
Dialogue: 0,1:20:39.13,1:20:40.12,Default,,0000,0000,0000,,MAGDALENA TODA:\NEven if they come,
Dialogue: 0,1:20:40.12,1:20:42.10,Default,,0000,0000,0000,,I'm going to stop\Nthem and talk to you,
Dialogue: 0,1:20:42.10,1:20:43.58,Default,,0000,0000,0000,,so don't worry about it.
Dialogue: 0,1:20:43.58,1:20:44.07,Default,,0000,0000,0000,,STUDENT: Thank you very much.
Dialogue: 0,1:20:44.07,1:20:44.57,Default,,0000,0000,0000,,I'll see you later.
Dialogue: 0,1:20:44.57,1:20:46.05,Default,,0000,0000,0000,,STUDENT: I just wanted to say\NI'm sorry for coming in late.
Dialogue: 0,1:20:46.05,1:20:47.04,Default,,0000,0000,0000,,I slept in a little\Nbit this morning--
Dialogue: 0,1:20:47.04,1:20:49.46,Default,,0000,0000,0000,,MAGDALENA TODA: Did you\Nget the chance to sign?
Dialogue: 0,1:20:49.46,1:20:50.00,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,1:20:50.00,1:20:50.99,Default,,0000,0000,0000,,MAGDALENA TODA:\NThere is no problem.
Dialogue: 0,1:20:50.99,1:20:51.49,Default,,0000,0000,0000,,I'm--
Dialogue: 0,1:20:51.49,1:20:55.93,Default,,0000,0000,0000,,STUDENT: I woke up at like\N12:30-- I woke up at like 11:30
Dialogue: 0,1:20:55.93,1:20:59.61,Default,,0000,0000,0000,,and I just fell right back\Nasleep, and then I got up
Dialogue: 0,1:20:59.61,1:21:01.36,Default,,0000,0000,0000,,and I looked at my\Nphone and it was 12:30,
Dialogue: 0,1:21:01.36,1:21:03.34,Default,,0000,0000,0000,,and I was like, I\Nhave class right now.
Dialogue: 0,1:21:03.34,1:21:04.82,Default,,0000,0000,0000,,And so what happened was like--
Dialogue: 0,1:21:04.82,1:21:05.81,Default,,0000,0000,0000,,MAGDALENA TODA: You were tired.
Dialogue: 0,1:21:05.81,1:21:06.83,Default,,0000,0000,0000,,You were doing\Nhomework until late.
Dialogue: 0,1:21:06.83,1:21:08.61,Default,,0000,0000,0000,,STUDENT: --homework\Nand like, I usually
Dialogue: 0,1:21:08.61,1:21:10.97,Default,,0000,0000,0000,,am on for an\Nearlier class, and I
Dialogue: 0,1:21:10.97,1:21:12.93,Default,,0000,0000,0000,,didn't go to bed earlier\Nthan I did last night,
Dialogue: 0,1:21:12.93,1:21:14.77,Default,,0000,0000,0000,,and so I just overslept.
Dialogue: 0,1:21:14.77,1:21:17.13,Default,,0000,0000,0000,,MAGDALENA TODA: I\Ndid the same, anyway.
Dialogue: 0,1:21:17.13,1:21:18.67,Default,,0000,0000,0000,,I have similar experience.
Dialogue: 0,1:21:18.67,1:21:20.09,Default,,0000,0000,0000,,STUDENT: You have\Na very nice day.
Dialogue: 0,1:21:20.09,1:21:21.17,Default,,0000,0000,0000,,MAGDALENA TODA: Thank you.
Dialogue: 0,1:21:21.17,1:21:21.89,Default,,0000,0000,0000,,You too.
Dialogue: 0,1:21:21.89,1:21:24.34,Default,,0000,0000,0000,,So, show me what\Nyou want to ask.
Dialogue: 0,1:21:24.34,1:21:25.53,Default,,0000,0000,0000,,STUDENT: There it was.
Dialogue: 0,1:21:25.53,1:21:27.20,Default,,0000,0000,0000,,I looked at that\Nproblem, and I thought,
Dialogue: 0,1:21:27.20,1:21:29.87,Default,,0000,0000,0000,,that's extremely\Nsimple, acceleration--
Dialogue: 0,1:21:29.87,1:21:31.75,Default,,0000,0000,0000,,MAGDALENA TODA: Are they\Nindependent, really?
Dialogue: 0,1:21:31.75,1:21:32.29,Default,,0000,0000,0000,,STUDENT: Huh?
Dialogue: 0,1:21:32.29,1:21:34.93,Default,,0000,0000,0000,,MAGDALENA TODA: Are they--\Nb and t are independent?
Dialogue: 0,1:21:34.93,1:21:36.66,Default,,0000,0000,0000,,I need to stop.
Dialogue: 0,1:21:36.66,1:21:39.11,Default,,0000,0000,0000,,STUDENT: But I\Ndidn't even bother.