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