0:00:00.000,0:00:01.708 MAGDALENA TODA: I'm[br]starting early, am I? 0:00:01.708,0:00:04.000 It's exactly 12:30. 0:00:04.000,0:00:07.000 The weather is getting[br]better, hopefully, 0:00:07.000,0:00:14.000 and not too many people[br]should miss class today. 0:00:14.000,0:00:18.000 Can you start an attendance[br]sheet for me [INAUDIBLE]? 0:00:18.000,0:00:22.000 I know I can count on you. 0:00:22.000,0:00:22.500 OK. 0:00:22.500,0:00:25.500 I have good markers today. 0:00:25.500,0:00:31.304 I'm going to go ahead[br]and talk about 12.3, 0:00:31.304,0:00:34.280 double integrals in[br]polar coordinates. 0:00:34.280,0:00:36.264 These are all friends of yours. 0:00:36.264,0:00:55.112 0:00:55.112,0:00:58.800 You've seen until now[br]only double integrals that 0:00:58.800,0:01:05.580 involve the rectangles, either[br]a rectangle, we saw [INAUDIBLE], 0:01:05.580,0:01:10.610 and we saw some type[br]of double integrals, 0:01:10.610,0:01:17.600 of course that involved[br]x and y, so-called type 0:01:17.600,0:01:21.210 1 and type 2[br]regions, which were-- 0:01:21.210,0:01:24.804 so we saw the rectangular case. 0:01:24.804,0:01:29.774 You have ab plus[br]cd, a rectangle. 0:01:29.774,0:01:33.253 You have what other kind[br]of a velocity [INAUDIBLE] 0:01:33.253,0:01:43.400 over the the main of the shape[br]x between a and be and y. 0:01:43.400,0:01:49.070 You write wild and happy[br]from bottom to top. 0:01:49.070,0:01:54.480 That's called the wild--[br]not wild, the vertical strip 0:01:54.480,0:01:58.994 method, where y will be[br]between the bottom function 0:01:58.994,0:02:02.906 f of x and the top[br]function f of x. 0:02:02.906,0:02:05.351 And last time I[br]took examples where 0:02:05.351,0:02:08.285 f and g were both positive, but[br]remember, you don't have to. 0:02:08.285,0:02:12.532 All you have to have is that[br]g is always greater than f, 0:02:12.532,0:02:14.105 or equal at some point. 0:02:14.105,0:02:16.810 0:02:16.810,0:02:21.190 And then what else do[br]we have for these cases? 0:02:21.190,0:02:24.922 These are all[br]continuous functions. 0:02:24.922,0:02:26.770 What else did we have? 0:02:26.770,0:02:29.030 We had two domains. 0:02:29.030,0:02:33.440 0:02:33.440,0:02:35.400 Had one and had two. 0:02:35.400,0:02:38.340 0:02:38.340,0:02:42.980 Where what was going on,[br]we have played a little bit 0:02:42.980,0:02:50.120 around with y between c[br]and d limits with points. 0:02:50.120,0:02:53.040 These are horizontal,[br]so we take the domain 0:02:53.040,0:02:58.662 as being defined by these[br]horizontal strips between let's 0:02:58.662,0:03:00.045 say a function. 0:03:00.045,0:03:03.840 Again, I need to rotate my head,[br]but I didn't do my yoga today, 0:03:03.840,0:03:07.140 so it's a little bit sticky. 0:03:07.140,0:03:07.930 I'll try. 0:03:07.930,0:03:19.354 x equals F of y, and x equals[br]G of y, assuming, of course, 0:03:19.354,0:03:24.070 that f of y is always greater[br]than or equal to g of y, 0:03:24.070,0:03:28.272 and the rest of the[br]apparatus is in place. 0:03:28.272,0:03:31.680 Those are not so[br]hard to understand. 0:03:31.680,0:03:33.080 We played around. 0:03:33.080,0:03:35.500 We switched the integrals. 0:03:35.500,0:03:38.920 We changed the order of[br]integration from dy dx 0:03:38.920,0:03:43.190 to dx dy, so we have[br]to change the domain. 0:03:43.190,0:03:45.785 We went from[br]vertical strip method 0:03:45.785,0:03:52.050 to horizontal strip method[br]or the other way around. 0:03:52.050,0:03:57.103 And for what kind of[br]example, something 0:03:57.103,0:03:59.940 like that-- I think it[br]was a leaf like that, 0:03:59.940,0:04:02.730 we said, let's compute[br]the area or compute 0:04:02.730,0:04:09.750 another kind of double[br]integral over this leaf in two 0:04:09.750,0:04:10.840 different ways. 0:04:10.840,0:04:14.440 And we did it with[br]vertical strips, 0:04:14.440,0:04:17.135 and we did the same[br]with horizontal strips. 0:04:17.135,0:04:20.329 0:04:20.329,0:04:22.650 So we reversed the[br]order of integration, 0:04:22.650,0:04:27.100 and we said, I'm having the[br]double integral over domain 0:04:27.100,0:04:31.555 of God knows what, f of[br]xy, continuous function, 0:04:31.555,0:04:37.440 positive, continuous whenever[br]you want, and we said da. 0:04:37.440,0:04:40.160 We didn't quite specify[br]the meaning of da. 0:04:40.160,0:04:43.025 We said that da is[br]the area element, 0:04:43.025,0:04:47.270 but that sounds a little[br]bit weird, because it makes 0:04:47.270,0:04:51.500 you think of surfaces,[br]and an area element 0:04:51.500,0:04:53.885 doesn't have to be a[br]little square in general. 0:04:53.885,0:04:59.290 It could be something like a[br]patch on a surface, bounded 0:04:59.290,0:05:04.100 by two curves within your[br]segments in each direction. 0:05:04.100,0:05:06.466 So you think, well, I[br]don't know what that is. 0:05:06.466,0:05:07.840 I'll tell you[br]today what that is. 0:05:07.840,0:05:11.130 It's a mysterious thing,[br]it's really beautiful, 0:05:11.130,0:05:12.620 and we'll talk about it. 0:05:12.620,0:05:15.440 Now, what did we do last time? 0:05:15.440,0:05:19.360 We applied the two[br]theorems that allowed 0:05:19.360,0:05:23.780 us to do this both ways. 0:05:23.780,0:05:29.360 Integral from a to b, what was[br]my usual [? wrist ?] is down, 0:05:29.360,0:05:32.426 f of x is in g of x, right? 0:05:32.426,0:05:36.130 0:05:36.130,0:05:37.340 dy dx. 0:05:37.340,0:05:39.750 So if you do it in[br]this order, it's 0:05:39.750,0:05:44.780 going to be the same as if[br]you do it in the other order. 0:05:44.780,0:05:53.464 ab are these guys, and then[br]this was cd on the y-axis. 0:05:53.464,0:05:56.810 This is the range between[br]c and d in altitudes. 0:05:56.810,0:06:00.740 So we have integral from[br]c to d, integral from, 0:06:00.740,0:06:02.910 I don't know what they will be. 0:06:02.910,0:06:07.140 This big guy I'm talking--[br]which one is the one? 0:06:07.140,0:06:11.335 This one, that's going to[br]be called x equals f of y, 0:06:11.335,0:06:17.596 or g of y, and let's put the[br]big one G and the smaller one, 0:06:17.596,0:06:19.560 x equals F of y. 0:06:19.560,0:06:24.038 So you have to [? re-denote ?][br]these functions, 0:06:24.038,0:06:31.430 these inverse functions, and[br]use them as functions of y. 0:06:31.430,0:06:34.640 So it makes sense to[br]say-- what did we do? 0:06:34.640,0:06:39.620 We first integrated respect to[br]x between two functions of y. 0:06:39.620,0:06:44.170 That was the so-called[br]horizontal strip method, dy. 0:06:44.170,0:06:48.320 So I have summarized[br]the ideas from last time 0:06:48.320,0:06:53.350 that we worked with, generally[br]with corners x and y. 0:06:53.350,0:06:55.953 We were very happy about them. 0:06:55.953,0:07:00.050 We had the rectangular[br]domain, where x was between ab 0:07:00.050,0:07:01.770 and y was between cd. 0:07:01.770,0:07:05.640 Then we went to type 1, not[br]diabetes, just type 1 region, 0:07:05.640,0:07:09.070 type 2, and those[br]guys are related. 0:07:09.070,0:07:12.325 So if you understood 1 and[br]understood the other one, 0:07:12.325,0:07:15.300 and if you have a[br]nice domain like that, 0:07:15.300,0:07:18.090 you can compute the[br]area or something. 0:07:18.090,0:07:21.070 The area will correspond[br]to x equals 1. 0:07:21.070,0:07:24.086 So if f is 1, then[br]that's the area. 0:07:24.086,0:07:28.850 That will also be a[br]volume of a cylinder based 0:07:28.850,0:07:33.130 on that region with height 1. 0:07:33.130,0:07:36.970 Imagine a can of Coke[br]that has height 1, 0:07:36.970,0:07:40.920 and-- maybe better,[br]chocolate cake, 0:07:40.920,0:07:43.820 that has the shape of[br]this leaf on the bottom, 0:07:43.820,0:07:47.710 and then its height[br]is 1 everywhere. 0:07:47.710,0:07:51.790 So if you put 1 here, and[br]you get the area element, 0:07:51.790,0:07:54.820 and then everything[br]else can be done 0:07:54.820,0:07:59.960 by reversing the order of[br]integration if f is continuous. 0:07:59.960,0:08:02.860 But for polar[br]coordinates, the situation 0:08:02.860,0:08:08.190 has to be reconsidered almost[br]entirely, because the area 0:08:08.190,0:08:17.736 element, da is called[br]the area element for us, 0:08:17.736,0:08:25.745 was equal to dx dy for the[br]cartesian coordinate case. 0:08:25.745,0:08:32.159 0:08:32.159,0:08:36.625 And here I'm making a[br]weird face, I'm weird, no? 0:08:36.625,0:08:39.950 Saying, what am I going[br]to do, what is this 0:08:39.950,0:08:44.415 going to become for[br]polar coordinates? 0:08:44.415,0:08:47.710 0:08:47.710,0:08:52.610 And now you go, oh my God,[br]not polar coordinates. 0:08:52.610,0:08:54.150 Those were my[br]enemies in Calc II. 0:08:54.150,0:08:55.870 Many people told me that. 0:08:55.870,0:09:01.870 And I tried to go[br]into my time machine 0:09:01.870,0:09:04.540 and go back something[br]like 25 years ago 0:09:04.540,0:09:07.980 and see how I felt about[br]them, and I remember that. 0:09:07.980,0:09:12.120 I didn't get them from[br]the first 48 hours 0:09:12.120,0:09:15.730 after I was exposed to them. 0:09:15.730,0:09:18.040 Therefore, let's[br]do some preview. 0:09:18.040,0:09:21.490 What were those[br]polar coordinates? 0:09:21.490,0:09:25.840 Polar coordinates were[br]a beautiful thing, 0:09:25.840,0:09:27.740 these guys from trig. 0:09:27.740,0:09:31.500 Trig was your friend hopefully. 0:09:31.500,0:09:34.660 And what did we have[br]in trigonometry? 0:09:34.660,0:09:38.710 In trigonometry, we had[br]a point on a circle. 0:09:38.710,0:09:41.270 This is not the unit[br]trigonometric circle, 0:09:41.270,0:09:45.205 it's a circle of--[br]bless you-- radius r. 0:09:45.205,0:09:49.760 I'm a little bit shifted[br]by a phase of phi 0. 0:09:49.760,0:09:54.510 So you have a radius r. 0:09:54.510,0:09:56.970 And let's call that little r. 0:09:56.970,0:10:02.710 0:10:02.710,0:10:06.372 And then, we say, OK,[br]how about the angle? 0:10:06.372,0:10:08.800 That's the second[br]polar coordinate. 0:10:08.800,0:10:16.395 The angle by measuring[br]from the, what 0:10:16.395,0:10:17.835 is this called, the x-axis. 0:10:17.835,0:10:21.240 0:10:21.240,0:10:25.710 Origin, x-axis, o, x,[br]going counterclockwise, 0:10:25.710,0:10:28.370 because we are mathemeticians. 0:10:28.370,0:10:30.900 Every normal person, when[br]they mix into a bowl, 0:10:30.900,0:10:32.930 they mix like that. 0:10:32.930,0:10:35.480 Well, I've seen that[br]most of my colleagues-- 0:10:35.480,0:10:37.670 this is just a[br]psychological test, OK? 0:10:37.670,0:10:39.560 I wanted to see[br]how they mix when 0:10:39.560,0:10:41.730 they cook, or mix[br]up-- most of them 0:10:41.730,0:10:44.020 mix in a trigonometric sense. 0:10:44.020,0:10:47.610 I don't know if this has[br]anything to do with the brain 0:10:47.610,0:10:51.366 connections, but I think[br]that's [? kind of weird. ?] 0:10:51.366,0:10:54.550 I don't have a statistical[br]result, but most of the people 0:10:54.550,0:10:58.590 I've seen that, and do[br]mathematics, mix like that. 0:10:58.590,0:11:02.530 So trigonometric sense. 0:11:02.530,0:11:09.010 What is the connection with the[br]actual Cartesian coordinates? 0:11:09.010,0:11:13.650 D you know what Cartesian[br]comes from as a word? 0:11:13.650,0:11:15.766 Cartesian, that sounds weird. 0:11:15.766,0:11:17.242 STUDENT: From Descartes. 0:11:17.242,0:11:18.242 MAGDALENA TODA: Exactly. 0:11:18.242,0:11:19.702 Who said that? 0:11:19.702,0:11:21.178 Roberto, thank you so much. 0:11:21.178,0:11:22.162 I'm impressed. 0:11:22.162,0:11:22.990 Descartes was-- 0:11:22.990,0:11:23.656 STUDENT: French. 0:11:23.656,0:11:26.010 MAGDALENA TODA: --a[br]French mathematician. 0:11:26.010,0:11:28.980 But actually, I mean,[br]he was everything. 0:11:28.980,0:11:30.912 He was a crazy lunatic. 0:11:30.912,0:11:34.780 He was a philosopher,[br]a mathematician, 0:11:34.780,0:11:37.360 a scientist in general. 0:11:37.360,0:11:40.850 He also knew a lot[br]about life science. 0:11:40.850,0:11:43.790 But at the time, I don't[br]know if this is true. 0:11:43.790,0:11:45.970 I should check with wiki,[br]or whoever can tell me. 0:11:45.970,0:11:50.440 One of my professors in college[br]told me that at that time, 0:11:50.440,0:11:53.240 there was a fashion[br]that people would 0:11:53.240,0:11:57.060 change their names like they[br]do on Facebook nowadays. 0:11:57.060,0:11:59.980 So they and change their[br]name from Francesca 0:11:59.980,0:12:04.780 to Frenchy, from Roberto[br]to Robby, from-- so 0:12:04.780,0:12:08.510 if they would have to[br]clean up Facebook and see 0:12:08.510,0:12:14.800 how many names correspond to[br]the ID, I think less than 20%. 0:12:14.800,0:12:16.860 At that time it was the same. 0:12:16.860,0:12:22.780 All of the scientists loved[br]to romanize their names. 0:12:22.780,0:12:26.150 And of course he was[br]of a romance language, 0:12:26.150,0:12:30.370 but he said, what if I[br]made my name a Latin name, 0:12:30.370,0:12:32.250 I changed my name[br]into a Latin name. 0:12:32.250,0:12:36.640 So he himself, this is what[br]my professor told me, he 0:12:36.640,0:12:39.785 himself changed his[br]name to Cartesius. 0:12:39.785,0:12:45.513 "Car-teh-see-yus" actually, in[br]Latin, the way it should be. 0:12:45.513,0:12:49.880 0:12:49.880,0:12:52.200 OK, very smart guy. 0:12:52.200,0:12:56.800 Now, when we look[br]a x and y, there 0:12:56.800,0:13:04.400 has to be a connection between[br]x, y as the couple, and r theta 0:13:04.400,0:13:09.230 as the same-- I mean a[br]couple, not the couple, 0:13:09.230,0:13:10.691 for the same point. 0:13:10.691,0:13:11.190 Yes, sir? 0:13:11.190,0:13:11.981 STUDENT: Cartesius. 0:13:11.981,0:13:14.751 Like meaning flat? 0:13:14.751,0:13:15.250 The name? 0:13:15.250,0:13:17.416 MAGDALENA TODA: These are[br]the Cartesian coordinates, 0:13:17.416,0:13:20.130 and it sounds like the word map. 0:13:20.130,0:13:22.016 I think he had meant 0:13:22.016,0:13:23.640 STUDENT: Because the[br]meaning of carte-- 0:13:23.640,0:13:24.760 STUDENT: But look, look. 0:13:24.760,0:13:27.950 Descartes means from the map. 0:13:27.950,0:13:30.250 From the books, or from the map. 0:13:30.250,0:13:33.350 So he thought what his[br]name would really mean, 0:13:33.350,0:13:36.260 and so he recalled himself. 0:13:36.260,0:13:39.000 There was no fun, no[br]Twitter, no Facebook. 0:13:39.000,0:13:43.550 So they had to do something[br]to enjoy themselves. 0:13:43.550,0:13:46.175 Now, when it comes[br]to these triangles, 0:13:46.175,0:13:49.780 we have to think of the[br]relationship between x, y 0:13:49.780,0:13:52.510 and r, theta. 0:13:52.510,0:13:56.160 And could somebody tell me what[br]the relationship between x, y 0:13:56.160,0:13:59.240 and r, theta is? 0:13:59.240,0:14:01.261 x represents 0:14:01.261,0:14:02.610 STUDENT: R cosine theta. 0:14:02.610,0:14:05.260 STUDENT: r cosine[br]theta, who says that? 0:14:05.260,0:14:07.900 Trigonometry taught us[br]that, because that's 0:14:07.900,0:14:14.390 the adjacent side over[br]the hypotenuse for cosine. 0:14:14.390,0:14:18.250 In terms of sine, you[br]know what you have, 0:14:18.250,0:14:22.586 so you're going to have[br]y equals r sine theta, 0:14:22.586,0:14:26.740 and we have to decide[br]if x and y are allowed 0:14:26.740,0:14:28.200 to be anywhere in plane. 0:14:28.200,0:14:31.160 0:14:31.160,0:14:35.470 For the plane,[br]I'll also write r2. 0:14:35.470,0:14:40.830 R2, not R2 from the movie,[br]just r2 is the plane, 0:14:40.830,0:14:44.100 and r3 is the space,[br]the [? intriguing ?] 0:14:44.100,0:14:46.850 space, three-dimensional one. 0:14:46.850,0:14:50.870 r theta, is a couple where? 0:14:50.870,0:14:52.440 That's a little bit tricky. 0:14:52.440,0:14:54.120 We have to make a restriction. 0:14:54.120,0:14:59.340 We allow r to be anywhere[br]between 0 and infinity. 0:14:59.340,0:15:03.950 So it has to be a[br]positive number. 0:15:03.950,0:15:13.050 And theta [INTERPOSING VOICES][br]between 0 and 2 pi. 0:15:13.050,0:15:14.900 STUDENT: I've been[br]sick since Tuesday. 0:15:14.900,0:15:16.400 MAGDALENA TODA: I[br]believe you, Ryan. 0:15:16.400,0:15:17.640 You sound sick to me. 0:15:17.640,0:15:20.780 Take your viruses away from me. 0:15:20.780,0:15:21.650 Take the germs away. 0:15:21.650,0:15:25.315 I don't even have[br]the-- I'm kidding, 0:15:25.315,0:15:27.746 Alex, I hope you[br]don't get offended. 0:15:27.746,0:15:31.529 So, I hope this works this time. 0:15:31.529,0:15:33.070 I'm making a[br]sarcastic-- it's really, 0:15:33.070,0:15:34.522 I hope you're feeling better. 0:15:34.522,0:15:35.974 I'm sorry about that. 0:15:35.974,0:15:38.900 0:15:38.900,0:15:41.760 So you haven't missed much. 0:15:41.760,0:15:42.630 Only the jokes. 0:15:42.630,0:15:46.730 So x equals r cosine theta,[br]y equals r sine theta. 0:15:46.730,0:15:49.640 Is that your[br]favorite change that 0:15:49.640,0:15:55.995 was a differential[br]mapping from the set x, 0:15:55.995,0:15:58.806 y to the set r,[br]theta back and forth. 0:15:58.806,0:16:02.300 0:16:02.300,0:16:05.380 And you are going[br]to probably say, OK 0:16:05.380,0:16:08.330 how do you denote such a map? 0:16:08.330,0:16:11.460 I mean, going from x,[br]y to r, theta and back, 0:16:11.460,0:16:14.690 let's suppose that we go[br]from r, theta to x, y, 0:16:14.690,0:16:17.280 and that's going to be a big if. 0:16:17.280,0:16:20.430 And going backwards is going[br]to be the inverse mapping. 0:16:20.430,0:16:23.710 So I'm going to[br]call it f inverse. 0:16:23.710,0:16:30.770 So that's a map from a couple[br]to another couple of number. 0:16:30.770,0:16:35.960 And you say, OK, but[br]why is that a map? 0:16:35.960,0:16:38.260 All right, guys,[br]now let me tell you. 0:16:38.260,0:16:43.326 So x, you can do x as a[br]function of r, theta, right? 0:16:43.326,0:16:45.750 It is a function of r and theta. 0:16:45.750,0:16:48.420 It's a function[br]of two variables. 0:16:48.420,0:16:51.960 And y is a function[br]of r and theta. 0:16:51.960,0:16:53.735 It's another function[br]of two variables. 0:16:53.735,0:16:58.170 They are both nice[br]and differentiable. 0:16:58.170,0:17:02.710 We assume not only that[br]they are differentiable, 0:17:02.710,0:17:07.356 but the partial derivatives[br]will be continuous. 0:17:07.356,0:17:10.640 So it's really[br]nice as a mapping. 0:17:10.640,0:17:14.660 And you think, could I[br]write the chain rule? 0:17:14.660,0:17:16.232 That is the whole idea. 0:17:16.232,0:17:18.040 What is the meaning[br]of differential? 0:17:18.040,0:17:20.050 dx differential dy. 0:17:20.050,0:17:23.390 Since I was chatting with[br]you, once, [? Yuniel ?], 0:17:23.390,0:17:28.600 and you asked me to[br]help you with homework, 0:17:28.600,0:17:31.480 I had to go over[br]differential again. 0:17:31.480,0:17:36.350 If you were to define,[br]like Mr. Leibniz did, 0:17:36.350,0:17:39.930 the differential of the[br]function x with respect 0:17:39.930,0:17:44.060 to both variables, that[br]was the sum, right? 0:17:44.060,0:17:45.450 You've done that[br]in the homework, 0:17:45.450,0:17:46.720 it's fresh in your mind. 0:17:46.720,0:17:53.505 So you get x sub r,[br]dr, plus f x sub what? 0:17:53.505,0:17:54.130 STUDENT: Theta. 0:17:54.130,0:17:56.940 MAGDALENA TODA:[br]Sub theta d-theta. 0:17:56.940,0:18:01.740 And somebody asked me,[br]what if I see skip the dr? 0:18:01.740,0:18:02.490 No, don't do that. 0:18:02.490,0:18:05.205 First of all, WeBWorK is not[br]going to take the answer. 0:18:05.205,0:18:09.310 But second of all, the[br]most important stuff 0:18:09.310,0:18:13.190 here to remember is that these[br]are small, infinitesimally 0:18:13.190,0:18:15.190 small, displacements. 0:18:15.190,0:18:32.130 Infinitesimally small[br]displacements in the directions 0:18:32.130,0:18:33.990 x and y, respectively. 0:18:33.990,0:18:37.435 So you would say, what does[br]that mean, infinitesimally? 0:18:37.435,0:18:39.790 It doesn't mean delta-x small. 0:18:39.790,0:18:43.950 Delta-x small would be like[br]me driving 7 feet, when 0:18:43.950,0:18:48.920 I know I have to drive fast to[br]Amarillo to be there in 1 hour. 0:18:48.920,0:18:49.655 Well, OK. 0:18:49.655,0:18:51.600 Don't tell anybody. 0:18:51.600,0:18:55.160 But, it's about 2 hours, right? 0:18:55.160,0:18:58.230 So I cannot be there in an hour. 0:18:58.230,0:19:01.940 But driving those seven[br]feet is like a delta x. 0:19:01.940,0:19:07.100 Imagine, however, me[br]measuring that speed of mine 0:19:07.100,0:19:10.130 in a much smaller[br]fraction of a second. 0:19:10.130,0:19:15.999 So shrink that time to[br]something infinitesimally small, 0:19:15.999,0:19:17.450 which is what you have here. 0:19:17.450,0:19:19.206 That kind of quantity. 0:19:19.206,0:19:25.440 And dy will be y sub r dr[br]plus y sub theta d-theta. 0:19:25.440,0:19:28.750 0:19:28.750,0:19:32.718 And now, I'm not going[br]to go by the book. 0:19:32.718,0:19:34.560 I'm going to go[br]a little bit more 0:19:34.560,0:19:39.630 in depth, because in the book--[br]First of all, let me tell you, 0:19:39.630,0:19:43.870 if I went by the book,[br]what I would come with. 0:19:43.870,0:19:48.790 And of course the way[br]we teach mathematics 0:19:48.790,0:19:52.430 all through K-12 and through[br]college is swallow this theorem 0:19:52.430,0:19:53.810 and believe it. 0:19:53.810,0:19:58.910 So practically you accept[br]whatever we give you 0:19:58.910,0:20:02.120 without controlling it, without[br]checking if we're right, 0:20:02.120,0:20:05.151 without trying to prove it. 0:20:05.151,0:20:06.650 Practically, the[br]theorem in the book 0:20:06.650,0:20:09.300 says that if you[br]have a bunch of x, 0:20:09.300,0:20:14.400 y that is continuous[br]over a domain, D, 0:20:14.400,0:20:21.322 and you do change[br]the variables over-- 0:20:21.322,0:20:22.714 STUDENT: I forgot my glasses. 0:20:22.714,0:20:25.040 So I'm going to sit very close. 0:20:25.040,0:20:28.570 MAGDALENA TODA:[br]What do you wear? 0:20:28.570,0:20:30.632 What [INAUDIBLE]? 0:20:30.632,0:20:31.840 STUDENT: I couldn't tell you. 0:20:31.840,0:20:33.104 I can see from here. 0:20:33.104,0:20:33.568 MAGDALENA TODA: You can? 0:20:33.568,0:20:34.032 STUDENT: Yeah. 0:20:34.032,0:20:35.073 My vision's not terrible. 0:20:35.073,0:20:41.602 MAGDALENA TODA: All[br]right. f of x, y da. 0:20:41.602,0:20:47.120 If I change this da[br]as dx dy, let's say, 0:20:47.120,0:20:49.820 to a perspective[br]of something else 0:20:49.820,0:20:52.470 in terms of polar[br]coordinates, then 0:20:52.470,0:20:57.380 the integral I'm going to get is[br]over the corresponding domain D 0:20:57.380,0:21:00.530 star, whatever that would be. 0:21:00.530,0:21:06.480 Then I'm going to have f of[br]x of r theta, y of r theta, 0:21:06.480,0:21:09.870 everything expressed[br]in terms of r theta. 0:21:09.870,0:21:13.990 And instead of[br]the a-- so we just 0:21:13.990,0:21:18.496 feed you this piece of[br]cake and say, believe it, 0:21:18.496,0:21:21.932 believe it and leave us alone. 0:21:21.932,0:21:22.432 OK? 0:21:22.432,0:21:26.780 That's what it does in[br]the book in section 11.3. 0:21:26.780,0:21:33.450 So without understanding why[br]you have to-- instead of the r 0:21:33.450,0:21:35.790 d theta and multiply it by an r. 0:21:35.790,0:21:36.480 What is that? 0:21:36.480,0:21:38.106 You don't know why you do that. 0:21:38.106,0:21:40.520 And I thought, that's[br]the way we thought it 0:21:40.520,0:21:42.580 for way too many years. 0:21:42.580,0:21:45.920 I'm sick and tired[br]of not explaining why 0:21:45.920,0:21:50.740 you multiply that with an r. 0:21:50.740,0:21:55.100 So I will tell you something[br]that's quite interesting, 0:21:55.100,0:21:58.290 and something that I learned[br]late in graduate school. 0:21:58.290,0:22:00.630 I was late already. 0:22:00.630,0:22:05.790 I was in my 20s when I[br]studied differential forms 0:22:05.790,0:22:07.586 for the first time. 0:22:07.586,0:22:12.880 And differential[br]forms have some sort 0:22:12.880,0:22:25.060 of special wedge product, which[br]is very physical in nature. 0:22:25.060,0:22:30.264 So if you love physics, you[br]will understand more or less 0:22:30.264,0:22:34.010 what I'm talking about. 0:22:34.010,0:22:40.660 Imagine that you have two[br]vectors, vector a and vector b. 0:22:40.660,0:22:44.170 0:22:44.170,0:22:48.250 For these vectors,[br]you go, oh my God. 0:22:48.250,0:22:54.086 If these would be vectors in[br]a tangent plane to a surface, 0:22:54.086,0:22:56.395 you think, some[br]of these would be 0:22:56.395,0:22:59.980 tangent vectors to a surface. 0:22:59.980,0:23:02.380 This is the tangent[br]plane and everything. 0:23:02.380,0:23:07.030 You go, OK, if these[br]were infinitesimally 0:23:07.030,0:23:11.630 small displacements-- which they[br]are not, but assume they would 0:23:11.630,0:23:19.490 be-- how would you do the area[br]of the infinitesimally small 0:23:19.490,0:23:22.370 parallelogram that[br]they have between them. 0:23:22.370,0:23:31.166 This is actually the area[br]element right here, ea. 0:23:31.166,0:23:35.150 So instead of dx dy, you're[br]not going to have dx dy, 0:23:35.150,0:23:39.998 you're going to have some[br]sort of, I don't know, 0:23:39.998,0:23:48.170 this is like a[br]d-something, d u, and this 0:23:48.170,0:23:55.180 is a d v. And when I compute[br]the area of the parallelogram, 0:23:55.180,0:23:58.120 I consider these to[br]be vectors, and I 0:23:58.120,0:24:02.180 say, how did we get[br]it from the vectors 0:24:02.180,0:24:06.259 to the area of[br]the parallelogram? 0:24:06.259,0:24:10.060 We took the vectors,[br]we shook them off. 0:24:10.060,0:24:19.320 We made a cross product[br]of them, and then we 0:24:19.320,0:24:23.370 took the norm, the[br]magnitude of that. 0:24:23.370,0:24:26.812 Does this makes sense,[br]compared to this parallelogram? 0:24:26.812,0:24:27.311 Yeah. 0:24:27.311,0:24:30.550 Remember, guys, this[br]was like, how big 0:24:30.550,0:24:33.170 is du, a small[br]infinitesimal displacement, 0:24:33.170,0:24:36.340 but that would be like the[br]width, one of the dimensions. 0:24:36.340,0:24:39.994 There's the other of the[br]dimension of the area element 0:24:39.994,0:24:44.050 times-- this area element[br]is that tiny pixel that 0:24:44.050,0:24:49.010 is sitting on the surface[br]in the tangent plane, yeah? 0:24:49.010,0:24:54.220 Sine of the angle[br]between the guys. 0:24:54.220,0:24:54.760 Oh, OK. 0:24:54.760,0:25:00.760 So if the guys are not[br]perpendicular to one another, 0:25:00.760,0:25:03.730 if the two displacements are not[br]perpendicular to one another, 0:25:03.730,0:25:07.340 you still have to multiply[br]the sine of theta. 0:25:07.340,0:25:09.187 Otherwise you don't[br]get the element 0:25:09.187,0:25:12.320 of the area of[br]this parallelogram. 0:25:12.320,0:25:17.530 So why did the Cartesian[br]coordinates not pose a problem? 0:25:17.530,0:25:19.560 For Cartesian[br]coordinates, it's easy. 0:25:19.560,0:25:22.615 0:25:22.615,0:25:23.490 It's a piece of cake. 0:25:23.490,0:25:24.365 Why? 0:25:24.365,0:25:32.130 Because this is the x, this is[br]the y, as little tiny measures 0:25:32.130,0:25:33.330 multiplied. 0:25:33.330,0:25:37.380 How much is sine of theta[br]between Cartesian coordinates? 0:25:37.380,0:25:37.880 STUDENT: 1. 0:25:37.880,0:25:40.970 MAGDALENA TODA: It's 1,[br]because its 90 degrees. 0:25:40.970,0:25:43.160 When they are[br]orthogonal coordinates, 0:25:43.160,0:25:46.884 it's a piece of cake,[br]because you have 1 there, 0:25:46.884,0:25:48.300 and then your life[br]becomes easier. 0:25:48.300,0:25:50.940 0:25:50.940,0:25:57.030 So in general, what[br]is the area limit? 0:25:57.030,0:26:02.026 The area limit for[br]arbitrary coordinates-- 0:26:02.026,0:26:17.020 So area limit for some[br]arbitrary coordinates 0:26:17.020,0:26:20.310 should be defined[br]as the sined area. 0:26:20.310,0:26:29.320 0:26:29.320,0:26:32.190 And you say, what do you[br]mean that's a sined area, 0:26:32.190,0:26:34.580 and why would you do that.? 0:26:34.580,0:26:38.280 Well, it's not so[br]hard to understand. 0:26:38.280,0:26:41.740 Imagine that you have a[br]convention, and you say, 0:26:41.740,0:26:54.810 OK, dx times dy equals[br]negative dy times dx. 0:26:54.810,0:26:56.920 And you say, what, what? 0:26:56.920,0:27:00.520 If you change the[br]order of dx dy, 0:27:00.520,0:27:06.597 this wedge stuff works exactly[br]like the-- what is that called? 0:27:06.597,0:27:07.790 Cross product. 0:27:07.790,0:27:13.150 So the wedge works just[br]like the cross product. 0:27:13.150,0:27:17.509 Just like the cross product. 0:27:17.509,0:27:23.320 In some other ways, suppose[br]that I am here, right? 0:27:23.320,0:27:27.720 And this is a vector, like an[br]infinitesimal displacement, 0:27:27.720,0:27:29.370 and that's the other one. 0:27:29.370,0:27:33.800 If I multiply them[br]one after the other, 0:27:33.800,0:27:38.060 and I use this strange wedge[br][INTERPOSING VOICES] the area, 0:27:38.060,0:27:40.970 I'm going to have an orientation[br]for that tangent line, 0:27:40.970,0:27:46.390 and it's going to go[br]up, the orientation. 0:27:46.390,0:27:48.330 The orientation is important. 0:27:48.330,0:27:50.990 But if dx dy and[br]I switched them, 0:27:50.990,0:27:56.050 I said, dy, swap with dx,[br]what's going to happen? 0:27:56.050,0:28:01.530 I have to change to[br]change to clockwise. 0:28:01.530,0:28:03.610 And then the[br]orientation goes down. 0:28:03.610,0:28:06.720 And that's what they use[br]in mechanics when it comes 0:28:06.720,0:28:09.130 to the normal to the surface. 0:28:09.130,0:28:12.773 So again, you guys remember,[br]we had 2 vector products, 0:28:12.773,0:28:16.370 and we did the cross product,[br]and we got the normal. 0:28:16.370,0:28:18.795 If it's from this[br]one to this one, 0:28:18.795,0:28:20.740 it's counterclockwise[br]and goes up, 0:28:20.740,0:28:23.855 but if it's from this[br]vector to this other vector, 0:28:23.855,0:28:26.990 it's clockwise and goes down. 0:28:26.990,0:28:29.820 This is how a[br]mechanical engineer 0:28:29.820,0:28:32.820 will know how the[br]surface is oriented 0:28:32.820,0:28:35.730 based on the partial[br]velocities, for example 0:28:35.730,0:28:39.370 He has the partial[br]velocities along a surface, 0:28:39.370,0:28:42.750 and somebody says, take the[br]normal, take the unit normal. 0:28:42.750,0:28:44.760 He goes, like, are[br]you a physicist? 0:28:44.760,0:28:46.440 No, I'm an engineer. 0:28:46.440,0:28:48.790 You don't know how[br]to take the normal. 0:28:48.790,0:28:50.110 And of course, he knows. 0:28:50.110,0:28:53.500 He knows the convention[br]by this right-hand rule, 0:28:53.500,0:28:55.190 whatever you guys call it. 0:28:55.190,0:28:57.260 I call it the faucet rule. 0:28:57.260,0:29:01.400 It goes like this,[br]or it goes like that. 0:29:01.400,0:29:04.272 It's the same for a faucet,[br]for any type of screw, 0:29:04.272,0:29:08.350 for the right-hand[br]rule, whatever. 0:29:08.350,0:29:11.360 What else do you have[br]to believe me are true? 0:29:11.360,0:29:14.560 dx wedge dx is 0. 0:29:14.560,0:29:17.740 Can somebody tell me why[br]that is natural to introduce 0:29:17.740,0:29:19.490 such a wedge product? 0:29:19.490,0:29:22.364 STUDENT: Because the sine of[br]the angle between those is 0. 0:29:22.364,0:29:23.280 MAGDALENA TODA: Right. 0:29:23.280,0:29:28.660 Once you flatten this, once[br]you flatten the parallelogram, 0:29:28.660,0:29:29.830 there is no area. 0:29:29.830,0:29:31.470 So the area is 0. 0:29:31.470,0:29:34.960 How about dy dy sined area? 0:29:34.960,0:29:35.930 0. 0:29:35.930,0:29:37.810 So these are all[br]the properties you 0:29:37.810,0:29:41.441 need to know of the[br]sine area, sined areas. 0:29:41.441,0:29:44.270 0:29:44.270,0:29:46.530 OK, so now let's[br]see what happens 0:29:46.530,0:29:51.150 if we take this element,[br]which is a differential, 0:29:51.150,0:29:55.350 and wedge it with this element,[br]which is also a differential. 0:29:55.350,0:29:56.370 OK. 0:29:56.370,0:29:59.920 Oh my God, I'm shaking[br]only thinking about it. 0:29:59.920,0:30:01.860 I'm going to get[br]something weird. 0:30:01.860,0:30:04.070 But I mean, mad weird. 0:30:04.070,0:30:06.663 Let's see what happens. 0:30:06.663,0:30:13.960 dx wedge dy equals-- do[br]you guys have questions? 0:30:13.960,0:30:18.334 Let's see what the mechanics are[br]for this type of computation. 0:30:18.334,0:30:21.250 0:30:21.250,0:30:27.690 I go-- this is like[br]a-- displacement wedge 0:30:27.690,0:30:29.590 this other displacement. 0:30:29.590,0:30:32.736 0:30:32.736,0:30:36.110 Think of them as true[br]vector displacements, 0:30:36.110,0:30:41.150 and as if you had a cross[br]product, or something. 0:30:41.150,0:30:42.080 OK. 0:30:42.080,0:30:43.657 How does this go? 0:30:43.657,0:30:44.820 It's distributed. 0:30:44.820,0:30:47.770 It's linear functions,[br]because we've 0:30:47.770,0:30:51.140 studied the[br]properties of vectors, 0:30:51.140,0:30:52.740 this acts by linearity. 0:30:52.740,0:30:58.182 So you go and say, first[br]first, times plus first times 0:30:58.182,0:31:02.640 second-- and times is[br]this guy, this weirdo-- 0:31:02.640,0:31:06.940 plus second times first,[br]plus second times second, 0:31:06.940,0:31:09.200 where the wedge is[br]the operator that 0:31:09.200,0:31:11.280 has to satisfy these functions. 0:31:11.280,0:31:14.060 It's similar to[br]the cross product. 0:31:14.060,0:31:15.190 OK. 0:31:15.190,0:31:21.370 Then let's go x[br]sub r, y sub r, dr 0:31:21.370,0:31:26.880 wedge dr. Oh, let's 0 go away. 0:31:26.880,0:31:30.340 I say, leave me alone,[br]you're making my life hard. 0:31:30.340,0:31:37.690 Then I go plus x sub r--[br]this is a small function. 0:31:37.690,0:31:40.520 y sub theta, another[br]small function. 0:31:40.520,0:31:44.050 What of this[br]displacement, dr d theta. 0:31:44.050,0:31:46.650 I'm like those d[br]something, d something, 0:31:46.650,0:31:49.350 two small displacements[br]in the cross product. 0:31:49.350,0:31:52.620 OK, plus. 0:31:52.620,0:31:55.271 Who is telling me what next? 0:31:55.271,0:31:56.020 STUDENT: x theta-- 0:31:56.020,0:32:05.550 MAGDALENA TODA: x theta[br]yr, d theta dr. Is it fair? 0:32:05.550,0:32:10.200 I did the second guy from the[br]first one with the first guy 0:32:10.200,0:32:11.790 from the second one. 0:32:11.790,0:32:14.720 And finally, I'm too[br]lazy to write it down. 0:32:14.720,0:32:15.964 What do I get? 0:32:15.964,0:32:16.860 STUDENT: 0. 0:32:16.860,0:32:17.020 MAGDALENA TODA: 0. 0:32:17.020,0:32:17.680 Why is that? 0:32:17.680,0:32:20.070 Because d theta,[br]always d theta is 0. 0:32:20.070,0:32:27.160 It's like you are flattening--[br]there is no more parallelogram. 0:32:27.160,0:32:27.940 OK? 0:32:27.940,0:32:32.330 So the two dimensions of[br]the parallelogram become 0. 0:32:32.330,0:32:37.070 The parallelogram would[br]become [? a secant. ?] 0:32:37.070,0:32:39.931 What you get is[br]something really weak. 0:32:39.931,0:32:42.210 And we don't talk[br]about it in the book, 0:32:42.210,0:32:45.022 but that's called the Jacobian. 0:32:45.022,0:32:51.150 dr d theta and d theta dr, once[br]you introduce the sine area, 0:32:51.150,0:32:55.920 you finally understand[br]why you get this r here, 0:32:55.920,0:32:57.740 what the Jacobian is. 0:32:57.740,0:32:59.370 If you don't introduce[br]the sine area, 0:32:59.370,0:33:02.340 you will never understand,[br]and you cannot explain it 0:33:02.340,0:33:06.140 to anybody, any student have. 0:33:06.140,0:33:11.530 OK, so this guy, d theta,[br]which the r is just 0:33:11.530,0:33:13.686 swapping the two displacements. 0:33:13.686,0:33:16.990 So it's going to be[br]minus dr d theta. 0:33:16.990,0:33:18.670 Why is that, guys? 0:33:18.670,0:33:23.030 Because that's how I said, every[br]time I swap two displacements, 0:33:23.030,0:33:25.440 I'm changing the orientation. 0:33:25.440,0:33:28.085 It's like the cross[br]product between a and b, 0:33:28.085,0:33:30.080 and the cross product[br]between b and a. 0:33:30.080,0:33:34.520 So I'm going up or I'm going[br]down, I'm changing orientation. 0:33:34.520,0:33:35.900 What's left in the end? 0:33:35.900,0:33:39.000 It's really just this[br]guy that's really weird. 0:33:39.000,0:33:41.286 I'm going to collect the terms. 0:33:41.286,0:33:44.930 One from here, one[br]from here, and a minus. 0:33:44.930,0:33:45.430 Go ahead. 0:33:45.430,0:33:49.030 STUDENT: Do the wedges[br]just cancel out? 0:33:49.030,0:33:50.350 MAGDALENA TODA: This was 0. 0:33:50.350,0:33:52.350 This was 0. 0:33:52.350,0:33:57.580 And this dr d theta is nonzero,[br]but is the common factor. 0:33:57.580,0:34:00.015 So I pull him out from here. 0:34:00.015,0:34:01.890 I pull him out from here. 0:34:01.890,0:34:02.390 Out. 0:34:02.390,0:34:08.469 Factor out, and what's[br]left is this guy over here 0:34:08.469,0:34:10.561 who is this guy over here. 0:34:10.561,0:34:14.576 And this guy over[br]here with a minus 0:34:14.576,0:34:20.320 who gives me minus d theta yr. 0:34:20.320,0:34:21.000 That's all. 0:34:21.000,0:34:25.270 So now you will understand[br]why I am going to get r. 0:34:25.270,0:34:30.440 So the general rule will[br]be that the area element dx 0:34:30.440,0:34:35.860 dy, the wedge sined[br]area, will be-- 0:34:35.860,0:34:39.210 you have to help me[br]with this individual, 0:34:39.210,0:34:42.989 because he really looks weird. 0:34:42.989,0:34:46.480 Do you know of a name for it? 0:34:46.480,0:34:49.909 Do you know what[br]this is going to be? 0:34:49.909,0:34:52.400 Linear algebra people,[br]only two of you. 0:34:52.400,0:34:56.650 Maybe you have an idea. 0:34:56.650,0:34:59.950 So it's like, I[br]take this fellow, 0:34:59.950,0:35:01.820 and I multiply by that fellow. 0:35:01.820,0:35:04.496 0:35:04.496,0:35:06.550 Multiply these two. 0:35:06.550,0:35:12.970 And I go minus this[br]fellow times that fellow. 0:35:12.970,0:35:14.820 STUDENT: [INAUDIBLE] 0:35:14.820,0:35:17.590 MAGDALENA TODA: It's like[br]a determinant of something. 0:35:17.590,0:35:23.380 So when people write[br]the differential system, 0:35:23.380,0:35:26.460 [INTERPOSING VOICES][br]51, you will understand 0:35:26.460,0:35:27.940 that this is a system. 0:35:27.940,0:35:28.440 OK? 0:35:28.440,0:35:29.935 It's a system of two equations. 0:35:29.935,0:35:32.155 0:35:32.155,0:35:34.030 The other little, like,[br]vector displacements, 0:35:34.030,0:35:36.370 you are going to[br]write it like that. 0:35:36.370,0:35:45.950 dx dy will be matrix[br]multiplication dr d theta. 0:35:45.950,0:35:50.210 And how do you multiply[br]x sub r x sub theta? 0:35:50.210,0:35:55.190 So you go first row times[br]first column give you that. 0:35:55.190,0:35:59.510 And second row times the[br]column gives you this. 0:35:59.510,0:36:02.150 y sub r, y sub theta. 0:36:02.150,0:36:06.340 This is a magic guy[br]called Jacobian. 0:36:06.340,0:36:09.880 We keep this a secret, and[br]most Professors don't even 0:36:09.880,0:36:13.050 cover 12.8, because[br]they don't want to tell 0:36:13.050,0:36:15.060 people what a Jacobian is. 0:36:15.060,0:36:16.890 This is little r. 0:36:16.890,0:36:20.855 I know you don't believe me, but[br]the determinant of this matrix 0:36:20.855,0:36:22.520 must be little r. 0:36:22.520,0:36:24.910 You have to help me prove that. 0:36:24.910,0:36:27.340 And this is the Jacobian. 0:36:27.340,0:36:30.385 Do you guys know why[br]it's called Jacobian? 0:36:30.385,0:36:33.355 It's the determinant[br]of this matrix. 0:36:33.355,0:36:43.255 Let's call this[br]matrix J. And this 0:36:43.255,0:36:49.210 is J, determinant[br]of [? scripture. ?] 0:36:49.210,0:36:50.480 This is called Jacobian. 0:36:50.480,0:36:54.160 0:36:54.160,0:36:55.080 Why is it r? 0:36:55.080,0:36:57.850 Let's-- I don't know. 0:36:57.850,0:36:59.710 Let's see how we do it. 0:36:59.710,0:37:03.540 0:37:03.540,0:37:06.900 This is r cosine theta, right? 0:37:06.900,0:37:09.790 This is r sine theta. 0:37:09.790,0:37:14.790 So dx must be what x sub r? 0:37:14.790,0:37:19.730 X sub r, x sub r, cosine theta. 0:37:19.730,0:37:21.750 d plus. 0:37:21.750,0:37:23.570 What is x sub t? 0:37:23.570,0:37:26.385 0:37:26.385,0:37:28.550 x sub theta. 0:37:28.550,0:37:31.601 I need to differentiate[br]this with respect to theta. 0:37:31.601,0:37:33.600 STUDENT: It's going to[br]be negative r sine theta. 0:37:33.600,0:37:36.390 MAGDALENA TODA: Minus r[br]sine theta, very good. 0:37:36.390,0:37:38.090 And d theta. 0:37:38.090,0:37:44.200 Then I go dy was[br]sine theta-- dr, 0:37:44.200,0:37:46.450 I'm looking at these[br]equations, and I'm 0:37:46.450,0:37:49.020 repeating them for my case. 0:37:49.020,0:37:52.890 This is true in general for[br]any kind of coordinates. 0:37:52.890,0:37:56.640 So it's a general equation[br]for any kind of coordinate, 0:37:56.640,0:37:58.830 two coordinates,[br]two coordinates, 0:37:58.830,0:38:00.630 any kind of[br]coordinates in plane, 0:38:00.630,0:38:04.940 you can choose any[br]functions, f of uv, g of uv, 0:38:04.940,0:38:06.600 whatever you want. 0:38:06.600,0:38:09.460 But for this particular[br]case of polar coordinates 0:38:09.460,0:38:12.270 is going to look really[br]pretty in the end. 0:38:12.270,0:38:15.610 What do I get when I do y theta? 0:38:15.610,0:38:17.485 r cosine theta. 0:38:17.485,0:38:18.830 Am I right, guys? 0:38:18.830,0:38:20.730 Keen an eye on it. 0:38:20.730,0:38:27.280 So this will become-- the[br]area element will become what? 0:38:27.280,0:38:31.310 The determinant of this matrix. 0:38:31.310,0:38:34.570 Red, red, red, red. 0:38:34.570,0:38:35.886 How do I compute a term? 0:38:35.886,0:38:39.410 Not everybody knows,[br]and it's this times 0:38:39.410,0:38:45.150 that minus this times that. 0:38:45.150,0:38:46.370 OK, let's do that. 0:38:46.370,0:38:53.440 So I get r cosine squared[br]theta minus minus plus r sine 0:38:53.440,0:38:56.490 squared theta. 0:38:56.490,0:38:58.870 dr, d theta, and our wedge. 0:38:58.870,0:39:00.000 What is this? 0:39:00.000,0:39:00.600 STUDENT: 1. 0:39:00.600,0:39:04.320 MAGDALENA TODA:[br]Jacobian is r times 1, 0:39:04.320,0:39:07.430 because that's the[br]Pythagorean theorem, right? 0:39:07.430,0:39:12.330 So we have r, and this is[br]the meaning of r, here. 0:39:12.330,0:39:16.830 So when I moved from dx dy,[br]I originally had the wedge 0:39:16.830,0:39:19.470 that I didn't tell you about. 0:39:19.470,0:39:23.090 And this wedge[br]becomes r dr d theta, 0:39:23.090,0:39:27.290 and that's the[br]correct way to explain 0:39:27.290,0:39:29.890 why you get the Jacobian there. 0:39:29.890,0:39:31.505 We don't do that in the book. 0:39:31.505,0:39:34.855 We do it later, and we[br]sort of smuggle through. 0:39:34.855,0:39:37.100 We don't do a very thorough job. 0:39:37.100,0:39:39.980 When you go into[br]advanced calculus, 0:39:39.980,0:39:43.237 you would see that again the[br]way I explained it to you. 0:39:43.237,0:39:47.370 If you ever want to[br]go to graduate school, 0:39:47.370,0:39:52.440 then you need to take the[br]Advanced Calculus I, 4350 0:39:52.440,0:39:57.530 and 4351 where you are[br]going to learn about this. 0:39:57.530,0:40:01.210 If you take those as a math[br]major or engineering major, 0:40:01.210,0:40:01.960 it doesn't matter. 0:40:01.960,0:40:03.920 When you go to[br]graduate school, they 0:40:03.920,0:40:07.470 don't make you take[br]advanced calculus again 0:40:07.470,0:40:09.380 at graduate school. 0:40:09.380,0:40:12.740 So it's somewhere borderline[br]between senior year 0:40:12.740,0:40:19.010 and graduate school, it's like[br]the first course you would take 0:40:19.010,0:40:22.020 in graduate school, for many. 0:40:22.020,0:40:22.670 OK. 0:40:22.670,0:40:29.890 So an example of[br]this transformation 0:40:29.890,0:40:33.270 where we know what[br]we are talking about. 0:40:33.270,0:40:39.130 Let's say I have[br]a picture, and I 0:40:39.130,0:40:42.730 have a domain D, which[br]is-- this is x squared 0:40:42.730,0:40:44.946 plus y squared equals 1. 0:40:44.946,0:40:48.369 I have the domain as being[br][INTERPOSING VOICES]. 0:40:48.369,0:40:51.800 0:40:51.800,0:40:58.120 And then I say, I would[br]like-- what would I like? 0:40:58.120,0:41:04.290 I would like the[br]volume of the-- this 0:41:04.290,0:41:10.220 is a paraboloid, z equals[br]x squared plus y squared. 0:41:10.220,0:41:12.616 I would like the[br]volume of this object. 0:41:12.616,0:41:13.820 This is my obsession. 0:41:13.820,0:41:17.580 I'm going to create a[br]vase some day like that. 0:41:17.580,0:41:22.560 So you want this[br]piece to be a solid. 0:41:22.560,0:41:25.420 In cross section,[br]it will just this. 0:41:25.420,0:41:26.250 In cross section. 0:41:26.250,0:41:27.830 And it's a solid of revolution. 0:41:27.830,0:41:30.300 In this cross section,[br]you have to imagine 0:41:30.300,0:41:36.100 revolving it around the z-axis,[br]then creating a heavy object. 0:41:36.100,0:41:38.440 From the outside, don't[br]see what's inside. 0:41:38.440,0:41:39.530 It looks like a cylinder. 0:41:39.530,0:41:42.460 But you go inside and[br]you see the valley. 0:41:42.460,0:41:46.260 So it's between a[br]paraboloid and a disc, 0:41:46.260,0:41:48.460 a unit disc on the floor. 0:41:48.460,0:41:51.400 How are we going[br]to try and do that? 0:41:51.400,0:41:53.960 And what did I[br]teach you last time? 0:41:53.960,0:42:02.020 Last time, I taught you that--[br]we have to go over a domain D. 0:42:02.020,0:42:04.190 But that domain[br]D, unfortunately, 0:42:04.190,0:42:05.780 is hard to express. 0:42:05.780,0:42:09.217 How would you express D[br]in Cartesian coordinates? 0:42:09.217,0:42:14.630 0:42:14.630,0:42:15.840 You can do it. 0:42:15.840,0:42:18.770 It's going to be a headache. 0:42:18.770,0:42:22.270 x is between minus 1 and 1. 0:42:22.270,0:42:23.770 Am I right, guys? 0:42:23.770,0:42:28.270 And y will be between--[br]now I have two branches. 0:42:28.270,0:42:30.230 One, and the other one. 0:42:30.230,0:42:33.100 One branch would be square--[br]I hate square roots. 0:42:33.100,0:42:36.250 I absolutely hate them. 0:42:36.250,0:42:40.330 y is between 1 minus[br]square root x squared, 0:42:40.330,0:42:43.300 minus square root[br]1 minus x squared. 0:42:43.300,0:42:47.650 So if I were to ask you to do[br]the integral like last time, 0:42:47.650,0:42:50.794 how would you set[br]up the integral? 0:42:50.794,0:42:53.380 You go, OK, I know what this is. 0:42:53.380,0:43:01.380 Integral over D of[br]f of x, y, dx dy. 0:43:01.380,0:43:02.900 This is actually a wedge. 0:43:02.900,0:43:06.060 In my case, we avoided that. 0:43:06.060,0:43:07.540 We said dh. 0:43:07.540,0:43:09.910 And we said, what is f of x, y? 0:43:09.910,0:43:11.770 x squared plus y[br]squared, because I 0:43:11.770,0:43:16.044 want everything that's under[br]the graph, not above the graph. 0:43:16.044,0:43:18.996 So everything that's[br]under the graph. 0:43:18.996,0:43:26.600 F of x, y is this guy. 0:43:26.600,0:43:28.430 And the I have to[br]start thinking, 0:43:28.430,0:43:31.540 because it's a type 1 or type 2? 0:43:31.540,0:43:35.700 It's a type 1 the[br]way I set it up, 0:43:35.700,0:43:39.060 but I can make it[br]type 2 by reversing 0:43:39.060,0:43:41.520 the order of integration[br]like I did last time. 0:43:41.520,0:43:44.035 If I treat it like[br]that, it's going 0:43:44.035,0:43:46.420 to be type 1, though, right? 0:43:46.420,0:43:50.640 So I have to put[br]dy first, and then 0:43:50.640,0:43:54.570 change the color of the dx. 0:43:54.570,0:43:58.280 And since mister y[br]is the purple guy, 0:43:58.280,0:44:03.000 y would be going between[br]these ugly square roots that 0:44:03.000,0:44:04.220 to go on my nerves. 0:44:04.220,0:44:10.360 0:44:10.360,0:44:17.485 And then x goes[br]between minus 1 and 1. 0:44:17.485,0:44:20.870 It's a little bit of a headache. 0:44:20.870,0:44:22.980 Why is it a headache, guys? 0:44:22.980,0:44:27.470 Let's anticipate what we need to[br]do if we do it like last time. 0:44:27.470,0:44:32.110 We need to integrate this[br]ugly fellow in terms of y, 0:44:32.110,0:44:35.510 and when we integrate this in[br]terms of y, what do we get? 0:44:35.510,0:44:38.450 Don't write it, because[br]it's going to be a mess. 0:44:38.450,0:44:44.870 We get x squared times[br]y plus y cubed over 3. 0:44:44.870,0:44:47.480 And then, instead of y, I have[br]to replace those square roots, 0:44:47.480,0:44:49.600 and I'll never get rid[br]of the square roots. 0:44:49.600,0:44:52.760 It's going to be a mess, indeed. 0:44:52.760,0:44:56.250 And I may even-- in[br]general, I may not even 0:44:56.250,0:44:58.860 be able to solve the[br]integral, and that's 0:44:58.860,0:45:00.780 a bit headache,[br]because I'll start 0:45:00.780,0:45:03.444 crying, I'll get depressed,[br]I'll take Prozac, whatever 0:45:03.444,0:45:04.815 you take for depression. 0:45:04.815,0:45:07.560 I don't know, I never took it,[br]because I'm never depressed. 0:45:07.560,0:45:10.960 So what do you do in that case? 0:45:10.960,0:45:12.220 STUDENT: Switch to polar. 0:45:12.220,0:45:13.720 MAGDALENA TODA: You[br]switch to polar. 0:45:13.720,0:45:18.610 So you use this big polar-switch[br]theorem, the theorem that 0:45:18.610,0:45:23.940 tells you, be smart,[br]apply this theorem, 0:45:23.940,0:45:30.700 and have to understand that[br]the D, which was this expressed 0:45:30.700,0:45:32.970 in [INTERPOSING VOICES][br]Cartesian coordinates 0:45:32.970,0:45:37.480 is D. If you want express[br]the same thing as D star, 0:45:37.480,0:45:39.600 D star will be in[br]polar coordinates. 0:45:39.600,0:45:44.010 You have to be a little bit[br]smarter, and say r theta, 0:45:44.010,0:45:48.980 where now you have to put[br]the bounds that limit-- 0:45:48.980,0:45:49.590 STUDENT: r. 0:45:49.590,0:45:50.694 MAGDALENA TODA: r from? 0:45:50.694,0:45:51.360 STUDENT: 0 to 1. 0:45:51.360,0:45:52.776 MAGDALENA TODA: 0[br]to 1, excellent. 0:45:52.776,0:45:56.899 You cannot let r go to[br]infinity, because the vase is 0:45:56.899,0:45:57.440 increasingly. 0:45:57.440,0:46:01.312 You only needs the vase that[br]has the radius 1 on the bottom. 0:46:01.312,0:46:08.723 So r is 0 to 1, and[br]theta is 0 to 1 pi. 0:46:08.723,0:46:10.640 And there you have[br]your domain this time. 0:46:10.640,0:46:15.746 So I need to be smart[br]and say integral. 0:46:15.746,0:46:18.000 Integral, what do[br]you want to do first? 0:46:18.000,0:46:21.850 Well, it doesn't matter, dr,[br]d theta, whatever you want. 0:46:21.850,0:46:26.310 So mister theta will[br]be the last of the two. 0:46:26.310,0:46:32.270 So theta will be between 0[br]and 2 pi, a complete rotation. 0:46:32.270,0:46:35.856 r between 0 and 1. 0:46:35.856,0:46:37.970 And inside here I[br]have to be smart 0:46:37.970,0:46:41.710 and see that life[br]can be fun when 0:46:41.710,0:46:44.320 I work with polar coordinates. 0:46:44.320,0:46:45.642 Why? 0:46:45.642,0:46:47.060 What is the integral? 0:46:47.060,0:46:48.110 x squared plus y squared. 0:46:48.110,0:46:50.680 I've seen him[br]somewhere before when 0:46:50.680,0:46:54.989 it came to polar coordinates. 0:46:54.989,0:46:55.780 STUDENT: R squared. 0:46:55.780,0:46:57.113 STUDENT: That will be r squared. 0:46:57.113,0:46:59.600 MAGDALENA TODA: That[br]will be r squared. 0:46:59.600,0:47:04.482 r squared times-- never[br]forget the Jacobian, 0:47:04.482,0:47:07.910 and the Jacobian is mister r. 0:47:07.910,0:47:13.030 And now I'm going to[br]take all this integral. 0:47:13.030,0:47:16.490 I'll finally compute[br]the volume of my vase. 0:47:16.490,0:47:19.960 Imagine if this vase[br]would be made of gold. 0:47:19.960,0:47:21.690 This is my dream. 0:47:21.690,0:47:24.970 So imagine that this[br]vase would have, 0:47:24.970,0:47:26.790 I don't know what dimensions. 0:47:26.790,0:47:29.390 I need to find the[br]volume, and multiply it 0:47:29.390,0:47:32.405 by the density of gold[br]and find out-- yes, sir? 0:47:32.405,0:47:35.660 STUDENT: Professor, like in this[br]question, b time is dt by dr, 0:47:35.660,0:47:38.062 but you can't switch it-- 0:47:38.062,0:47:39.270 MAGDALENA TODA: Yes, you can. 0:47:39.270,0:47:41.320 That's exactly my point. 0:47:41.320,0:47:42.690 I'll tell you in a second. 0:47:42.690,0:47:47.980 When can you replace d theta dr? 0:47:47.980,0:47:52.450 You can always do that when[br]you have something under here, 0:47:52.450,0:47:55.690 which is a big[br]function of theta times 0:47:55.690,0:48:01.630 a bit function of r, because[br]you can treat them differently. 0:48:01.630,0:48:05.050 We will work about this later. 0:48:05.050,0:48:08.640 Now, this has no theta. 0:48:08.640,0:48:13.720 So actually, the[br]theta is not going 0:48:13.720,0:48:18.700 to affect your computation. 0:48:18.700,0:48:22.410 Let's not even think about[br]theta for the time being. 0:48:22.410,0:48:29.904 What you have inside is Calculus[br]I. When you have a product, 0:48:29.904,0:48:31.395 you can always switch. 0:48:31.395,0:48:33.880 And I'll give you[br]a theorem later. 0:48:33.880,0:48:39.150 0 over 1, r cubed,[br]thank God, this 0:48:39.150,0:48:42.500 is Calc I. Integral[br]from 0 to 1, r 0:48:42.500,0:48:47.000 cubed dr. That's Calc[br]I. How much is that? 0:48:47.000,0:48:47.620 I'm lazy. 0:48:47.620,0:48:50.110 I don't want to do it. 0:48:50.110,0:48:51.179 STUDENT: 1/4. 0:48:51.179,0:48:52.220 MAGDALENA TODA: It's 1/4. 0:48:52.220,0:48:52.720 Very good. 0:48:52.720,0:48:53.910 Thank you. 0:48:53.910,0:48:58.460 And if I get further, and I'm a[br]little bi lazy, what do I get? 0:48:58.460,0:49:01.500 1/4 is the constant,[br]it pulls out. 0:49:01.500,0:49:03.140 STUDENT: So, they don't-- 0:49:03.140,0:49:09.780 MAGDALENA TODA: So I get 2 pi[br]over 4, which is pi over 2. 0:49:09.780,0:49:10.535 Am I right? 0:49:10.535,0:49:11.118 STUDENT: Yeah. 0:49:11.118,0:49:12.867 MAGDALENA TODA: So[br]this constant gets out, 0:49:12.867,0:49:14.200 integral comes in through 2 pi. 0:49:14.200,0:49:16.225 It will be 2 pi, and[br]this is my answer. 0:49:16.225,0:49:19.520 So pi over 2 is the volume. 0:49:19.520,0:49:22.570 If I have a 1-inch[br]diameter, and I 0:49:22.570,0:49:26.536 have this vase made of gold,[br]which is a piece of jewelry, 0:49:26.536,0:49:34.160 really beautiful, then I'm going[br]to have pi over 2 the volume. 0:49:34.160,0:49:36.330 That will be a little[br]bit hard to see 0:49:36.330,0:49:38.930 what we have in square inches. 0:49:38.930,0:49:43.920 We have 1.5-something[br]square inches, and then-- 0:49:43.920,0:49:45.105 STUDENT: More. 0:49:45.105,0:49:46.480 MAGDALENA TODA:[br]And then multiply 0:49:46.480,0:49:50.350 by the density of[br]gold, and estimate, 0:49:50.350,0:49:57.730 based on the mass, how much[br]money that's going to be. 0:49:57.730,0:49:59.880 What did I want to[br]tell [? Miteish? ?] 0:49:59.880,0:50:02.633 I don't want to forget what[br]he asked me, because that 0:50:02.633,0:50:04.240 was a smart question. 0:50:04.240,0:50:08.620 When can we reverse the[br]order of integration? 0:50:08.620,0:50:11.995 In general, it's[br]hard to compute. 0:50:11.995,0:50:14.540 But in this case, I'm you[br]are the luckiest person 0:50:14.540,0:50:16.790 in the world, because[br]just take a look at me. 0:50:16.790,0:50:22.180 I have, let's see, my[br]r between 0 and 2 pi, 0:50:22.180,0:50:29.470 and my theta between 0 and 2[br]pi, and my r between 0 and 1. 0:50:29.470,0:50:31.970 Whatever, it doesn't matter,[br]it could be anything. 0:50:31.970,0:50:36.390 And here I have a function of r[br]and a function g of theta only. 0:50:36.390,0:50:38.060 And it's a product. 0:50:38.060,0:50:40.790 The variables are separate. 0:50:40.790,0:50:45.800 When I do-- what do I[br]do for dr or d theta? 0:50:45.800,0:50:49.240 dr. When I do dr--[br]with respect to dr, 0:50:49.240,0:50:52.702 this fellow goes, I[br]don't belong in here. 0:50:52.702,0:50:55.650 I'm mister theta that[br]doesn't belong in here. 0:50:55.650,0:50:56.930 I'm independent. 0:50:56.930,0:50:59.160 I want to go out. 0:50:59.160,0:51:01.600 And he wants out. 0:51:01.600,0:51:10.480 So you have some integrals[br]that you got out a g of theta, 0:51:10.480,0:51:16.440 and another integral, and you[br]have f of r dr in a bracket, 0:51:16.440,0:51:20.880 and then you go d theta. 0:51:20.880,0:51:23.080 What is going to happen next? 0:51:23.080,0:51:26.790 You solve this integral, and[br]it's going to be a number. 0:51:26.790,0:51:30.400 This number could be 8,[br]7, 9.2, God knows what. 0:51:30.400,0:51:33.230 Why don't you pull that[br]constant out right now? 0:51:33.230,0:51:35.480 So you say, OK, I can do that. 0:51:35.480,0:51:37.130 It's just a number. 0:51:37.130,0:51:37.630 Whatever. 0:51:37.630,0:51:41.610 That's going to be[br]integral f dr, times 0:51:41.610,0:51:44.320 what do you have left[br]when you pull that out? 0:51:44.320,0:51:44.820 A what? 0:51:44.820,0:51:45.623 STUDENT: Integral. 0:51:45.623,0:51:49.460 MAGDALENA TODA: Integral of[br]G, the integral of g of theta, 0:51:49.460,0:51:51.000 d theta. 0:51:51.000,0:51:54.440 So we just proved a theorem[br]that is really pretty. 0:51:54.440,0:51:59.238 If you have to integrate,[br]and I will try to do it here. 0:51:59.238,0:52:03.201 0:52:03.201,0:52:03.700 No-- 0:52:03.700,0:52:06.241 STUDENT: So essentially, when[br]you're integrating with respect 0:52:06.241,0:52:11.243 to r, you can treat any function[br]of only theta as a constant? 0:52:11.243,0:52:12.230 MAGDALENA TODA: Yeah. 0:52:12.230,0:52:15.050 I'll tell you in a second[br]what it means, because-- 0:52:15.050,0:52:15.809 STUDENT: Sorry. 0:52:15.809,0:52:16.975 MAGDALENA TODA: You're fine. 0:52:16.975,0:52:21.601 Integrate for domain,[br]rectangular domains, 0:52:21.601,0:52:25.770 let's say u between alpha,[br]beta, u between gamma, 0:52:25.770,0:52:29.710 delta, then what's[br]going to happen? 0:52:29.710,0:52:35.383 As you said very well,[br]integral from-- what 0:52:35.383,0:52:38.332 do you want first, dv or du? 0:52:38.332,0:52:41.132 dv, du, it doesn't matter. 0:52:41.132,0:52:44.108 v is between gamma, delta. 0:52:44.108,0:52:47.084 v is the first guy inside, OK. 0:52:47.084,0:52:48.572 Gamma, delta. 0:52:48.572,0:52:50.060 I should have cd. 0:52:50.060,0:52:51.080 It's all Greek to me. 0:52:51.080,0:52:55.060 Why did I pick[br]that three people? 0:52:55.060,0:52:59.600 If this is going to be a product[br]of two functions, one is in u 0:52:59.600,0:53:06.210 and one is in v. Let's[br]say A of u and B of v, 0:53:06.210,0:53:11.100 I can go ahead and say[br]product of two constants. 0:53:11.100,0:53:14.040 And who are those two[br]constants I was referring to? 0:53:14.040,0:53:16.000 You can do that directly. 0:53:16.000,0:53:18.940 If the two variables are[br]separated through a product, 0:53:18.940,0:53:22.730 you have a product of[br]two separate variables. 0:53:22.730,0:53:26.320 A is only in u, it[br]depends only on u. 0:53:26.320,0:53:30.820 And B is only on v. They have[br]nothing to do with one another. 0:53:30.820,0:53:35.152 Then you can go ahead and do[br]the first integral with respect 0:53:35.152,0:53:43.310 to u only of a of u, du,[br]u between alpha, beta. 0:53:43.310,0:53:45.940 That was your first variable. 0:53:45.940,0:53:48.615 Times this other constant. 0:53:48.615,0:53:54.490 Integral of B of v,[br]where v is moving, 0:53:54.490,0:53:59.070 v is moving between[br]gamma, delta. 0:53:59.070,0:54:00.980 Instead of alpha,[br]beta, gamma, delta, 0:54:00.980,0:54:03.970 put any numbers you want. 0:54:03.970,0:54:04.854 OK? 0:54:04.854,0:54:06.180 This is the lucky case. 0:54:06.180,0:54:09.200 So you're always hoping[br]that on the final, 0:54:09.200,0:54:12.840 you can get something[br]where you can separate. 0:54:12.840,0:54:13.950 Here you have no theta. 0:54:13.950,0:54:16.330 This is the luckiest[br]case in the world. 0:54:16.330,0:54:18.550 So it's just r[br]cubed times theta. 0:54:18.550,0:54:21.440 But you can still[br]have a lucky case 0:54:21.440,0:54:24.530 when you have something[br]like a function of r 0:54:24.530,0:54:25.940 times a function of theta. 0:54:25.940,0:54:28.550 And then you have[br]another beautiful polar 0:54:28.550,0:54:31.600 coordinate integral[br]that you're not going 0:54:31.600,0:54:35.000 to struggle with for very long. 0:54:35.000,0:54:37.450 OK, I'm going to erase here. 0:54:37.450,0:54:56.100 0:54:56.100,0:55:01.560 For example, let me[br]give you another one. 0:55:01.560,0:55:04.110 Suppose that somebody[br]was really mean to you, 0:55:04.110,0:55:08.399 and wanted to kill[br]you in the final, 0:55:08.399,0:55:10.065 and they gave you the[br]following problem. 0:55:10.065,0:55:12.590 0:55:12.590,0:55:17.370 Assume the domain D-- they[br]don't even tell you what it is. 0:55:17.370,0:55:19.350 They just want to[br]challenge you-- 0:55:19.350,0:55:25.472 will be x, y with the[br]property that x squared plus y 0:55:25.472,0:55:32.080 squared is between a 1 and a 4. 0:55:32.080,0:55:36.370 0:55:36.370,0:55:52.922 Compute the integral over D of[br]r [? pan ?] of y over x and da, 0:55:52.922,0:55:57.330 where bi would be ds dy. 0:55:57.330,0:56:00.710 So you look at this[br]cross-eyed and say, gosh, 0:56:00.710,0:56:04.220 whoever-- we don't do that. 0:56:04.220,0:56:05.310 But I've seen schools. 0:56:05.310,0:56:08.600 I've seen this given at a[br]school, when they covered 0:56:08.600,0:56:11.620 this particular[br]example, they've covered 0:56:11.620,0:56:14.710 something like the previous[br]one that I showed you. 0:56:14.710,0:56:16.200 But they never covered this. 0:56:16.200,0:56:18.460 And they said,[br]OK, they're smart, 0:56:18.460,0:56:19.990 let them figure this out. 0:56:19.990,0:56:23.360 And I think it was Texas A&M.[br]They gave something like that 0:56:23.360,0:56:26.350 without working this in class. 0:56:26.350,0:56:28.576 They assumed that[br]the students should 0:56:28.576,0:56:31.120 be good enough to[br]figure out what 0:56:31.120,0:56:35.360 this is in polar coordinates. 0:56:35.360,0:56:39.790 So in polar coordinates,[br]what does the theorem say? 0:56:39.790,0:56:44.360 We should switch to a domain[br]D star that corresponds to D. 0:56:44.360,0:56:48.220 Now, D was given like that. 0:56:48.220,0:56:50.660 But we have to say[br]the corresponding D 0:56:50.660,0:56:55.090 star, reinterpreted[br]in polar coordinates, 0:56:55.090,0:56:59.710 r theta has to be also[br]written beautifully out. 0:56:59.710,0:57:03.910 Unless you draw the picture,[br]first of all, you cannot do it. 0:57:03.910,0:57:07.790 So the prof at Texas A&M didn't[br]even say, draw the picture, 0:57:07.790,0:57:10.700 and think of the[br]meaning of that. 0:57:10.700,0:57:14.950 What is the meaning of[br]this set, geometric set, 0:57:14.950,0:57:17.072 geometric locus of points. 0:57:17.072,0:57:18.880 STUDENT: You've[br]got a circle sub- 0:57:18.880,0:57:21.580 MAGDALENA TODA: You[br]have concentric circles, 0:57:21.580,0:57:26.950 sub-radius 1 and 2, and it's[br]like a ring, it's an annulus. 0:57:26.950,0:57:30.020 And he said, well,[br]I didn't do it. 0:57:30.020,0:57:33.020 I mean they were smart. 0:57:33.020,0:57:35.450 I gave it to them to do. 0:57:35.450,0:57:40.670 So if the students don't see[br]at least an example like that, 0:57:40.670,0:57:44.550 they have difficulty,[br]in my experience. 0:57:44.550,0:57:47.300 OK, for this kind[br]of annulus, you 0:57:47.300,0:57:50.810 see the radius would start[br]here, but the dotted part 0:57:50.810,0:57:53.490 is not included in your domain. 0:57:53.490,0:57:57.107 So you have to be smart and[br]say, wait a minute, my radius 0:57:57.107,0:57:58.550 is not starting at 0. 0:57:58.550,0:58:01.534 It's starting at 1[br]and it's ending at 2. 0:58:01.534,0:58:05.980 And I put that here. 0:58:05.980,0:58:11.016 And theta is the whole[br]ring, so from 0 to 2 pi. 0:58:11.016,0:58:14.490 0:58:14.490,0:58:18.250 Whether you do that[br]over the open set, 0:58:18.250,0:58:21.360 that's called annulus[br]without the boundaries, 0:58:21.360,0:58:25.265 or you do it about the[br]one with the boundaries, 0:58:25.265,0:58:28.235 it doesn't matter, the integral[br]is not going to change. 0:58:28.235,0:58:33.185 And you are going to learn[br]that in Advanced Calculus, why 0:58:33.185,0:58:36.650 it doesn't matter that if[br]you remove the boundary, 0:58:36.650,0:58:38.630 you put back the boundary. 0:58:38.630,0:58:42.970 That is a certain set of a[br]measure 0 for your integration. 0:58:42.970,0:58:46.000 It's not going to[br]change your results. 0:58:46.000,0:58:48.740 So no matter how you[br]express it-- maybe 0:58:48.740,0:58:51.590 you want to express[br]it like an open set. 0:58:51.590,0:58:55.362 You still have[br]the same integral. 0:58:55.362,0:58:57.870 Double integral[br]of D star, this is 0:58:57.870,0:59:01.684 going to give me a headache,[br]unless you help me. 0:59:01.684,0:59:05.661 What is this in[br]polar coordinates? 0:59:05.661,0:59:06.494 STUDENT: [INAUDIBLE] 0:59:06.494,0:59:09.785 0:59:09.785,0:59:11.410 MAGDALENA TODA: I[br]know when-- once I've 0:59:11.410,0:59:13.240 figured out the[br]integrand, I'm going 0:59:13.240,0:59:16.730 to remember to always[br]multiply by an r, 0:59:16.730,0:59:18.620 because if I don't,[br]I'm in big trouble. 0:59:18.620,0:59:23.540 And then I go dr d theta. 0:59:23.540,0:59:26.310 But I don't know what this is. 0:59:26.310,0:59:28.276 STUDENT: r. 0:59:28.276,0:59:34.310 MAGDALENA TODA: Nope, but[br]you're-- so r cosine theta is 0:59:34.310,0:59:37.680 x, r sine theta is y. 0:59:37.680,0:59:41.260 When you do y over[br]x, what do you get? 0:59:41.260,0:59:43.710 Always tangent of theta. 0:59:43.710,0:59:47.680 And if you do arctangent[br]of tangent, you get theta. 0:59:47.680,0:59:50.516 So that was not hard,[br]but the students did 0:59:50.516,0:59:53.130 not-- in that[br]class, I was talking 0:59:53.130,0:59:56.600 to whoever gave the exam,[br]70-something percent 0:59:56.600,0:59:58.920 of the students did[br]not know how to do it, 0:59:58.920,1:00:01.490 because they had never[br]seen something similar, 1:00:01.490,1:00:07.230 and they didn't think how[br]to express this theta in r. 1:00:07.230,1:00:08.860 So what do we mean to do? 1:00:08.860,1:00:11.630 We mean, is this a product? 1:00:11.630,1:00:13.170 It's a beautiful product. 1:00:13.170,1:00:17.620 They are separate variables like[br][INAUDIBLE] [? shafts. ?] Now, 1:00:17.620,1:00:19.830 you see, you can separate them. 1:00:19.830,1:00:26.730 The r is between 1 and 2,[br]so I can do-- eventually I 1:00:26.730,1:00:27.980 can do the r first. 1:00:27.980,1:00:33.320 And theta is between 0 and[br]2 pi, and as I taught you 1:00:33.320,1:00:37.650 by the previous theorem, you[br]can separate the two integrals, 1:00:37.650,1:00:39.967 because this one gets out. 1:00:39.967,1:00:41.280 It's a constant. 1:00:41.280,1:00:46.580 So you're left with integral[br]from 0 to 2 pi theta d 1:00:46.580,1:01:04.930 theta, and the integral from 1[br]to 2 r dr. r dr theta d theta. 1:01:04.930,1:01:06.400 This should be a piece of cake. 1:01:06.400,1:01:13.940 The only thing we have to[br]do is some easy Calculus I. 1:01:13.940,1:01:18.440 So what is integral[br]of theta d theta? 1:01:18.440,1:01:20.480 I'm not going to rush anywhere. 1:01:20.480,1:01:27.160 Theta squared over 2[br]between theta equals 0 down 1:01:27.160,1:01:30.945 and theta equals 2 pi up. 1:01:30.945,1:01:32.352 Right? 1:01:32.352,1:01:33.735 STUDENT: [INAUDIBLE] 1:01:33.735,1:01:34.610 MAGDALENA TODA: Yeah. 1:01:34.610,1:01:35.560 I'll do that later. 1:01:35.560,1:01:36.520 I don't care. 1:01:36.520,1:01:41.140 This is going to be r squared[br]over 2 between 1 and 2. 1:01:41.140,1:01:44.400 So the numerical[br]answer, if I know 1:01:44.400,1:01:51.055 how to do any math like[br]that, is going to be-- 1:01:51.055,1:01:52.125 STUDENT: 2 pi squared. 1:01:52.125,1:01:53.750 MAGDALENA TODA: 2 pi[br]squared, because I 1:01:53.750,1:01:57.770 have 4 pi squared over[br]2, so the first guy 1:01:57.770,1:02:07.890 is 2 pi squared, times-- I[br]get a 4 and 4 minus 1-- are 1:02:07.890,1:02:09.250 you guys with me? 1:02:09.250,1:02:12.620 So I get a-- let me[br]write it like that. 1:02:12.620,1:02:16.530 4 over 2 minus 1 over 2. 1:02:16.530,1:02:18.775 What's going to[br]happen to the over 2? 1:02:18.775,1:02:20.200 We'll simplify. 1:02:20.200,1:02:23.540 So this is going[br]to be 3 pi squared. 1:02:23.540,1:02:24.870 Okey Dokey? 1:02:24.870,1:02:25.375 Yes, sir? 1:02:25.375,1:02:28.380 STUDENT: How did you split it[br]into two integrals, right here? 1:02:28.380,1:02:31.100 MAGDALENA TODA: That's exactly[br]what I taught you before. 1:02:31.100,1:02:34.040 So if I had not[br]taught you before, 1:02:34.040,1:02:36.830 how did I prove that theorem? 1:02:36.830,1:02:41.430 The theorem that was[br]before was like that. 1:02:41.430,1:02:44.380 What was it? 1:02:44.380,1:02:48.700 Suppose I have a function of[br]theta, and a function of r, 1:02:48.700,1:02:52.543 and I have d theta[br]dr. And I think 1:02:52.543,1:02:55.780 this weather got to us,[br]because several people have 1:02:55.780,1:02:57.772 the cold and the flu. 1:02:57.772,1:02:59.266 Wash your hands a lot. 1:02:59.266,1:03:03.620 It's full of--[br]mathematicians full of germs. 1:03:03.620,1:03:08.559 So theta, you want theta to[br]be between whatever you want. 1:03:08.559,1:03:11.030 Any two numbers. 1:03:11.030,1:03:12.290 Alpha and beta. 1:03:12.290,1:03:14.840 And r between gamma, delta. 1:03:14.840,1:03:17.500 This is what I[br]explained last time. 1:03:17.500,1:03:22.450 So when you integrate with[br]respect to theta first inside, 1:03:22.450,1:03:26.420 g of r says I have nothing[br]to do with these guys. 1:03:26.420,1:03:28.390 They're not my type,[br]they're not my gang. 1:03:28.390,1:03:31.360 I'm going out, have[br]a beer by myself. 1:03:31.360,1:03:39.220 So he goes out and[br]joins the r group, 1:03:39.220,1:03:41.420 because theta and r[br]have nothing in common. 1:03:41.420,1:03:44.560 They are separate variables. 1:03:44.560,1:03:46.336 This is a function[br]of r only, and that's 1:03:46.336,1:03:48.100 a function of theta only. 1:03:48.100,1:03:50.410 This is what I'm talking about. 1:03:50.410,1:03:52.416 OK, so that's a constant. 1:03:52.416,1:03:55.620 That constant pulls out. 1:03:55.620,1:03:59.515 So in the end, what you have is[br]that constant that pulled out 1:03:59.515,1:04:06.270 is going to be alpha, beta, f of[br]beta d theta as a number, times 1:04:06.270,1:04:07.660 what's left inside? 1:04:07.660,1:04:11.250 Integral from gamma[br]to delta g of r 1:04:11.250,1:04:17.780 dr. So when the two functions[br]F and G are functions of theta, 1:04:17.780,1:04:22.170 respectively, r only, they have[br]nothing to do with one another, 1:04:22.170,1:04:24.740 and you can write[br]the original integral 1:04:24.740,1:04:28.570 as the product of integrals,[br]and it's really a lucky case. 1:04:28.570,1:04:33.260 But you are going to encounter[br]this lucky case many times 1:04:33.260,1:04:38.900 in your final, in the midterm,[br]in-- OK, now thinking of what 1:04:38.900,1:04:41.368 I wanted to put on the midterm. 1:04:41.368,1:04:45.310 1:04:45.310,1:04:47.885 Somebody asked me if I'm going[br]to put-- they looked already 1:04:47.885,1:04:52.180 at the homework and at the[br]book, and they asked me, 1:04:52.180,1:04:57.570 are we going to have something[br]like the area of the cardioid? 1:04:57.570,1:05:01.130 Maybe not necessarily[br]that-- or area 1:05:01.130,1:05:05.430 between a cardioid and a circle[br]that intersect each other. 1:05:05.430,1:05:10.130 Those were doable[br]even with Calc II. 1:05:10.130,1:05:12.710 Something like that, that[br]was doable with Calc II, 1:05:12.710,1:05:16.370 I don't want to do it with a[br]double integral in Calc III, 1:05:16.370,1:05:22.585 and I want to give some problems[br]that are relevant to you guys. 1:05:22.585,1:05:26.590 1:05:26.590,1:05:29.220 The question, what's going[br]to be on the midterm? 1:05:29.220,1:05:32.820 is not-- OK, what's going[br]to be on the midterm? 1:05:32.820,1:05:36.060 It's going to be something[br]very similar to the sample 1:05:36.060,1:05:37.816 that I'm going to write. 1:05:37.816,1:05:40.690 And I have already[br]included in that sample 1:05:40.690,1:05:44.730 the volume of a[br]sphere of radius r. 1:05:44.730,1:05:50.390 So how do you compute out[br]the weight-- exercise 3 or 4, 1:05:50.390,1:06:07.410 whatever that is-- we compute[br]the volume of a sphere using 1:06:07.410,1:06:08.357 double integrals. 1:06:08.357,1:06:16.640 1:06:16.640,1:06:20.210 I don't know if we have time to[br]do this problem, but if we do, 1:06:20.210,1:06:25.390 that will be the last problem--[br]when you ask you teacher, 1:06:25.390,1:06:28.996 why is the volume inside the[br]sphere, volume of a ball, 1:06:28.996,1:06:29.890 actually. 1:06:29.890,1:06:33.210 Well, the size-- the solid ball. 1:06:33.210,1:06:35.830 Why is it 4 pi r cubed over 2? 1:06:35.830,1:06:38.440 Your, did she tell[br]you, or she told 1:06:38.440,1:06:42.840 you something that you asked,[br]Mr. [? Jaime ?], for example? 1:06:42.840,1:06:47.512 They were supposed to tell[br]you that you can prove that 1:06:47.512,1:06:49.020 with Calc II or Calc III. 1:06:49.020,1:06:51.060 It's not easy. 1:06:51.060,1:06:52.885 It's not an elementary formula. 1:06:52.885,1:06:54.260 In the ancient[br]times, they didn't 1:06:54.260,1:06:57.030 know how to do it, because[br]they didn't know calculus. 1:06:57.030,1:07:00.496 So what they tried to is[br]to approximate it and see 1:07:00.496,1:07:02.770 how it goes. 1:07:02.770,1:07:07.300 Assume you have the[br]sphere of radius r, 1:07:07.300,1:07:09.490 and r is from here[br]to here, and I'm 1:07:09.490,1:07:12.932 going to go ahead and draw the[br]equator, the upper hemisphere, 1:07:12.932,1:07:18.510 the lower hemisphere, and[br]you shouldn't help me, 1:07:18.510,1:07:25.420 because isn't enough to say[br]it's twice the upper hemisphere 1:07:25.420,1:07:28.640 volume, right? 1:07:28.640,1:07:34.275 So if I knew the--[br]what is this called? 1:07:34.275,1:07:36.560 If I knew the[br]expression z equals 1:07:36.560,1:07:41.145 f of x, y of the spherical[br]cap of the hemisphere, 1:07:41.145,1:07:45.390 of the northern hemisphere,[br]I would be in business. 1:07:45.390,1:07:49.750 So if somebody even[br]tries-- one of my students, 1:07:49.750,1:07:53.220 I gave him that, he didn't know[br]polar coordinates very well, 1:07:53.220,1:07:57.620 so what he tried to do,[br]he was trying to do, 1:07:57.620,1:08:03.870 let's say z is going[br]to be square root of r 1:08:03.870,1:08:09.770 squared minus z squared minus[br]y squared over the domain. 1:08:09.770,1:08:13.300 So D will be what[br]domain? x squared 1:08:13.300,1:08:21.689 plus y squared between 0 and[br]r squared, am I right guys? 1:08:21.689,1:08:25.892 So the D is on[br]the floor, means x 1:08:25.892,1:08:28.620 squared plus y squared[br]between 0 and r squared. 1:08:28.620,1:08:32.345 This is the D that we have. 1:08:32.345,1:08:35.890 This is D So twice what? 1:08:35.890,1:08:37.109 f of x, y. 1:08:37.109,1:08:40.420 1:08:40.420,1:08:42.010 The volume of the[br]upper hemisphere 1:08:42.010,1:08:44.965 is the volume of everything[br]under this graph, which 1:08:44.965,1:08:46.380 is like a half. 1:08:46.380,1:08:49.910 It's the northern hemisphere. 1:08:49.910,1:08:52.819 dx dy, whatever is dx. 1:08:52.819,1:08:55.149 So he tried to do[br]it, and he came up 1:08:55.149,1:08:58.456 with something very ugly. 1:08:58.456,1:09:02.080 Of course you can imagine[br]what he came up with. 1:09:02.080,1:09:03.260 What would it be? 1:09:03.260,1:09:04.180 I don't know. 1:09:04.180,1:09:05.859 Oh, God. 1:09:05.859,1:09:10.336 x between minus r to r. 1:09:10.336,1:09:30.685 y would be between 0[br]and-- you have to draw it. 1:09:30.685,1:09:32.060 STUDENT: It's[br]going to be 0 or r. 1:09:32.060,1:09:32.319 STUDENT: Yeah. 1:09:32.319,1:09:33.080 STUDENT: Oh, no. 1:09:33.080,1:09:35.337 MAGDALENA TODA: So x[br]is between minus r-- 1:09:35.337,1:09:36.322 STUDENT: It's going to[br]be as a function of x. 1:09:36.322,1:09:37.798 MAGDALENA TODA: And this is x. 1:09:37.798,1:09:39.375 And it's a function of x. 1:09:39.375,1:09:44.555 And then you go square root[br]r squared minus x squared. 1:09:44.555,1:09:47.046 It looks awful in[br]Cartesian coordinates. 1:09:47.046,1:09:53.609 And then for f, he just[br]plugged in that thingy, 1:09:53.609,1:09:55.570 and he said dy dx. 1:09:55.570,1:09:58.060 And he would be[br]right, except that I 1:09:58.060,1:09:59.530 would get a headache[br]just looking 1:09:59.530,1:10:03.590 at it, because it's a mess. 1:10:03.590,1:10:05.930 It's a horrible, horrible mess. 1:10:05.930,1:10:09.100 I don't like it. 1:10:09.100,1:10:13.860 So how am I going to solve[br]this in polar coordinates? 1:10:13.860,1:10:15.539 I still have the 2. 1:10:15.539,1:10:16.810 I cannot get rid of the 2. 1:10:16.810,1:10:21.350 How do I express--[br]in polar coordinates, 1:10:21.350,1:10:25.770 the 2 would be one for the upper[br]part, one for the lower part-- 1:10:25.770,1:10:29.337 How do I express in polar[br]coordinates the disc? 1:10:29.337,1:10:31.213 Rho or r. 1:10:31.213,1:10:37.970 r between 0 to R, and theta,[br]all the way from 0 to 2 pi. 1:10:37.970,1:10:41.140 So I'm still sort of lucky[br]that I'm in business. 1:10:41.140,1:10:46.620 I go 0 to 2 pi[br]integral from 0 to r, 1:10:46.620,1:10:51.030 and for that guy, that[br]is in the integrand, 1:10:51.030,1:10:54.260 I'm going to say squared of z. 1:10:54.260,1:11:03.600 z is r squared minus-- who[br]is z squared plus y squared 1:11:03.600,1:11:06.682 in polar coordinates? 1:11:06.682,1:11:10.270 r squared. very good. r squared. 1:11:10.270,1:11:13.640 Don't forget that[br]instead of dy dx, 1:11:13.640,1:11:19.575 you have to say times r,[br]the Jacobian, dr d theta. 1:11:19.575,1:11:23.565 Can we solve this, and[br]find the correct formula? 1:11:23.565,1:11:25.840 That's what I'm talking about. 1:11:25.840,1:11:27.410 We need the u substitution. 1:11:27.410,1:11:30.800 Without the u substitution,[br]we will be dead meat. 1:11:30.800,1:11:33.060 But I don't know how[br]to do u substitution, 1:11:33.060,1:11:35.384 so I need your help. 1:11:35.384,1:11:37.769 Of course you can help me. 1:11:37.769,1:11:39.200 Who is the constant? 1:11:39.200,1:11:41.108 R is the constant. 1:11:41.108,1:11:43.030 It's a number. 1:11:43.030,1:11:46.312 Little r is a variable. 1:11:46.312,1:11:48.260 Little r is a variable. 1:11:48.260,1:11:53.617 1:11:53.617,1:11:55.570 STUDENT: r squared[br]is going to be the u. 1:11:55.570,1:11:56.780 MAGDALENA TODA: u, very good. 1:11:56.780,1:11:58.880 r squared minus r squared. 1:11:58.880,1:12:01.735 How come this is[br]working so well? 1:12:01.735,1:12:07.345 Look why du will be[br]constant prime 0 minus 2rdr. 1:12:07.345,1:12:10.010 1:12:10.010,1:12:18.430 So I take this couple[br]called rdr, this block, 1:12:18.430,1:12:21.778 and I identify the[br]block over here. 1:12:21.778,1:12:31.110 And rdr represents du[br]over minus 2, right? 1:12:31.110,1:12:32.820 So I have to be[br]smart and attentive, 1:12:32.820,1:12:36.622 because if I make a mistake[br]at the end, it's all over. 1:12:36.622,1:12:41.330 So 2 tiomes integral[br]from 0 to 2 pi. 1:12:41.330,1:12:44.620 I could get rid of[br]that and say just 2 pi. 1:12:44.620,1:12:46.280 Are you guys with me? 1:12:46.280,1:12:53.380 I could say 1 is theta-- as[br]the product, go out-- times-- 1:12:53.380,1:12:57.381 and this is my integral that[br]I'm worried about, the one only 1:12:57.381,1:13:00.327 in r. 1:13:00.327,1:13:01.800 Let me review it. 1:13:01.800,1:13:06.720 1:13:06.720,1:13:09.140 This is the only one[br]I'm worried about. 1:13:09.140,1:13:10.840 This is a piece of cake. 1:13:10.840,1:13:12.610 This is 2, this is 2 pi. 1:13:12.610,1:13:14.010 This whole thing is 4 pi a. 1:13:14.010,1:13:18.360 At least I got some 4 pi out. 1:13:18.360,1:13:19.970 What have I done in here? 1:13:19.970,1:13:23.300 I've applied the u[br]substitution, and I 1:13:23.300,1:13:25.100 have to be doing a better job. 1:13:25.100,1:13:30.690 I get 4 pi times what is[br]it after u substitution. 1:13:30.690,1:13:37.080 This guy was minus[br]1/2 du, right? 1:13:37.080,1:13:40.295 This fellow is squared[br]u, [? squared ?] 1:13:40.295,1:13:42.106 squared u as a power. 1:13:42.106,1:13:43.110 STUDENT: u to the 1/2. 1:13:43.110,1:13:44.820 MAGDALENA TODA: u[br]to the one half. 1:13:44.820,1:13:51.877 And for the integral, what[br]in the world do I write? 1:13:51.877,1:13:52.710 STUDENT: r squared-- 1:13:52.710,1:13:54.460 MAGDALENA TODA: OK. 1:13:54.460,1:14:03.232 So when little r is 0, u[br]is going to be r squared. 1:14:03.232,1:14:08.790 When little r is[br]big R, you get 0. 1:14:08.790,1:14:11.020 Now you have to[br]help me finish this. 1:14:11.020,1:14:12.710 It should be a piece of cake. 1:14:12.710,1:14:15.650 I cannot believe it's hard. 1:14:15.650,1:14:19.442 What is the integral of 4 pi? 1:14:19.442,1:14:20.858 Copy and paste. 1:14:20.858,1:14:25.188 Minus 1/2, integrate[br]y to the 1/2. 1:14:25.188,1:14:27.172 STUDENT: 2/3u to the 3/2. 1:14:27.172,1:14:34.315 MAGDALENA TODA: 2/3 u to the[br]3/2, between u equals 0 up, 1:14:34.315,1:14:37.568 and u equals r squared down. 1:14:37.568,1:14:38.610 It still looks bad, but-- 1:14:38.610,1:14:40.109 STUDENT: You've got[br]a negative sign. 1:14:40.109,1:14:41.960 MAGDALENA TODA: I've[br]got a negative sign. 1:14:41.960,1:14:42.940 STUDENT: Where is it-- 1:14:42.940,1:14:46.090 MAGDALENA TODA: So when[br]I go this minus that, 1:14:46.090,1:14:47.780 it's going to be very nice. 1:14:47.780,1:14:48.466 Why? 1:14:48.466,1:14:56.390 I'm going to say minus 4[br]pi over 2 times 2 over 3. 1:14:56.390,1:14:59.190 I should have simplified[br]them from the beginning. 1:14:59.190,1:15:05.220 I have minus 5 pi over[br]3 times at 0 I have 0. 1:15:05.220,1:15:09.265 At r squared, I have r[br]squared, and the square root 1:15:09.265,1:15:11.738 is r, r cubed. 1:15:11.738,1:15:12.730 r cubed. 1:15:12.730,1:15:19.690 1:15:19.690,1:15:22.060 Oh my God, look how[br]beautiful it is. 1:15:22.060,1:15:24.000 Two minuses in a row. 1:15:24.000,1:15:27.154 Multiply, give me a plus. 1:15:27.154,1:15:28.320 STUDENT: This is the answer. 1:15:28.320,1:15:29.770 MAGDALENA TODA: Plus. 1:15:29.770,1:15:37.150 4 pi up over 3 down, r cubed. 1:15:37.150,1:15:40.670 So we proved something[br]that is essential, 1:15:40.670,1:15:42.900 and we knew it from[br]when we were in school, 1:15:42.900,1:15:46.140 but they told us that[br]we cannot prove it, 1:15:46.140,1:15:50.555 because we couldn't prove that[br]the volume of a ball was 4 pi r 1:15:50.555,1:15:51.700 cubed over 3. 1:15:51.700,1:15:52.760 Yes, sir? 1:15:52.760,1:15:55.719 STUDENT: Why are the limits[br]of integration reversed? 1:15:55.719,1:15:57.010 Why is r squared on the bottom? 1:15:57.010,1:16:02.350 MAGDALENA TODA: Because[br]first comes little r, 0, 1:16:02.350,1:16:06.310 and then comes little r to[br]be big R. When I plug them 1:16:06.310,1:16:09.640 in in this order-- so[br]let's plug them in first, 1:16:09.640,1:16:11.050 little r equals 0. 1:16:11.050,1:16:15.570 I get, for the bottom part,[br]I get u equals r squared, 1:16:15.570,1:16:18.930 and when little[br]r equals big R, I 1:16:18.930,1:16:21.806 get big R squared minus[br]big R squared equals 0. 1:16:21.806,1:16:24.060 And that's the good[br]thing, because when 1:16:24.060,1:16:28.750 I do that, I get a minus, and[br]with the minus I already had, 1:16:28.750,1:16:29.800 I get a plus. 1:16:29.800,1:16:33.470 And the volume is a positive[br]volume, like every volume. 1:16:33.470,1:16:36.110 4 pi [INAUDIBLE]. 1:16:36.110,1:16:39.380 So that's it for today. 1:16:39.380,1:16:42.050 We finished 12-- what is that? 1:16:42.050,1:16:44.074 12.3, polar coordinates. 1:16:44.074,1:16:49.541 And we will next time[br]do some homework. 1:16:49.541,1:16:52.026 Ah, I opened the[br]homework for you. 1:16:52.026,1:16:55.008 So go ahead and do at least[br]the first 10 problems. 1:16:55.008,1:16:57.990 If you have difficulties,[br]let me know on Tuesday, 1:16:57.990,1:17:02.463 so we can work some in class. 1:17:02.463,1:17:04.948 STUDENT: [? You do ?] so much. 1:17:04.948,1:17:09.582 STUDENT: So, I went to the[br][INAUDIBLE], and I asked them, 1:17:09.582,1:17:10.415 [INTERPOSING VOICES] 1:17:10.415,1:17:13.894 1:17:13.894,1:17:16.379 [SIDE CONVERSATION] 1:17:16.379,1:18:34.302 1:18:34.302,1:18:35.799 STUDENT: Can you[br]imagine two years 1:18:35.799,1:18:38.294 of a calculus that's the[br]equivalent to [? American ?] 1:18:38.294,1:18:39.664 and only two credits? 1:18:39.664,1:18:41.288 MAGDALENA TODA:[br]Because in your system, 1:18:41.288,1:18:44.282 everything was pretty[br]much accelerated. 1:18:44.282,1:18:46.777 STUDENT: Yeah, and[br]they say, no, no, no-- 1:18:46.777,1:18:48.274 I had to ask him again. 1:18:48.274,1:18:52.765 [INAUDIBLE] calculus,[br]in two years, 1:18:52.765,1:18:56.258 that is only equivalent[br]to two credits. 1:18:56.258,1:18:58.251 I was like-- 1:18:58.251,1:18:59.751 MAGDALENA TODA:[br]Anyway, what happens 1:18:59.751,1:19:03.244 is that we used to have[br]very good evaluators 1:19:03.244,1:19:06.238 in the registrar's office, and[br]most of those people retired 1:19:06.238,1:19:09.232 or they got promoted in other[br]administrative positions. 1:19:09.232,1:19:11.727 So they have three new hires. 1:19:11.727,1:19:14.599 Those guys, they don't[br]know what they are doing. 1:19:14.599,1:19:17.064 Imagine, you would[br]finish, graduate, today, 1:19:17.064,1:19:19.529 next week, you go[br]for the registrar. 1:19:19.529,1:19:21.501 You don't know[br]what you're doing. 1:19:21.501,1:19:22.480 You need time. 1:19:22.480,1:19:22.980 Yes? 1:19:22.980,1:19:25.445 STUDENT: I had a question[br]about the homework. 1:19:25.445,1:19:27.403 I'll wait for [INAUDIBLE]. 1:19:27.403,1:19:28.403 MAGDALENA TODA: It's OK. 1:19:28.403,1:19:30.375 Do you have secrets? 1:19:30.375,1:19:31.854 STUDENT: No, I don't. 1:19:31.854,1:19:33.826 MAGDALENA TODA: Homework[br]is due the 32st. 1:19:33.826,1:19:34.812 STUDENT: No, I had a[br]question from the homework. 1:19:34.812,1:19:35.305 Like I had a problem that I[br]was working on, and I was like 1:19:35.305,1:19:36.721 MAGDALENA TODA:[br]From the homework. 1:19:36.721,1:19:39.249 OK You can wait. 1:19:39.249,1:19:42.207 You guys have other,[br]more basic questions? 1:19:42.207,1:19:43.040 [INTERPOSING VOICES] 1:19:43.040,1:19:49.211 1:19:49.211,1:19:50.960 MAGDALENA TODA: There[br]is only one meeting. 1:19:50.960,1:19:53.930 Oh, you mean-- Ah. 1:19:53.930,1:19:55.415 Yes, I do. 1:19:55.415,1:19:59.870 I have the following[br]three-- Tuesday, 1:19:59.870,1:20:04.992 Wednesday, and Friday-[br]no, Tuesday, Wednesday, 1:20:04.992,1:20:05.620 and Thursday. 1:20:05.620,1:20:09.870 On Friday we can have something,[br]some special arrangement. 1:20:09.870,1:20:12.410 This Friday? 1:20:12.410,1:20:16.750 OK, how about like 11:15. 1:20:16.750,1:20:22.574 Today, I have--[br]I have right now. 1:20:22.574,1:20:23.516 2:00. 1:20:23.516,1:20:26.784 And I think the grad[br]students will come later. 1:20:26.784,1:20:28.768 So you can just right now. 1:20:28.768,1:20:32.240 And tomorrow around 11:15,[br]because I have meetings 1:20:32.240,1:20:34.730 before 11 at the college. 1:20:34.730,1:20:37.260 STUDENT: Do you mind if I go[br]get something to eat first? 1:20:37.260,1:20:39.134 Or how long do you think[br]they'll be in your office? 1:20:39.134,1:20:40.122 MAGDALENA TODA:[br]Even if they come, 1:20:40.122,1:20:42.098 I'm going to stop[br]them and talk to you, 1:20:42.098,1:20:43.580 so don't worry about it. 1:20:43.580,1:20:44.074 STUDENT: Thank you very much. 1:20:44.074,1:20:44.568 I'll see you later. 1:20:44.568,1:20:46.050 STUDENT: I just wanted to say[br]I'm sorry for coming in late. 1:20:46.050,1:20:47.038 I slept in a little[br]bit this morning-- 1:20:47.038,1:20:49.461 MAGDALENA TODA: Did you[br]get the chance to sign? 1:20:49.461,1:20:50.002 STUDENT: Yes. 1:20:50.002,1:20:50.990 MAGDALENA TODA:[br]There is no problem. 1:20:50.990,1:20:51.490 I'm-- 1:20:51.490,1:20:55.930 STUDENT: I woke up at like[br]12:30-- I woke up at like 11:30 1:20:55.930,1:20:59.614 and I just fell right back[br]asleep, and then I got up 1:20:59.614,1:21:01.364 and I looked at my[br]phone and it was 12:30, 1:21:01.364,1:21:03.340 and I was like, I[br]have class right now. 1:21:03.340,1:21:04.822 And so what happened was like-- 1:21:04.822,1:21:05.810 MAGDALENA TODA: You were tired. 1:21:05.810,1:21:06.830 You were doing[br]homework until late. 1:21:06.830,1:21:08.610 STUDENT: --homework[br]and like, I usually 1:21:08.610,1:21:10.972 am on for an[br]earlier class, and I 1:21:10.972,1:21:12.930 didn't go to bed earlier[br]than I did last night, 1:21:12.930,1:21:14.774 and so I just overslept. 1:21:14.774,1:21:17.134 MAGDALENA TODA: I[br]did the same, anyway. 1:21:17.134,1:21:18.674 I have similar experience. 1:21:18.674,1:21:20.090 STUDENT: You have[br]a very nice day. 1:21:20.090,1:21:21.173 MAGDALENA TODA: Thank you. 1:21:21.173,1:21:21.890 You too. 1:21:21.890,1:21:24.338 So, show me what[br]you want to ask. 1:21:24.338,1:21:25.534 STUDENT: There it was. 1:21:25.534,1:21:27.200 I looked at that[br]problem, and I thought, 1:21:27.200,1:21:29.871 that's extremely[br]simple, acceleration-- 1:21:29.871,1:21:31.746 MAGDALENA TODA: Are they[br]independent, really? 1:21:31.746,1:21:32.287 STUDENT: Huh? 1:21:32.287,1:21:34.930 MAGDALENA TODA: Are they--[br]b and t are independent? 1:21:34.930,1:21:36.660 I need to stop. 1:21:36.660,1:21:39.110 STUDENT: But I[br]didn't even bother.